Bola (geometri) - Wikipedia bahasa Indonesia, ensiklopedia bebas

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class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Daftar isi" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Awal</div> </a> </li> <li id="toc-Persamaan_dalam_tiga_dimensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Persamaan_dalam_tiga_dimensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Persamaan dalam tiga dimensi</span> </div> </a> <ul id="toc-Persamaan_dalam_tiga_dimensi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rumus_bola" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rumus_bola"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Rumus bola</span> </div> </a> <button aria-controls="toc-Rumus_bola-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Rumus bola</span> </button> <ul id="toc-Rumus_bola-sublist" class="vector-toc-list"> <li id="toc-Luas_permukaan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Luas_permukaan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Luas permukaan</span> </div> </a> <ul id="toc-Luas_permukaan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Volume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volume"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Volume</span> </div> </a> <ul id="toc-Volume-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kurva_pada_bola" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kurva_pada_bola"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Kurva pada bola</span> </div> </a> <button aria-controls="toc-Kurva_pada_bola-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Kurva pada bola</span> </button> <ul id="toc-Kurva_pada_bola-sublist" class="vector-toc-list"> <li id="toc-Lingkaran" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lingkaran"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Lingkaran</span> </div> </a> <ul id="toc-Lingkaran-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kurva_Clelia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kurva_Clelia"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Kurva Clelia</span> </div> </a> <ul id="toc-Kurva_Clelia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Loksodrom" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Loksodrom"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Loksodrom</span> </div> </a> <ul id="toc-Loksodrom-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Persimpangan_bola_dengan_permukaan_yang_umum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Persimpangan_bola_dengan_permukaan_yang_umum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Persimpangan bola dengan permukaan yang umum</span> </div> </a> <ul id="toc-Persimpangan_bola_dengan_permukaan_yang_umum-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sifat_geometris" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sifat_geometris"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sifat geometris</span> </div> </a> <button aria-controls="toc-Sifat_geometris-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Sifat geometris</span> </button> <ul id="toc-Sifat_geometris-sublist" class="vector-toc-list"> <li id="toc-Pensil_bola" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pensil_bola"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Pensil bola</span> </div> </a> <ul id="toc-Pensil_bola-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalisasi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalisasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Generalisasi</span> </div> </a> <button aria-controls="toc-Generalisasi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Generalisasi</span> </button> <ul id="toc-Generalisasi-sublist" class="vector-toc-list"> <li id="toc-Dimensi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Dimensi</span> </div> </a> <ul id="toc-Dimensi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ruang_metrik" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ruang_metrik"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Ruang metrik</span> </div> </a> <ul id="toc-Ruang_metrik-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geometri_bola" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometri_bola"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Geometri bola</span> </div> </a> <ul id="toc-Geometri_bola-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lokus_jumlah_konstan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lokus_jumlah_konstan"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Lokus jumlah konstan</span> </div> </a> <ul id="toc-Lokus_jumlah_konstan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gambar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gambar"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Gambar</span> </div> </a> <ul id="toc-Gambar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bagian" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bagian"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bagian</span> </div> </a> <ul id="toc-Bagian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Catatan_dan_referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Catatan_dan_referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Catatan dan referensi</span> </div> </a> <button aria-controls="toc-Catatan_dan_referensi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Catatan dan referensi</span> </button> <ul id="toc-Catatan_dan_referensi-sublist" class="vector-toc-list"> <li id="toc-Catatan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Catatan"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Catatan</span> </div> </a> <ul id="toc-Catatan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Referensi</span> </div> </a> <ul id="toc-Referensi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bacaan_lebih_lanjut" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bacaan_lebih_lanjut"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Bacaan lebih lanjut</span> </div> </a> <ul id="toc-Bacaan_lebih_lanjut-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pranala_luar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pranala_luar"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Pranala luar</span> </div> </a> <ul id="toc-Pranala_luar-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Daftar isi" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Gulingkan daftar isi" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Gulingkan daftar isi</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Bola (geometri)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Pergi ke artikel dalam bahasa lain. Terdapat 106 bahasa" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-106" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">106 bahasa</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Sfeer" title="Sfeer – Afrikaans" lang="af" hreflang="af" data-title="Sfeer" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%89%E1%88%8D" title="ሉል – Amharik" lang="am" hreflang="am" data-title="ሉል" data-language-autonym="አማርኛ" data-language-local-name="Amharik" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D8%B1%D8%A9" title="كرة – Arab" lang="ar" hreflang="ar" data-title="كرة" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%83%D9%88%D8%B1%D8%A9" title="كورة – Arab Maroko" lang="ary" hreflang="ary" data-title="كورة" data-language-autonym="الدارجة" data-language-local-name="Arab Maroko" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Esfera" title="Esfera – Asturia" lang="ast" hreflang="ast" data-title="Esfera" data-language-autonym="Asturianu" data-language-local-name="Asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Sfera" title="Sfera – Azerbaijani" lang="az" hreflang="az" data-title="Sfera" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D9%88%D8%B1%D9%87_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="کوره (هندسه) – South Azerbaijani" lang="azb" hreflang="azb" data-title="کوره (هندسه)" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Bashkir" lang="ba" hreflang="ba" data-title="Сфера" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Belarusia" lang="be" hreflang="be" data-title="Сфера" data-language-autonym="Беларуская" data-language-local-name="Belarusia" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Bulgaria" lang="bg" hreflang="bg" data-title="Сфера" data-language-autonym="Български" data-language-local-name="Bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%97%E0%A5%8B%E0%A4%B2%E0%A4%BE" title="गोला – Bhojpuri" lang="bh" hreflang="bh" data-title="गोला" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8B%E0%A6%B2%E0%A6%95" title="গোলক – Bengali" lang="bn" hreflang="bn" data-title="গোলক" data-language-autonym="বাংলা" data-language-local-name="Bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Sfera" title="Sfera – Bosnia" lang="bs" hreflang="bs" data-title="Sfera" data-language-autonym="Bosanski" data-language-local-name="Bosnia" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Esfera" title="Esfera – Katalan" lang="ca" hreflang="ca" data-title="Esfera" data-language-autonym="Català" data-language-local-name="Katalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8E%A6%E1%8F%90%E1%8F%86%E1%8E%B8" title="ᎦᏐᏆᎸ – Cherokee" lang="chr" hreflang="chr" data-title="ᎦᏐᏆᎸ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="Cherokee" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%DB%86" title="گۆ – Kurdi Sorani" lang="ckb" hreflang="ckb" data-title="گۆ" data-language-autonym="کوردی" data-language-local-name="Kurdi Sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Sf%C3%A9ra_(matematika)" title="Sféra (matematika) – Cheska" lang="cs" hreflang="cs" data-title="Sféra (matematika)" data-language-autonym="Čeština" data-language-local-name="Cheska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Chuvash" lang="cv" hreflang="cv" data-title="Сфера" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Sff%C3%AAr" title="Sffêr – Welsh" lang="cy" hreflang="cy" data-title="Sffêr" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kugle" title="Kugle – Dansk" lang="da" hreflang="da" data-title="Kugle" data-language-autonym="Dansk" data-language-local-name="Dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kugel" title="Kugel – Jerman" lang="de" hreflang="de" data-title="Kugel" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%86%CE%B1%CE%AF%CF%81%CE%B1" title="Σφαίρα – Yunani" lang="el" hreflang="el" data-title="Σφαίρα" data-language-autonym="Ελληνικά" data-language-local-name="Yunani" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Sphere" title="Sphere – Inggris" lang="en" hreflang="en" data-title="Sphere" data-language-autonym="English" data-language-local-name="Inggris" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Sfero" title="Sfero – Esperanto" lang="eo" hreflang="eo" data-title="Sfero" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Esfera" title="Esfera – Spanyol" lang="es" hreflang="es" data-title="Esfera" data-language-autonym="Español" data-language-local-name="Spanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Sf%C3%A4%C3%A4r" title="Sfäär – Esti" lang="et" hreflang="et" data-title="Sfäär" data-language-autonym="Eesti" data-language-local-name="Esti" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Esfera" title="Esfera – Basque" lang="eu" hreflang="eu" data-title="Esfera" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%87_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="کره (هندسه) – Persia" lang="fa" hreflang="fa" data-title="کره (هندسه)" data-language-autonym="فارسی" data-language-local-name="Persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pallo_(geometria)" title="Pallo (geometria) – Suomi" lang="fi" hreflang="fi" data-title="Pallo (geometria)" data-language-autonym="Suomi" data-language-local-name="Suomi" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Vuravura_(Geometry)" title="Vuravura (Geometry) – Fiji" lang="fj" hreflang="fj" data-title="Vuravura (Geometry)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fiji" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sph%C3%A8re" title="Sphère – Prancis" lang="fr" hreflang="fr" data-title="Sphère" data-language-autonym="Français" data-language-local-name="Prancis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kuugel" title="Kuugel – Frisia Utara" lang="frr" hreflang="frr" data-title="Kuugel" data-language-autonym="Nordfriisk" data-language-local-name="Frisia Utara" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sf%C3%A9ar" title="Sféar – Irlandia" lang="ga" hreflang="ga" data-title="Sféar" data-language-autonym="Gaeilge" data-language-local-name="Irlandia" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – Gan" lang="gan" hreflang="gan" data-title="球面" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Sf%C3%A8r" title="Sfèr – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Sfèr" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Cruinne" title="Cruinne – Gaelik Skotlandia" lang="gd" hreflang="gd" data-title="Cruinne" data-language-autonym="Gàidhlig" data-language-local-name="Gaelik Skotlandia" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Esfera" title="Esfera – Galisia" lang="gl" hreflang="gl" data-title="Esfera" data-language-autonym="Galego" data-language-local-name="Galisia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%97%E0%AB%8B%E0%AA%B3%E0%AB%8B" title="ગોળો – Gujarat" lang="gu" hreflang="gu" data-title="ગોળો" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarat" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%A4%D7%99%D7%A8%D7%94_(%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94)" title="ספירה (גאומטריה) – Ibrani" lang="he" hreflang="he" data-title="ספירה (גאומטריה)" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A5%8B%E0%A4%B2%E0%A4%BE" title="गोला – Hindi" lang="hi" hreflang="hi" data-title="गोला" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Sfera" title="Sfera – Kroasia" lang="hr" hreflang="hr" data-title="Sfera" data-language-autonym="Hrvatski" data-language-local-name="Kroasia" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Esf%C3%A8" title="Esfè – Kreol Haiti" lang="ht" hreflang="ht" data-title="Esfè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Kreol Haiti" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/G%C3%B6mb" title="Gömb – Hungaria" lang="hu" hreflang="hu" data-title="Gömb" data-language-autonym="Magyar" data-language-local-name="Hungaria" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Sphera" title="Sphera – Interlingua" lang="ia" hreflang="ia" data-title="Sphera" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Sfero" title="Sfero – Ido" lang="io" hreflang="io" data-title="Sfero" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/K%C3%BAla" title="Kúla – Islandia" lang="is" hreflang="is" data-title="Kúla" data-language-autonym="Íslenska" data-language-local-name="Islandia" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sfera" title="Sfera – Italia" lang="it" hreflang="it" data-title="Sfera" data-language-autonym="Italiano" data-language-local-name="Italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – Jepang" lang="ja" hreflang="ja" data-title="球面" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Sfier" title="Sfier – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Sfier" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%A4%E1%83%94%E1%83%A0%E1%83%9D" title="სფერო – Georgia" lang="ka" hreflang="ka" data-title="სფერო" data-language-autonym="ქართული" data-language-local-name="Georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tasegla" title="Tasegla – Kabyle" lang="kab" hreflang="kab" data-title="Tasegla" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Kazakh" lang="kk" hreflang="kk" data-title="Сфера" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B3%8B%E0%B2%B3" title="ಗೋಳ – Kannada" lang="kn" hreflang="kn" data-title="ಗೋಳ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%AC_(%EA%B8%B0%ED%95%98%ED%95%99)" title="구 (기하학) – Korea" lang="ko" hreflang="ko" data-title="구 (기하학)" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Sphaera" title="Sphaera – Latin" lang="la" hreflang="la" data-title="Sphaera" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sfera" title="Sfera – Lituavi" lang="lt" hreflang="lt" data-title="Sfera" data-language-autonym="Lietuvių" data-language-local-name="Lituavi" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Sf%C4%93ra" title="Sfēra – Latvi" lang="lv" hreflang="lv" data-title="Sfēra" data-language-autonym="Latviešu" data-language-local-name="Latvi" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0%D1%81%D1%8C" title="Сферась – Moksha" lang="mdf" hreflang="mdf" data-title="Сферась" data-language-autonym="Мокшень" data-language-local-name="Moksha" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Bola_(je%C3%B4metria)" title="Bola (jeômetria) – Malagasi" lang="mg" hreflang="mg" data-title="Bola (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="Malagasi" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Makedonia" lang="mk" hreflang="mk" data-title="Сфера" data-language-autonym="Македонски" data-language-local-name="Makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8B%E0%B4%B3%E0%B4%82" title="ഗോളം – Malayalam" lang="ml" hreflang="ml" data-title="ഗോളം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D3%A9%D0%BC%D0%B1%D3%A9%D0%BB%D3%A9%D0%B3" title="Бөмбөлөг – Mongolia" lang="mn" hreflang="mn" data-title="Бөмбөлөг" data-language-autonym="Монгол" data-language-local-name="Mongolia" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sfera" title="Sfera – Melayu" lang="ms" hreflang="ms" data-title="Sfera" data-language-autonym="Bahasa Melayu" data-language-local-name="Melayu" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%85%E1%80%80%E1%80%BA%E1%80%9C%E1%80%AF%E1%80%B6%E1%80%B8" title="စက်လုံး – Burma" lang="my" hreflang="my" data-title="စက်လုံး" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burma" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Sfeer_(wiskunde)" title="Sfeer (wiskunde) – Belanda" lang="nl" hreflang="nl" data-title="Sfeer (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kule" title="Kule – Nynorsk Norwegia" lang="nn" hreflang="nn" data-title="Kule" data-language-autonym="Norsk nynorsk" data-language-local-name="Nynorsk Norwegia" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kule" title="Kule – Bokmål Norwegia" lang="nb" hreflang="nb" data-title="Kule" data-language-autonym="Norsk bokmål" data-language-local-name="Bokmål Norwegia" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Esf%C3%A8ra" title="Esfèra – Ositania" lang="oc" hreflang="oc" data-title="Esfèra" data-language-autonym="Occitan" data-language-local-name="Ositania" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Duqunqula" title="Duqunqula – Oromo" lang="om" hreflang="om" data-title="Duqunqula" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%8B%E0%A8%B2%E0%A8%BC%E0%A8%BE" title="ਗੋਲ਼ਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੋਲ਼ਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sfera" title="Sfera – Polski" lang="pl" hreflang="pl" data-title="Sfera" data-language-autonym="Polski" data-language-local-name="Polski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sfera" title="Sfera – Piedmontese" lang="pms" hreflang="pms" data-title="Sfera" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Esfera" title="Esfera – Portugis" lang="pt" hreflang="pt" data-title="Esfera" data-language-autonym="Português" data-language-local-name="Portugis" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Lunq%27u" title="Lunq'u – Quechua" lang="qu" hreflang="qu" data-title="Lunq'u" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sfer%C4%83" title="Sferă – Rumania" lang="ro" hreflang="ro" data-title="Sferă" data-language-autonym="Română" data-language-local-name="Rumania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rsk mw-list-item"><a href="https://rsk.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B1%D0%B4%D0%B0_(%D2%91%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Лабда (ґеометрия) – Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Лабда (ґеометрия)" data-language-autonym="Руски" data-language-local-name="Pannonian Rusyn" class="interlanguage-link-target"><span>Руски</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Rusia" lang="ru" hreflang="ru" data-title="Сфера" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Sakha" lang="sah" hreflang="sah" data-title="Сфера" data-language-autonym="Саха тыла" data-language-local-name="Sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Sfera" title="Sfera – Sisilia" lang="scn" hreflang="scn" data-title="Sfera" data-language-autonym="Sicilianu" data-language-local-name="Sisilia" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Sfera" title="Sfera – Serbo-Kroasia" lang="sh" hreflang="sh" data-title="Sfera" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Kroasia" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B7%9D%E0%B6%BD%E0%B6%BA" title="ගෝලය – Sinhala" lang="si" hreflang="si" data-title="ගෝලය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Sphere" title="Sphere – Simple English" lang="en-simple" hreflang="en-simple" data-title="Sphere" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Gu%C4%BEa_(matematika)" title="Guľa (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Guľa (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Sfera" title="Sfera – Sloven" lang="sl" hreflang="sl" data-title="Sfera" data-language-autonym="Slovenščina" data-language-local-name="Sloven" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mburungwa" title="Mburungwa – Shona" lang="sn" hreflang="sn" data-title="Mburungwa" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Kubad" title="Kubad – Somalia" lang="so" hreflang="so" data-title="Kubad" data-language-autonym="Soomaaliga" data-language-local-name="Somalia" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sfera" title="Sfera – Albania" lang="sq" hreflang="sq" data-title="Sfera" data-language-autonym="Shqip" data-language-local-name="Albania" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Serbia" lang="sr" hreflang="sr" data-title="Сфера" data-language-autonym="Српски / srpski" data-language-local-name="Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Buleudan" title="Buleudan – Sunda" lang="su" hreflang="su" data-title="Buleudan" data-language-autonym="Sunda" data-language-local-name="Sunda" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sf%C3%A4r" title="Sfär – Swedia" lang="sv" hreflang="sv" data-title="Sfär" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Tufe" title="Tufe – Swahili" lang="sw" hreflang="sw" data-title="Tufe" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%8B%E0%AE%B3%E0%AE%AE%E0%AF%8D" title="கோளம் – Tamil" lang="ta" hreflang="ta" data-title="கோளம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%97%E0%B1%8B%E0%B0%B3%E0%B0%82" title="గోళం – Telugu" lang="te" hreflang="te" data-title="గోళం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A3%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1" title="ทรงกลม – Thai" lang="th" hreflang="th" data-title="ทรงกลม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espera" title="Espera – Tagalog" lang="tl" hreflang="tl" data-title="Espera" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCre" title="Küre – Turki" lang="tr" hreflang="tr" data-title="Küre" data-language-autonym="Türkçe" data-language-local-name="Turki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Tatar" lang="tt" hreflang="tt" data-title="Сфера" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Ukraina" lang="uk" hreflang="uk" data-title="Сфера" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sfera" title="Sfera – Uzbek" lang="uz" hreflang="uz" data-title="Sfera" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BA%B7t_c%E1%BA%A7u" title="Mặt cầu – Vietnam" lang="vi" hreflang="vi" data-title="Mặt cầu" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Espira" title="Espira – Warai" lang="war" hreflang="war" data-title="Espira" data-language-autonym="Winaray" data-language-local-name="Warai" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – Wu Tionghoa" lang="wuu" hreflang="wuu" data-title="球面" data-language-autonym="吴语" data-language-local-name="Wu Tionghoa" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – Tionghoa" lang="zh" hreflang="zh" data-title="球面" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%90%83" title="球 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="球" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ki%C3%BB-b%C4%ABn" title="Kiû-bīn – Minnan" lang="nan" hreflang="nan" data-title="Kiû-bīn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%90%83%E9%AB%94" title="球體 – Kanton" lang="yue" hreflang="yue" data-title="球體" data-language-autonym="粵語" data-language-local-name="Kanton" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12507#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ruang nama"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> 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sumber</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bola_(geometri)&action=history" title="Revisi sebelumnya dari halaman ini. 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vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Perkakas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">sembunyikan</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Opsi lainnya" > <div class="vector-menu-heading"> Tindakan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Bola_(geometri)"><span>Baca</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bola_(geometri)&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bola_(geometri)&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bola_(geometri)&action=history"><span>Lihat riwayat</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Umum </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Istimewa:Pranala_balik/Bola_(geometri)" title="Daftar semua halaman wiki yang memiliki pranala ke halaman ini [j]" accesskey="j"><span>Pranala balik</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terkait/Bola_(geometri)" rel="nofollow" title="Perubahan terbaru halaman-halaman yang memiliki pranala ke halaman ini [k]" accesskey="k"><span>Perubahan terkait</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_istimewa" title="Daftar semua halaman istimewa [q]" accesskey="q"><span>Halaman istimewa</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Bola_(geometri)&oldid=25994388" title="Pranala permanen untuk revisi halaman ini"><span>Pranala permanen</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Bola_(geometri)&action=info" title="Informasi lanjut tentang halaman ini"><span>Informasi halaman</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Kutip&page=Bola_%28geometri%29&id=25994388&wpFormIdentifier=titleform" title="Informasi tentang bagaimana mengutip halaman ini"><span>Kutip halaman ini</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Istimewa:UrlShortener&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FBola_%28geometri%29"><span>Lihat URL pendek</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Istimewa:QrCode&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FBola_%28geometri%29"><span>Unduh kode QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Cetak/ekspor </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Buku&bookcmd=book_creator&referer=Bola+%28geometri%29"><span>Buat buku</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Istimewa:DownloadAsPdf&page=Bola_%28geometri%29&action=show-download-screen"><span>Unduh versi PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Bola_(geometri)&printable=yes" title="Versi cetak halaman ini [p]" accesskey="p"><span>Versi cetak</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dalam proyek lain </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Sphere" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12507" title="Pranala untuk menghubungkan butir pada ruang penyimpanan data [g]" accesskey="g"><span>Butir di Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Tampilan</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sembunyikan</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="id" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r18844875">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Untuk kegunaan lain, lihat <a href="/wiki/Bola_(disambiguasi)" class="mw-disambig" title="Bola (disambiguasi)">Bola (disambiguasi)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">"Globosa" beralih ke halaman ini. Untuk struktur neuroanatomik, lihat <a href="/w/index.php?title=Nukelus_globosa&action=edit&redlink=1" class="new" title="Nukelus globosa (halaman belum tersedia)">nukelus globosa</a>.</div><style data-mw-deduplicate="TemplateStyles:r26333525">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Periksaterjemahan plainlinks metadata ambox ambox-content ambox-rough_translation" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/Berkas:Translation_to_english_arrow.svg" class="mw-file-description" title="Translation arrow icon"><img alt="Translation arrow icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Translation_to_english_arrow.svg/50px-Translation_to_english_arrow.svg.png" decoding="async" width="50" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Translation_to_english_arrow.svg/75px-Translation_to_english_arrow.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Translation_to_english_arrow.svg/100px-Translation_to_english_arrow.svg.png 2x" data-file-width="60" data-file-height="20" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Artikel atau sebagian dari artikel ini mungkin diterjemahkan dari <i><a href="https://en.wikipedia.org/wiki/Sphere_(geometry)" class="extiw" title="en:Sphere (geometry)">Sphere (geometry)</a></i> di en.wikipedia.org. <b>Isinya masih belum akurat</b>, karena bagian yang diterjemahkan masih perlu diperhalus dan disempurnakan. Jika Anda menguasai bahasa aslinya, harap pertimbangkan untuk menelusuri referensinya dan menyempurnakan terjemahan ini. Anda juga dapat ikut bergotong royong pada <a href="/wiki/Wikipedia:ProyekWiki_Perbaikan_Terjemahan" title="Wikipedia:ProyekWiki Perbaikan Terjemahan">ProyekWiki Perbaikan Terjemahan</a>.<br /> <small>(Pesan ini dapat dihapus jika terjemahan dirasa sudah cukup tepat. Lihat pula: <a href="/wiki/Wikipedia:Panduan_dalam_menerjemahkan_artikel" title="Wikipedia:Panduan dalam menerjemahkan artikel">panduan penerjemahan artikel</a>)</small></div></td></tr></tbody></table><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333525"><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">artikel ini <b>perlu <a href="/wiki/Wikipedia:Merapikan_artikel" title="Wikipedia:Merapikan artikel">dirapikan</a></b> agar memenuhi <a href="/wiki/Wikipedia:Pedoman_gaya" title="Wikipedia:Pedoman gaya">standar Wikipedia</a>.<span class="hide-when-compact"> Tidak ada alasan yang diberikan. Silakan <a class="external text" href="https://id.wikipedia.org/w/index.php?title=Bola_(geometri)&action=edit">kembangkan artikel</a> ini semampu Anda. Merapikan artikel dapat dilakukan dengan <a href="/wiki/Bantuan:Wikifikasi" title="Bantuan:Wikifikasi">wikifikasi</a> atau membagi artikel ke paragraf-paragraf. Jika sudah dirapikan, silakan hapus templat ini.</span><span class="hide-when-compact"><i> (<small><a href="/wiki/Bantuan:Penghapusan_templat_pemeliharaan" class="mw-redirect" title="Bantuan:Penghapusan templat pemeliharaan">Pelajari cara dan kapan saatnya untuk menghapus pesan templat ini</a></small>)</i></span></div></td></tr></tbody></table> <p><b><style data-mw-deduplicate="TemplateStyles:r22657712">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}</style></b></p><b><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">Bola</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Berkas:Sphere_wireframe_10deg_6r.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/220px-Sphere_wireframe_10deg_6r.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/330px-Sphere_wireframe_10deg_6r.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/440px-Sphere_wireframe_10deg_6r.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><div class="infobox-caption">Sebuah perspektif 3 dimensi dari bola</div></td></tr><tr><th scope="row" class="infobox-label"><a href="/w/index.php?title=Daftar_grup_simetri_sferis&action=edit&redlink=1" class="new" title="Daftar grup simetri sferis (halaman belum tersedia)">Grup simetri</a></th><td class="infobox-data"><a href="/w/index.php?title=Orthogonal_group&action=edit&redlink=1" class="new" title="Orthogonal group (halaman belum tersedia)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6131f8527575c6355eb54266653bb7f8ed9fb14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(3)}"></span></a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Luas_permukaan" title="Luas permukaan">Luas permukaan</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81fcce302776a01dc66fc186a1ce0a616b4d772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.597ex; height:2.676ex;" alt="{\displaystyle 4\pi r^{2}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Volume" title="Volume">Volume</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4}{3}}\pi r^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{3}}\pi r^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7112cb504e1399319cb9edbc622d1c4609fd56e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.433ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{3}}\pi r^{3}}"></span></td></tr></tbody></table></b><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">Bola</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Berkas:Sphere_wireframe_10deg_6r.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/220px-Sphere_wireframe_10deg_6r.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/330px-Sphere_wireframe_10deg_6r.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/440px-Sphere_wireframe_10deg_6r.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><div class="infobox-caption">Sebuah perspektif 3 dimensi dari bola</div></td></tr><tr><th scope="row" class="infobox-label"><a href="/w/index.php?title=Daftar_grup_simetri_sferis&action=edit&redlink=1" class="new" title="Daftar grup simetri sferis (halaman belum tersedia)">Grup simetri</a></th><td class="infobox-data"><a href="/w/index.php?title=Orthogonal_group&action=edit&redlink=1" class="new" title="Orthogonal group (halaman belum tersedia)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6131f8527575c6355eb54266653bb7f8ed9fb14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(3)}"></span></a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Luas_permukaan" title="Luas permukaan">Luas permukaan</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81fcce302776a01dc66fc186a1ce0a616b4d772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.597ex; height:2.676ex;" alt="{\displaystyle 4\pi r^{2}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Volume" title="Volume">Volume</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4}{3}}\pi r^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{3}}\pi r^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7112cb504e1399319cb9edbc622d1c4609fd56e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.433ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{3}}\pi r^{3}}"></span></td></tr></tbody></table> <p><br /> <b>Bola</b> adalah objek <a href="/wiki/Geometri" title="Geometri">geometri</a> <a href="/wiki/Geometri_padat" title="Geometri padat">tiga dimensi</a> yang serupa dengan objek melingkar dua dimensi, yaitu "<a href="/wiki/Lingkaran" title="Lingkaran">lingkaran</a>" adalah batas dari <a href="/wiki/Cakram_(matematika)" title="Cakram (matematika)">"cakram"</a>. Pada umumnya, bola didefinisikan sebagai <a href="/wiki/Lokus_(matematika)" title="Lokus (matematika)">himpunan titik</a> yang memiliki jarak sama dari pusat bola ke permukaan bola. Jarak yang sama dalam bola bisa dikenal dengan <a href="/wiki/Jari-jari" title="Jari-jari">jari-jari (radius)</a> dan disimbolkan dengan huruf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>.<sup id="cite_ref-Albert54_1-0" class="reference"><a href="#cite_note-Albert54-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Ruas garis lurus terpanjang melalui bola, menghubungkan dua titik di permukaan bola, melewati pusat dan panjangnya dua kali jari-jari disebut sebagai<a href="/wiki/Diameter" title="Diameter">diameter</a>. </p><p>Sementara di luar matematika istilah "bola" terkadang digunakan secara bergantian. Dalam <a href="/wiki/Matematika" title="Matematika">matematika</a>, perbedaan di atas dibuat dengan antara <i>bola</i> yang merupakan <a href="/w/index.php?title=Permukaan_tertutup&action=edit&redlink=1" class="new" title="Permukaan tertutup (halaman belum tersedia)">permukaan tertutup</a> dua dimensi <a href="/w/index.php?title=Pembenaman&action=edit&redlink=1" class="new" title="Pembenaman (halaman belum tersedia)">pembenaman</a> dalam <a href="/wiki/Ruang_Euklides" title="Ruang Euklides">ruang Euklides</a> tiga dimensi, dan <i>bola</i> yang merupakan bentuk tiga dimensi yang mencakup bola dan segala sesuatu <i>di dalam</i> bola (<i>bola tertutup</i>), atau, lebih sering, hanya titik <i>di dalam</i>, namun <i>bukan di</i> antara bola (<i>bola terbuka</i>). Ini sejalan dengan situasi dalam <a href="/wiki/Bidang_(geometri)" title="Bidang (geometri)">bidang</a>, dimana istilah "lingkaran" dan "cakram" juga dapat dikacaukan. </p><p>Bola adalah objek fundamental dalam banyak bidang matematika. Bentuk bola dan hampir bulat juga muncul di alam dan industri. Gelembung seperti gelembung sabun berbentuk bola dalam keadaan seimbang. Bumi sering kali didekati sebagai bola dalam geografi, dan bola langit merupakan konsep penting dalam astronomi. Barang-barang yang diproduksi termasuk bejana tekan dan sebagian besar cermin dan lensa melengkung didasarkan pada bola. Bola menggelinding dengan mulus ke segala arah, sehingga sebagian besar bola yang digunakan dalam olahraga dan mainan berbentuk bola, begitu pula bantalan bola. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Persamaan_dalam_tiga_dimensi">Persamaan dalam tiga dimensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=1" title="Sunting bagian: Persamaan dalam tiga dimensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=1" title="Sunting kode sumber bagian: Persamaan dalam tiga dimensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Sphere_and_Ball.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/220px-Sphere_and_Ball.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/330px-Sphere_and_Ball.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/440px-Sphere_and_Ball.png 2x" data-file-width="1548" data-file-height="1536" /></a><figcaption>Dua jari-jari ortogonal (tegak lurus) dari suatu bola</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/wiki/Fungsi_trigonometri" title="Fungsi trigonometri">Fungsi trigonometri</a> dan <a href="/wiki/Sistem_koordinat_bola" title="Sistem koordinat bola">Sistem koordinat bola</a></div> <p>Dalam geometri analitik, bola dengan pusat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}"></span> dan jari-jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> adalah lokus titik <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> sedemikian rupa sehingga </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a476c6003522e221e5620b363ad446c25c0044b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.927ex; height:3.176ex;" alt="{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.}"></span></dd></dl> <p>Jika variabel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a503f107a7c104e40e484cee9e1f5993d28ffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\textstyle a}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\textstyle b}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d411ca19645ddd4fff0704de95ec770681093bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\textstyle c}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252135f29da0e9f9e130ff2d53be5df2f7044d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\textstyle d}"></span>, dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd2bd906fc55b6e7cef6d70b52a9fbd1df8f8b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\textstyle e}"></span> adalah <a href="/wiki/Bilangan_real" class="mw-redirect" title="Bilangan real">bilangan real</a> dengan nilai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"></span> dan nilai titik tengah <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}"></span> didefinisikan sebagai: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}={\frac {-b}{a}},\quad y_{0}={\frac {-c}{a}},\quad z_{0}={\frac {-d}{a}},\quad \rho ={\frac {b^{2}+c^{2}+d^{2}-ae}{a^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>c</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>d</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>e</mi> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}={\frac {-b}{a}},\quad y_{0}={\frac {-c}{a}},\quad z_{0}={\frac {-d}{a}},\quad \rho ={\frac {b^{2}+c^{2}+d^{2}-ae}{a^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/836ec8f22b47f49bcc6dd7dfdf439aa1229fa65b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:60.234ex; height:6.009ex;" alt="{\displaystyle x_{0}={\frac {-b}{a}},\quad y_{0}={\frac {-c}{a}},\quad z_{0}={\frac {-d}{a}},\quad \rho ={\frac {b^{2}+c^{2}+d^{2}-ae}{a^{2}}}.}"></span></dd></dl> <p>Lalu persamaan </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,z)=a(x^{2}+y^{2}+z^{2})+2(bx+cy+dz)+e=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mi>y</mi> <mo>+</mo> <mi>d</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>e</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,z)=a(x^{2}+y^{2}+z^{2})+2(bx+cy+dz)+e=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a133703df70ba0dab184e3fcf4be2cd38c74eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.762ex; height:3.176ex;" alt="{\displaystyle f(x,y,z)=a(x^{2}+y^{2}+z^{2})+2(bx+cy+dz)+e=0}"></span></dd></dl> <p>tidak memiliki poin nyata sebagai solusi jika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befc67bb24ee0793a950953cd8b7464bdde924e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho <0}"></span> dan disebut persamaan <b>bola imajiner</b>. Jika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba6310b27df5f9c9b0b1732e08cce27b99d68cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =0}"></span>, satu-satunya solusi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,z)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9dc9c0f7052aaefb6f7194bb0d9e419086a4bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.99ex; height:2.843ex;" alt="{\displaystyle f(x,y,z)=0}"></span> adalah titik tengah bolah <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2728b2a274122fbaf50539fa2dd9c885afca413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}"></span> dan persamaannya disebut persamaan <b>titik bola</b>. Terakhir, dalam kasus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11bd697f113e3e1bd7c76f2f441fd102eca99cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho >0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,z)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9dc9c0f7052aaefb6f7194bb0d9e419086a4bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.99ex; height:2.843ex;" alt="{\displaystyle f(x,y,z)=0}"></span> adalah persamaan bola yang pusatnya adalah <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span> dan yang radiusnya adalah <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae01a7de75247c1dabce17e09101772823066e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.138ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\rho }}}"></span>.<sup id="cite_ref-Albert54_1-1" class="reference"><a href="#cite_note-Albert54-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Jika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> dalam persamaan di atas adalah nol maka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,z)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9dc9c0f7052aaefb6f7194bb0d9e419086a4bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.99ex; height:2.843ex;" alt="{\displaystyle f(x,y,z)=0}"></span> adalah persamaan suatu bidang. Dengan demikian, sebuah bidang dapat dianggap sebagai bola jari-jari tak terbatas yang pusatnya adalah titik tak terhingga.<sup id="cite_ref-Woods266_2-0" class="reference"><a href="#cite_note-Woods266-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Titik-titik di bola dengan jari-jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23cbbcd53bd13620bc53490e3eec42790850b452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r>0}"></span> dan pusat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}"></span> dapat diparameterisasi via </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \qquad (0\leq \theta \leq \pi ,\;0\leq \varphi <2\pi )\\z&=z_{0}+r\cos \theta \,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thickmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \qquad (0\leq \theta \leq \pi ,\;0\leq \varphi <2\pi )\\z&=z_{0}+r\cos \theta \,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451a8eec53942d0a86038a61ae4eacc72705bd47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:51.067ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \qquad (0\leq \theta \leq \pi ,\;0\leq \varphi <2\pi )\\z&=z_{0}+r\cos \theta \,\end{aligned}}}"></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></dd></dl> <p><a href="/wiki/Keliling" title="Keliling">Keliling</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> dapat dikaitkan dengan sudut yang dihitung dari arah <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> positif melalui pusat ke vektor radius, dan keliling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> dapat dikaitkan dengan sudut yang dihitung dari arah <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> positif melalui pusat ke proyeksi vektor-jari-jari pada bidang <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db9936ddb2761b76fa640fb275cb5d1fa4d6fa23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\textstyle y}"></span></i>. </p><p>Bola dari jari-jari yang berpusat di nol adalah permukaan <a href="/wiki/Integral" title="Integral">integral</a> dari bentuk <a href="/wiki/Diferensial" class="mw-redirect" title="Diferensial">diferensial</a> berikut: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,dx+y\,dy+z\,dz=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,dx+y\,dy+z\,dz=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cfcc379e60de9faca3aa371dc3f6b3ea23e965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.544ex; height:2.509ex;" alt="{\displaystyle x\,dx+y\,dy+z\,dz=0.}"></span></dd></dl> <p>Persamaan ini mencerminkan bahwa vektor posisi dan kecepatan suatu titik, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (dx,dy,dz)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mi>d</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (dx,dy,dz)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb129e59a737995413326e53a661713ae7720d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.098ex; height:2.843ex;" alt="{\displaystyle (dx,dy,dz)}"></span>, yang berjalan di bola selalu ortogonal satu sama lain. </p><p>Sebuah bola juga dapat dibangun sebagai permukaan yang dibentuk dengan memutar <a href="/wiki/Lingkaran" title="Lingkaran">lingkaran</a> tentang semua <a href="/wiki/Diameter" title="Diameter">diameternya</a>. Karena lingkaran adalah jenis <a href="/wiki/Elips" title="Elips">elips</a> khusus, maka bola adalah jenis elips khusus revolusi. Mengganti lingkaran dengan elips yang diputar pada sumbu utamanya, bentuknya menjadi <a href="/w/index.php?title=Spheroid_prolate&action=edit&redlink=1" class="new" title="Spheroid prolate (halaman belum tersedia)">spheroid prolate</a>; jika diputar terhadap sumbu minor, bentuknya akan menjadi sebuah <a href="/w/index.php?title=Spheroid_oblate&action=edit&redlink=1" class="new" title="Spheroid oblate (halaman belum tersedia)">spheroid oblate</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Rumus_bola">Rumus bola</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=2" title="Sunting bagian: Rumus bola" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=2" title="Sunting kode sumber bagian: Rumus bola"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Luas_permukaan">Luas permukaan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=3" title="Sunting bagian: Luas permukaan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=3" title="Sunting kode sumber bagian: Luas permukaan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Luas permukaan pada bola yaitu. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=4\pi r^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=4\pi r^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ce66d9acf8c5a2ce122b0ae4455e04150bc68a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.666ex; height:2.676ex;" alt="{\displaystyle L=4\pi r^{2}\,}"></span></dd></dl> <p><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> pertama kali memperoleh rumus ini<sup id="cite_ref-MathWorld_Sphere_5-0" class="reference"><a href="#cite_note-MathWorld_Sphere-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> dari fakta bahwa proyeksi ke permukaan lateral dari silinder yang dibatasi adalah pengawet area.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Pendekatan lain untuk memperoleh rumus berasal dari fakta bahwa rumus tersebut sama dengan turunan rumus untuk volume sehubungan dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> karena volume total di dalam bola jari-jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> dapat dianggap sebagai penjumlahan dari luas permukaan jumlah yang tidak terbatas dari cangkang bola dengan ketebalan sangat kecil yang ditumpuk secara konseptual di dalam satu sama lain dari jari jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> hingga jari jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. Pada ketebalan sangat kecil perbedaan antara luas permukaan bagian dalam dan luar setiap cangkang yang diberikan sangat kecil, dan volume unsur pada jari-jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> hanyalah produk dari luas permukaan pada jari-jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> dan ketebalan sangat kecil. </p><p>Pada jari-jari tertentu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, volume tambahan (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c26e906a0088049ad05b9350aa1b7666576b94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.836ex; height:2.343ex;" alt="{\displaystyle \delta V}"></span>) sama dengan produk dari luas permukaan pada jari-jari (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4afc6672b04c66b9eca5e7290259d7f5b4e1df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.441ex; height:2.843ex;" alt="{\displaystyle L(r)}"></span>) dan ketebalan cangkang (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d0990c9c28625a24ddd1c9db5791698c889d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.097ex; height:2.343ex;" alt="{\displaystyle \delta r}"></span>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta V\approx L(r)\cdot \delta r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>V</mi> <mo>≈</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta V\approx L(r)\cdot \delta r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167c6b5cac5bc50704e12b3523af7de0fb61774c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.798ex; height:2.843ex;" alt="{\displaystyle \delta V\approx L(r)\cdot \delta r.}"></span></dd></dl> <p>Volume total adalah penjumlahan dari semua volume cangkang: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\approx \sum L(r)\cdot \delta r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>≈</mo> <mo>∑</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\approx \sum L(r)\cdot \delta r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52e97ae40eaf3f91dd0f05dec67ffcdfff661d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.492ex; height:3.843ex;" alt="{\displaystyle V\approx \sum L(r)\cdot \delta r.}"></span></dd></dl> <p>Dalam batas ketika ketebalan cangkang <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d0990c9c28625a24ddd1c9db5791698c889d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.097ex; height:2.343ex;" alt="{\displaystyle \delta r}"></span> mendekati nol <sup id="cite_ref-delta_7-0" class="reference"><a href="#cite_note-delta-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> persamaan ini menjadi: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\int _{0}^{r}L(r)\,dr.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mi>L</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\int _{0}^{r}L(r)\,dr.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdf09f4bc6eed0cb9aa90b0da68660e49d6b5dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.45ex; height:5.843ex;" alt="{\displaystyle V=\int _{0}^{r}L(r)\,dr.}"></span></dd></dl> <p>Masukkan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> (lihat bagian <a class="mw-selflink-fragment" href="#Volume">rumus volume bola</a>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4}{3}}\pi r^{3}=\int _{0}^{r}L(r)\,dr.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mi>L</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{3}}\pi r^{3}=\int _{0}^{r}L(r)\,dr.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b472cb148e238a9f936c94c3fcb2cd29ac2c731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.096ex; height:5.843ex;" alt="{\displaystyle {\frac {4}{3}}\pi r^{3}=\int _{0}^{r}L(r)\,dr.}"></span></dd></dl> <p>Mengambil turunan dari kedua sisi persamaan ini berdasarkan dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> akan menghasilkan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> sebagai fungsi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi r^{2}=L(r).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi r^{2}=L(r).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6a56324cff512ad387b00c08b247ad0296f671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.783ex; height:3.176ex;" alt="{\displaystyle 4\pi r^{2}=L(r).}"></span></dd></dl> <p>di mana <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> sekarang dianggap sebagai jari-jari bola yang tetap. </p><p>Atau, elemen luas pada bola diberikan dalam <a href="/w/index.php?title=Koordinat_bola&action=edit&redlink=1" class="new" title="Koordinat bola (halaman belum tersedia)">koordinat bola</a> oleh <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dA=r^{2}\sin(\theta )\;d\theta \;d\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>A</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thickmathspace" /> <mi>d</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dA=r^{2}\sin(\theta )\;d\theta \;d\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed689cd63b26448955a3734d5b3afe18ad34702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.501ex; height:3.176ex;" alt="{\displaystyle dA=r^{2}\sin(\theta )\;d\theta \;d\phi }"></span>. Dalam <a href="/w/index.php?title=Sistem_kordinat_Kartesius&action=edit&redlink=1" class="new" title="Sistem kordinat Kartesius (halaman belum tersedia)">Kordinat Kartesius</a>, elemen luas adalah </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dS={\frac {r}{\sqrt {r^{2}-{\displaystyle \sum _{i\neq k}x_{i}^{2}}}}}\prod _{i\neq k}dx_{i},\;\forall k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>k</mi> </mrow> </munder> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> </msqrt> </mfrac> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>k</mi> </mrow> </munder> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS={\frac {r}{\sqrt {r^{2}-{\displaystyle \sum _{i\neq k}x_{i}^{2}}}}}\prod _{i\neq k}dx_{i},\;\forall k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78277b9aae2fc2a99ab712de135bfefe1aba6e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:31.961ex; height:10.176ex;" alt="{\displaystyle dS={\frac {r}{\sqrt {r^{2}-{\displaystyle \sum _{i\neq k}x_{i}^{2}}}}}\prod _{i\neq k}dx_{i},\;\forall k.}"></span></dd></dl> <p>Total luas dengan demikian dapat diperoleh dengan <a href="/wiki/Integral" title="Integral">integral</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\int _{0}^{2\pi }\int _{0}^{\pi }r^{2}\sin \theta \,d\theta \,d\varphi =4\pi r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\int _{0}^{2\pi }\int _{0}^{\pi }r^{2}\sin \theta \,d\theta \,d\varphi =4\pi r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995221f096f1e650ed15d055c097df2cd7980f36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.536ex; height:6.176ex;" alt="{\displaystyle L=\int _{0}^{2\pi }\int _{0}^{\pi }r^{2}\sin \theta \,d\theta \,d\varphi =4\pi r^{2}.}"></span></dd></dl> <p>Bola memiliki <a href="/wiki/Luas_permukaan" title="Luas permukaan">luas permukaan</a> terkecil dari semua permukaan yang membungkus <a href="/wiki/Volume" title="Volume">volume</a> tertentu, dan melingkupi volume terbesar di antara semua permukaan tertutup dengan luas permukaan tertentu.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Karenanya bola muncul di alam: misalnya, gelembung dan tetesan air kecil secara kasar berbentuk bola karena tegangan permukaan secara lokal meminimalkan luas permukaan. </p><p><a href="/wiki/Luas_permukaan" title="Luas permukaan">Luas permukaan</a> relatif terhadap massa bola disebut <b>luas permukaan spesifik</b> dan dapat dinyatakan dari persamaan yang dinyatakan di atas sebagai </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {LPS} ={\frac {A}{V\rho }}={\frac {3}{r\rho }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mrow> <mi>V</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mi>r</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {LPS} ={\frac {A}{V\rho }}={\frac {3}{r\rho }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8587bab63f4d81707e207acf6af9239b2b949213" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.084ex; height:5.843ex;" alt="{\displaystyle \mathrm {LPS} ={\frac {A}{V\rho }}={\frac {3}{r\rho }},}"></span></dd></dl> <p>di mana ρ adalah kepadatan (rasio massa terhadap volume). </p> <div class="mw-heading mw-heading3"><h3 id="Volume">Volume</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=4" title="Sunting bagian: Volume" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=4" title="Sunting kode sumber bagian: Volume"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Volume pada bola yaitu: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {4}{3}}\pi r^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {4}{3}}\pi r^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07808adf2ca4b2aeb69b6c15fb6251a3a3617c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.319ex; height:5.176ex;" alt="{\displaystyle V={\frac {4}{3}}\pi r^{3}}"></span></dd></dl> <p>Pada setiap <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> yang diberikan, volume tambahan (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c26e906a0088049ad05b9350aa1b7666576b94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.836ex; height:2.343ex;" alt="{\displaystyle \delta V}"></span>) sama dengan produk dari luas penampang disk pada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> dan ketebalannya (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22318bef6d7358b79bd993321d65d7c1d3db9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.343ex;" alt="{\displaystyle \delta x}"></span>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta V\approx \pi y^{2}\cdot \delta x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>V</mi> <mo>≈</mo> <mi>π</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>δ</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta V\approx \pi y^{2}\cdot \delta x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f5294b86738de4a6370477df266e31adea58e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.185ex; height:3.009ex;" alt="{\displaystyle \delta V\approx \pi y^{2}\cdot \delta x.}"></span></dd></dl> <p>Volume total adalah penjumlahan dari semua volume tambahan: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\approx \sum \pi y^{2}\cdot \delta x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>≈</mo> <mo>∑</mo> <mi>π</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>δ</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\approx \sum \pi y^{2}\cdot \delta x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d25825c42ffeebfbf63349381f1bb020398910a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.879ex; height:3.843ex;" alt="{\displaystyle V\approx \sum \pi y^{2}\cdot \delta x.}"></span></dd></dl> <p>Dalam batas ketika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22318bef6d7358b79bd993321d65d7c1d3db9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.343ex;" alt="{\displaystyle \delta x}"></span> mendekati nol,<sup id="cite_ref-delta_7-1" class="reference"><a href="#cite_note-delta-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Persamaan" title="Persamaan">persamaan</a> ini menjadi: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\int _{-r}^{r}\pi y^{2}dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mi>π</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\int _{-r}^{r}\pi y^{2}dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d1ea760eaab5ec19ef3372ea4c9ee24addee22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.557ex; height:6.009ex;" alt="{\displaystyle V=\int _{-r}^{r}\pi y^{2}dx.}"></span></dd></dl> <p>Pada setiap <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> yang diberikan, segitiga siku-siku menghubungkan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> ke titik asal; karenanya, dengan menerapkan <a href="/wiki/Teorema_Pythagoras" title="Teorema Pythagoras">Teorema Pythagoras</a> akan menghasilkan: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=r^{2}-x^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=r^{2}-x^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61fa971d8c5410f0cbbbb2f948b29db288e26550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.287ex; height:3.009ex;" alt="{\displaystyle y^{2}=r^{2}-x^{2}.}"></span></dd></dl> <p>Menggunakan substitusi ini memberi </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\int _{-r}^{r}\pi \left(r^{2}-x^{2}\right)dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\int _{-r}^{r}\pi \left(r^{2}-x^{2}\right)dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2040d2da3a61d453ac509a262a4efaf566ba484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.573ex; height:6.009ex;" alt="{\displaystyle V=\int _{-r}^{r}\pi \left(r^{2}-x^{2}\right)dx,}"></span></dd></dl> <p>yang dapat dievaluasi untuk memberikan hasilnya </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\pi \left[r^{2}x-{\frac {x^{3}}{3}}\right]_{-r}^{r}=\pi \left(r^{3}-{\frac {r^{3}}{3}}\right)-\pi \left(-r^{3}+{\frac {r^{3}}{3}}\right)={\frac {4}{3}}\pi r^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>π</mi> <msubsup> <mrow> <mo>[</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mo>=</mo> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\pi \left[r^{2}x-{\frac {x^{3}}{3}}\right]_{-r}^{r}=\pi \left(r^{3}-{\frac {r^{3}}{3}}\right)-\pi \left(-r^{3}+{\frac {r^{3}}{3}}\right)={\frac {4}{3}}\pi r^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c081de9760153a5ab7e59be1b9de1aa97d08dec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:63.389ex; height:6.509ex;" alt="{\displaystyle V=\pi \left[r^{2}x-{\frac {x^{3}}{3}}\right]_{-r}^{r}=\pi \left(r^{3}-{\frac {r^{3}}{3}}\right)-\pi \left(-r^{3}+{\frac {r^{3}}{3}}\right)={\frac {4}{3}}\pi r^{3}.}"></span></dd></dl> <p>Rumus alternatif ditemukan menggunakan koordinat bola , dengan elemen volume </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>V</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7baab55bb4d5559e61d50df77cca1d7f6befc27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.393ex; height:3.176ex;" alt="{\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\varphi }"></span></dd></dl> <p>begitu </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{r}r'^{2}\sin \theta \,dr'\,d\theta \,d\varphi =2\pi \int _{0}^{\pi }\int _{0}^{r}r'^{2}\sin \theta \,dr'\,d\theta =4\pi \int _{0}^{r}r'^{2}\,dr'\ ={\frac {4}{3}}\pi r^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>r</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>r</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>r</mi> <mo>′</mo> </msup> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{r}r'^{2}\sin \theta \,dr'\,d\theta \,d\varphi =2\pi \int _{0}^{\pi }\int _{0}^{r}r'^{2}\sin \theta \,dr'\,d\theta =4\pi \int _{0}^{r}r'^{2}\,dr'\ ={\frac {4}{3}}\pi r^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b51d4bd953c2d8ddb0b746770be5d790eb6e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:86.604ex; height:6.176ex;" alt="{\displaystyle V=\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{r}r'^{2}\sin \theta \,dr'\,d\theta \,d\varphi =2\pi \int _{0}^{\pi }\int _{0}^{r}r'^{2}\sin \theta \,dr'\,d\theta =4\pi \int _{0}^{r}r'^{2}\,dr'\ ={\frac {4}{3}}\pi r^{3}.}"></span></dd></dl> <p>Untuk tujuan paling praktis, volume di dalam bola yang tertulis dalam kubus dapat diperkirakan sekitar 52,4% dari volume kubus, karena <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V={\frac {\pi }{6}}d^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle V={\frac {\pi }{6}}d^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae46a7aa965bf16890fe92269426ccc9db018bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.936ex; height:3.509ex;" alt="{\textstyle V={\frac {\pi }{6}}d^{3}}"></span>, di mana <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> adalah diameter bola dan juga panjang sisi kubus dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\pi }{6}}\approx 0,5236}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0</mn> <mo>,</mo> <mn>5236</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{6}}\approx 0,5236}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c082ff188e4109be1c74262d843554df19c11c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.723ex; height:3.343ex;" alt="{\textstyle {\frac {\pi }{6}}\approx 0,5236}"></span>. Sebagai contoh, bola dengan diameter 1 m memiliki 52,4% volume kubus dengan panjang tepi 1 m, atau sekitar 0,524 m3 </p> <div class="mw-heading mw-heading2"><h2 id="Kurva_pada_bola">Kurva pada bola <span id="Kurva"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=5" title="Sunting bagian: Kurva pada bola" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=5" title="Sunting kode sumber bagian: Kurva pada bola"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Ellipso-eb-ku.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Ellipso-eb-ku.svg/260px-Ellipso-eb-ku.svg.png" decoding="async" width="260" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Ellipso-eb-ku.svg/390px-Ellipso-eb-ku.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Ellipso-eb-ku.svg/520px-Ellipso-eb-ku.svg.png 2x" data-file-width="331" data-file-height="176" /></a><figcaption>Bagian bidang dari sebuah bola: 1 lingkaran</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Kugel-zylinder-kk.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Kugel-zylinder-kk.svg/180px-Kugel-zylinder-kk.svg.png" decoding="async" width="180" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Kugel-zylinder-kk.svg/270px-Kugel-zylinder-kk.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Kugel-zylinder-kk.svg/360px-Kugel-zylinder-kk.svg.png 2x" data-file-width="441" data-file-height="288" /></a><figcaption>Perpotongan koaksial bola dan silinder: 2 lingkaran</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Lingkaran">Lingkaran</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=6" title="Sunting bagian: Lingkaran" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=6" title="Sunting kode sumber bagian: Lingkaran"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Lingkaran_dari_sebuah_bola&action=edit&redlink=1" class="new" title="Lingkaran dari sebuah bola (halaman belum tersedia)">Lingkaran dari sebuah bola</a></div> <ul><li>Perpotongan bola dan bidang adalah lingkaran, titik atau kosong.</li></ul> <p>Dalam kasus lingkaran, lingkaran tersebut dapat dijelaskan dengan <a href="/wiki/Persamaan_parametrik" title="Persamaan parametrik">persamaan parametrik</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;{\vec {x}}=({\vec {e}}_{0}+{\vec {e}}_{1}\cos t+{\vec {e}}_{2}\sin t)^{T}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;{\vec {x}}=({\vec {e}}_{0}+{\vec {e}}_{1}\cos t+{\vec {e}}_{2}\sin t)^{T}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/305388634dd3ebf0f1558db37256743788c7d3a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.624ex; height:3.176ex;" alt="{\displaystyle \;{\vec {x}}=({\vec {e}}_{0}+{\vec {e}}_{1}\cos t+{\vec {e}}_{2}\sin t)^{T}\;}"></span>: lihat penampang bidang dari <a href="/w/index.php?title=Ellipsoid&action=edit&redlink=1" class="new" title="Ellipsoid (halaman belum tersedia)">ellipsoid</a>. </p><p>Namun permukaan yang lebih rumit juga dapat memotong sebuah bola dalam lingkaran: </p> <ul><li>Perpotongan bola yang tidak kosong dengan <a href="/w/index.php?title=Permukaan_revolusi&action=edit&redlink=1" class="new" title="Permukaan revolusi (halaman belum tersedia)">permukaan revolusi</a>, porosnya berisi pusat bola yaitu <i>koaksial</i> yang terdiri dari lingkaran dan/atau titik.</li></ul> <p>Diagram menunjukkan kasus, dimana perpotongan tabung dan bola terdiri dari dua lingkaran. Jika jari-jari tabung sama dengan jari-jari bola, perpotongannya menjadi satu lingkaran, dimana kedua permukaan bersinggungan. </p><p>Dalam kasus sferoid dengan pusat dan sumbu utama yang sama dengan bola, persimpangan akan terdiri dari dua titik (simpul), dimana permukaannya bersinggungan. </p> <div class="mw-heading mw-heading3"><h3 id="Kurva_Clelia">Kurva Clelia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=7" title="Sunting bagian: Kurva Clelia" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=7" title="Sunting kode sumber bagian: Kurva Clelia"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Kugel-spirale-1-2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/240px-Kugel-spirale-1-2.svg.png" decoding="async" width="240" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/360px-Kugel-spirale-1-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/480px-Kugel-spirale-1-2.svg.png 2x" data-file-width="759" data-file-height="377" /></a><figcaption>spiral bulat dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0977619356f0caa405a5f40070bed06c655db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=8}"></span></figcaption></figure> <p>Jika bola dijelaskan dengan wakilan parametrik </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}=(r\cos \theta \cos \varphi ,r\cos \theta \sin \varphi ,r\sin \theta )^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}=(r\cos \theta \cos \varphi ,r\cos \theta \sin \varphi ,r\sin \theta )^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6508500e7b1a08a4fa5bcd7c861b2fd384ae5dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.068ex; height:3.176ex;" alt="{\displaystyle {\vec {x}}=(r\cos \theta \cos \varphi ,r\cos \theta \sin \varphi ,r\sin \theta )^{T}}"></span></dd></dl> <p>maka akan mendapat <a href="/w/index.php?title=Cl%C3%A9lie&action=edit&redlink=1" class="new" title="Clélie (halaman belum tersedia)">kurva Clelia</a>, jika sudut-sudutnya dihubungkan dengan persamaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =c\;\theta \;,\ c>0\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>c</mi> <mspace width="thickmathspace" /> <mi>θ</mi> <mspace width="thickmathspace" /> <mo>,</mo> <mtext> </mtext> <mi>c</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =c\;\theta \;,\ c>0\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e991a3a009a864904ef03e8ffbde262f1392e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.181ex; height:2.676ex;" alt="{\displaystyle \varphi =c\;\theta \;,\ c>0\;.}"></span> </p><p>Kasus khususnya adalah: <a href="/w/index.php?title=Kurva_Viviani&action=edit&redlink=1" class="new" title="Kurva Viviani (halaman belum tersedia)">kurva Viviani</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3467f9e219a5ea38a30da5c3a02c2c23f61a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=1}"></span>) dan <a href="/w/index.php?title=Spiral_bola&action=edit&redlink=1" class="new" title="Spiral bola (halaman belum tersedia)">spiral bola</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23589bfa9e6bcefa64f663a435c2338fee9eca15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>2}"></span>), sebagai contohnya <a href="/w/index.php?title=Spiral_Seiffert&action=edit&redlink=1" class="new" title="Spiral Seiffert (halaman belum tersedia)">spiral Seiffert</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Loksodrom">Loksodrom</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=8" title="Sunting bagian: Loksodrom" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=8" title="Sunting kode sumber bagian: Loksodrom"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Loxodrome.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Loxodrome.png/150px-Loxodrome.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Loxodrome.png/225px-Loxodrome.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Loxodrome.png/300px-Loxodrome.png 2x" data-file-width="693" data-file-height="694" /></a><figcaption>Loxodrome</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Loksodrom" title="Loksodrom">Loksodrom</a></div> <p>Dalam <a href="/wiki/Navigasi" title="Navigasi">navigasi</a>, <b>loksodrom</b> adalah busur yang melintasi semua <a href="/wiki/Meridian_(geografi)" title="Meridian (geografi)">meridian</a> dari <a href="/wiki/Garis_bujur" title="Garis bujur">garis bujur</a> pada sudut yang sama. Garis Rhumb bukanlah spiral bola. Tidak ada hubungan sederhana antara sudut <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Persimpangan_bola_dengan_permukaan_yang_umum">Persimpangan bola dengan permukaan yang umum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=9" title="Sunting bagian: Persimpangan bola dengan permukaan yang umum" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=9" title="Sunting kode sumber bagian: Persimpangan bola dengan permukaan yang umum"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Is-spherecyl5-s.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Is-spherecyl5-s.svg/180px-Is-spherecyl5-s.svg.png" decoding="async" width="180" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Is-spherecyl5-s.svg/270px-Is-spherecyl5-s.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Is-spherecyl5-s.svg/360px-Is-spherecyl5-s.svg.png 2x" data-file-width="432" data-file-height="351" /></a><figcaption>Tabung bola persimpangan umum</figcaption></figure> <p>Jika sebuah bola berpotongan dengan permukaan lain, mungkin ada kurva bola yang lebih rumit. </p> <dl><dt>Contoh</dt> <dd>bola-tabung</dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Persimpangan_bola%E2%80%93tabung&action=edit&redlink=1" class="new" title="Persimpangan bola–tabung (halaman belum tersedia)">Persimpangan bola–tabung</a></div> <p>Perpotongan bola dengan persamaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;x^{2}+y^{2}+z^{2}=r^{2}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;x^{2}+y^{2}+z^{2}=r^{2}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88cd888f9226fcfcb71012e96f40a279d8a2c29c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.916ex; height:3.009ex;" alt="{\displaystyle \;x^{2}+y^{2}+z^{2}=r^{2}\;}"></span> dan tabung dengan persamaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;(y-y_{0})^{2}+z^{2}=a^{2},\;y_{0}\neq 0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;(y-y_{0})^{2}+z^{2}=a^{2},\;y_{0}\neq 0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec812f01589696b6b1db88003c9a3746214d6311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.844ex; height:3.176ex;" alt="{\displaystyle \;(y-y_{0})^{2}+z^{2}=a^{2},\;y_{0}\neq 0\;}"></span> bukan hanya satu atau dua lingkaran. Ini adalah solusi dari sistem persamaan non linear </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}-r^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}+z^{2}-r^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91803efe99c5b176a4bccd39b154fc1c398a8510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.628ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}-r^{2}=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y-y_{0})^{2}+z^{2}-a^{2}=0\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y-y_{0})^{2}+z^{2}-a^{2}=0\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a647e39f51682c5f61442adba20602d76042669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.651ex; height:3.176ex;" alt="{\displaystyle (y-y_{0})^{2}+z^{2}-a^{2}=0\ .}"></span></dd></dl> <p>lihat <a href="/w/index.php?title=Kurva_implisit&action=edit&redlink=1" class="new" title="Kurva implisit (halaman belum tersedia)">kurva implisit</a> dan diagram </p> <div class="mw-heading mw-heading2"><h2 id="Sifat_geometris">Sifat geometris</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=10" title="Sunting bagian: Sifat geometris" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=10" title="Sunting kode sumber bagian: Sifat geometris"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bola secara unik ditentukan oleh empat titik yang bukan <a href="/w/index.php?title=Koplanar&action=edit&redlink=1" class="new" title="Koplanar (halaman belum tersedia)">koplanar</a>. Secara lebih umum, bola secara unik ditentukan oleh empat kondisi seperti melewati suatu titik, bersinggungan dengan bidang, dan lain-lain.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Sifat ini analog dengan properti bahwa tiga titik <a href="/w/index.php?title=Kollinear&action=edit&redlink=1" class="new" title="Kollinear (halaman belum tersedia)">non-kollinear</a> menentukan lingkaran unik dalam sebuah bidang. </p><p>Maka, sebuah bola unik ditentukan oleh sebuah lingkaran dan sebuah titik yang tidak berada di bidang lingkaran itu. </p><p>Dengan memeriksa <a href="/w/index.php?title=Lingkaran_bola&action=edit&redlink=1" class="new" title="Lingkaran bola (halaman belum tersedia)">solusi umum dari persamaan dua bola</a>, dapat dilihat bahwa dua bola berpotongan dalam satu lingkaran dan bidang yang mengandung lingkaran itu disebut <b>bidang radikal</b> dari bola berpotongan.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Meskipun bidang radikal adalah bidang riil, lingkaran mungkin imajiner yaitu bola tidak memiliki titik yang sama atau terdiri dari satu titik sebagai bola bersinggungan pada titik itu.<sup id="cite_ref-Woods267_11-0" class="reference"><a href="#cite_note-Woods267-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Sudut antara dua bola pada titik perpotongan sebenarnya adalah <a href="/wiki/Sudut_dihedral" title="Sudut dihedral">sudut dihedral</a> yang ditentukan oleh bidang bersinggungan dengan bola pada titik tersebut. Dua bola berpotongan pada sudut yang sama di semua titik perpotongan lingkaran.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Potongan pada sudut siku-siku adalah <a href="/w/index.php?title=Ortogonalitas&action=edit&redlink=1" class="new" title="Ortogonalitas (halaman belum tersedia)">ortogonal</a> jika dan hanya jika kuadrat jarak antara pusatnya sama dengan jumlah kuadrat jari-jarinya.<sup id="cite_ref-Woods266_2-1" class="reference"><a href="#cite_note-Woods266-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Pensil_bola">Pensil bola</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=11" title="Sunting bagian: Pensil bola" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=11" title="Sunting kode sumber bagian: Pensil bola"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Pensil_(matematika)&action=edit&redlink=1" class="new" title="Pensil (matematika) (halaman belum tersedia)">Pensil (matematika) § Pensil bola</a></div> <p>Jika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x,y,z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x,y,z)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c708cc026410661d1c4fef508920c812d8e325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.99ex; height:2.843ex;" alt="{\textstyle f(x,y,z)=0}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g(x,y,z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle g(x,y,z)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7234fd41cfdc821edf443cfe8351d4c717da830e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.827ex; height:2.843ex;" alt="{\textstyle g(x,y,z)=0}"></span>adalah persamaan dari dua bidang yang berbeda </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sf(x,y,z)+tg(x,y,z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>t</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sf(x,y,z)+tg(x,y,z)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163ef7689e46b6f2d09c1e04048b990192fef39d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.327ex; height:2.843ex;" alt="{\displaystyle sf(x,y,z)+tg(x,y,z)=0}"></span></dd></dl> <p>juga persamaan bola untuk nilai arbitrer dari parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bc926f90178739fccd01a96c6fa778ab3535d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\textstyle t}"></span>. Himpunan semua bola memenuhi persamaan ini disebut <b>pensil bola</b> yang ditentukan oleh dua bola asli. Dalam definisi ini bola dijadikan menjadi bidang (jari-jari tak hingga, berpusat pada tak hingga) dan jika kedua bola asli adalah bidang maka semua bidang pensil adalah bidang, jika tidak, hanya ada satu bidang (bidang akar) dalam pensil.<sup id="cite_ref-Woods266_2-2" class="reference"><a href="#cite_note-Woods266-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalisasi">Generalisasi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=12" title="Sunting bagian: Generalisasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=12" title="Sunting kode sumber bagian: Generalisasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=N-bola&action=edit&redlink=1" class="new" title="N-bola (halaman belum tersedia)">n-bola</a> dan <a href="/wiki/Ruang_metrik" title="Ruang metrik">Ruang metrik</a></div> <div class="mw-heading mw-heading3"><h3 id="Dimensi">Dimensi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=13" title="Sunting bagian: Dimensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=13" title="Sunting kode sumber bagian: Dimensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bola dapat digeneralisasikan ke ruang dengan jumlah <a href="/wiki/Dimensi" title="Dimensi">dimensi</a> berapa pun. Untuk <a href="/wiki/Bilangan_asli" title="Bilangan asli">bilangan asli</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span>, sebuah "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span>-bola," sering kali ditulis sebagai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966657e05b67213e20c638305e8feebc21da751c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.176ex;" alt="{\textstyle S^{n}}"></span>, adalah titik himpunan dalam (dimensi-(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0362ca5ff26cdd1c0e0ea03aaeb811993037cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\textstyle n+1}"></span>)). Ruang Euklides yang berada pada jarak tetap <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\textstyle r}"></span> dari titik pusat ruang itu, dimana <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\textstyle r}"></span>, seperti sebelumnya, adalah bilangan riil positif. Khususnya: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff67a8d534f5a7b926df33be0320cb5a9c5f607d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.509ex;" alt="{\textstyle S^{0}}"></span>: bola 0 adalah sepasang titik akhir dari sebuah interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [-r,r]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mi>r</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [-r,r]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750a4d55d2fc4083f03a3810ed335d059d1fb096" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.233ex; height:2.843ex;" alt="{\textstyle [-r,r]}"></span> dari garis sebenarnya</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d16e4b102c372bf4a1fa8acbe491af07e0c92c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.509ex;" alt="{\textstyle S^{1}}"></span>: 1 bola adalah <a href="/wiki/Lingkaran" title="Lingkaran">lingkaran</a> dengan jari-jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\textstyle r}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c2a1c638c40bff27beb1c3e05d423e1f24210a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.509ex;" alt="{\textstyle S^{2}}"></span>: 2-bola adalah bola biasa</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a024b8f56024a03dc36f7678d5e4dfa7d8274974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.509ex;" alt="{\textstyle S^{3}}"></span>: <a href="/w/index.php?title=3-bola&action=edit&redlink=1" class="new" title="3-bola (halaman belum tersedia)">3-bola</a> adalah bola dalam ruang Euclidean 4-dimensi.</li></ul> <p>Bola untuk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302f53c3fc56fd109d32f8703895c58fec8bd2a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\textstyle n>2}"></span> terkadang disebut <a href="/wiki/Hiperbola" title="Hiperbola">hiperbola</a>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span>-bola dengan radius unit yang berpusat di titik asal dilambangkan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966657e05b67213e20c638305e8feebc21da751c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.176ex;" alt="{\textstyle S^{n}}"></span> dan sering disebut sebagai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span>-bola. Perhatikan bahwa bola biasa adalah bola 2, karena permukaannya 2 dimensi yang tertanam dalam ruang 3 dimensi. </p><p>Luas permukaan unit (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960c88fa1831b7505d9672de66058532fa5d4053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\textstyle n-1}"></span>)-bola adalah </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e11c413d3b1c2217860dd64b619fef0e8e27cfbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.628ex; height:7.676ex;" alt="{\displaystyle {\frac {2\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}\right)}}}"></span></dd></dl> <p>dimana <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Gamma (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Gamma (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52882da13de8263a2e94c29b3e774bcee43f7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.35ex; height:2.843ex;" alt="{\textstyle \Gamma (z)}"></span> adalah <a href="/wiki/Fungsi_gamma" title="Fungsi gamma">fungsi gamma</a> Euler. </p><p>Ekspresi lain untuk luas permukaan adalah </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\displaystyle {\frac {(2\pi )^{n/2}\,r^{n-1}}{2\cdot 4\cdots (n-2)}},&{\text{jika }}n{\text{ genap}};\\\\\displaystyle {\frac {2(2\pi )^{(n-1)/2}\,r^{n-1}}{1\cdot 3\cdots (n-2)}},&{\text{jika }}n{\text{ ganjil}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>jika </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> genap</mtext> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>jika </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> ganjil</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\displaystyle {\frac {(2\pi )^{n/2}\,r^{n-1}}{2\cdot 4\cdots (n-2)}},&{\text{jika }}n{\text{ genap}};\\\\\displaystyle {\frac {2(2\pi )^{(n-1)/2}\,r^{n-1}}{1\cdot 3\cdots (n-2)}},&{\text{jika }}n{\text{ ganjil}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f12d065d68dccbdc6498fb7b42776a812a8c091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:35.98ex; height:16.176ex;" alt="{\displaystyle {\begin{cases}\displaystyle {\frac {(2\pi )^{n/2}\,r^{n-1}}{2\cdot 4\cdots (n-2)}},&{\text{jika }}n{\text{ genap}};\\\\\displaystyle {\frac {2(2\pi )^{(n-1)/2}\,r^{n-1}}{1\cdot 3\cdots (n-2)}},&{\text{jika }}n{\text{ ganjil}}.\end{cases}}}"></span></dd></dl> <p>dan volume adalah kali luas permukaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {r}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {r}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a8f2e6d2ffcd132365ad4e2cdd901731a5b480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.009ex;" alt="{\textstyle {\frac {r}{n}}}"></span> atau </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\displaystyle {\frac {(2\pi )^{n/2}\,r^{n}}{2\cdot 4\cdots n}},&{\text{jika }}n{\text{ genap}};\\\\\displaystyle {\frac {2(2\pi )^{(n-1)/2}\,r^{n}}{1\cdot 3\cdots n}},&{\text{jika }}n{\text{ ganjil}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋯</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>jika </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> genap</mtext> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋯</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>jika </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> ganjil</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\displaystyle {\frac {(2\pi )^{n/2}\,r^{n}}{2\cdot 4\cdots n}},&{\text{jika }}n{\text{ genap}};\\\\\displaystyle {\frac {2(2\pi )^{(n-1)/2}\,r^{n}}{1\cdot 3\cdots n}},&{\text{jika }}n{\text{ ganjil}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce5459227c1007aec66c9279b55c9c3e5ced39d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:33.88ex; height:14.843ex;" alt="{\displaystyle {\begin{cases}\displaystyle {\frac {(2\pi )^{n/2}\,r^{n}}{2\cdot 4\cdots n}},&{\text{jika }}n{\text{ genap}};\\\\\displaystyle {\frac {2(2\pi )^{(n-1)/2}\,r^{n}}{1\cdot 3\cdots n}},&{\text{jika }}n{\text{ ganjil}}.\end{cases}}}"></span></dd></dl> <p>Rumus rekursif umum juga ada untuk <a href="/w/index.php?title=Volume_bola-n&action=edit&redlink=1" class="new" title="Volume bola-n (halaman belum tersedia)">volume dari <span class="texhtml mvar" style="font-style:italic;">n</span>-bola</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Ruang_metrik">Ruang metrik</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=14" title="Sunting bagian: Ruang metrik" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=14" title="Sunting kode sumber bagian: Ruang metrik"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Secara lebih umum, dalam <a href="/wiki/Ruang_metrik" title="Ruang metrik">ruang metrik</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (E,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (E,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/553607181c987b5ac1a85e0a44acc8fb9e4e353d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.835ex; height:2.843ex;" alt="{\textstyle (E,d)}"></span>, bola pusat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span> dan jari-jari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ecde4f6901db44d3a87dd8c22ba1d4d09f95f58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\textstyle r>0}"></span> adalah titik himpunan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db9936ddb2761b76fa640fb275cb5d1fa4d6fa23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\textstyle y}"></span> sedemikian rupa maka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d(x,y)=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d(x,y)=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa4b2e8286ffc99cce603d530857d6a8b62e4792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.691ex; height:2.843ex;" alt="{\textstyle d(x,y)=r}"></span>. </p><p>Jika pusatnya adalah titik dibedakan yang dianggap sebagai asal dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d934f67126f64e9c061b598b8941b8767a8d343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\textstyle E}"></span>, seperti dalam ruang <a href="/wiki/Norma_(matematika)" title="Norma (matematika)">norma</a>, itu tidak disebutkan dalam definisi dan notasi. Hal yang sama berlaku untuk jari-jari jika dianggap sama dengan satu, seperti dalam kasus <a href="/w/index.php?title=Bola_unit&action=edit&redlink=1" class="new" title="Bola unit (halaman belum tersedia)">bola unit</a>. </p><p>Tidak dengan <a href="/wiki/Bola_(matematika)" class="mw-redirect" title="Bola (matematika)">bola</a>, bahkan sebuah bola besar dapat berupa himpunan kosong. Misalnya, dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathbf {Z}}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathbf {Z}}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff55a359c8bab1fcb67632704ca79dc472e82c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.343ex;" alt="{\textstyle {\mathbf {Z}}^{n}}"></span> dengan <a href="/w/index.php?title=Metrik_Eullides&action=edit&redlink=1" class="new" title="Metrik Eullides (halaman belum tersedia)">metrik Eullides</a>, radius radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\textstyle r}"></span> tidak kosong hanya jika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2ef4294c503d7f5861ab12e7e7432cc8d14038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.103ex; height:2.509ex;" alt="{\textstyle r^{2}}"></span> bisa ditulis sebagai jumlah dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span> kuadrat dari <a href="/wiki/Bilangan_bulat" title="Bilangan bulat">bilangan bulat</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Geometri_bola">Geometri bola</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=15" title="Sunting bagian: Geometri bola" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=15" title="Sunting kode sumber bagian: Geometri bola"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Sphere_halve.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Sphere_halve.png/220px-Sphere_halve.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Sphere_halve.png/330px-Sphere_halve.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Sphere_halve.png/440px-Sphere_halve.png 2x" data-file-width="960" data-file-height="960" /></a><figcaption><a href="/wiki/Lingkaran_besar" title="Lingkaran besar">Lingkaran besar</a> pada bola</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Geometri_bola" title="Geometri bola">Geometri bola</a></div> <p>Elemen dasar geometri bidang Euclidean adalah titik dan garis. Di bola, titik didefinisikan dalam arti biasa. Analog dari "garis" adalah geodesik, yang merupakan lingkaran besar; ciri utama dari lingkaran besar adalah bahwa bidang yang berisi semua titiknya juga melewati pusat bola. Mengukur dengan panjang busur menunjukkan bahwa jalur terpendek antara dua titik yang terletak di bola adalah segmen yang lebih pendek dari lingkaran besar yang mencakup titik-titik tersebut. </p><p>Banyak teorema dari geometri klasik juga berlaku untuk geometri bola, tetapi tidak semua melakukannya karena bola gagal memenuhi beberapa postulat geometri klasik, termasuk postulat paralel. Dalam trigonometri bola, sudut didefinisikan antara lingkaran besar. <a href="/wiki/Trigonometri" title="Trigonometri">Trigonometri</a> bola berbeda dari trigonometri biasa dalam banyak hal. Misalnya, jumlah sudut interior segitiga bulat selalu melebihi 180 derajat. Juga, dua segitiga bundar yang serupa adalah kongruen. </p> <div class="mw-heading mw-heading2"><h2 id="Lokus_jumlah_konstan">Lokus jumlah konstan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=16" title="Sunting bagian: Lokus jumlah konstan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=16" title="Sunting kode sumber bagian: Lokus jumlah konstan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lokus titik dalam ruang sedemikian rupa sehingga jumlah ke <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e100f32f96dc84bf0591df4f5c5bd40d71189f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle 2m}"></span> pangkat jarak <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}"></span> ke simpul dari <a href="/w/index.php?title=Padatan_Platonis&action=edit&redlink=1" class="new" title="Padatan Platonis (halaman belum tersedia)">padatan Platonis</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9241493be76739f2400f258f32c24f9689161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}"></span> dengan sirkumradius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> konstan adalah sebuah bola, jika </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msubsup> <mo>></mo> <mi>n</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aba8e5893caef016f476abbc446aacbf8045c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.211ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m}}"></span>,</dd></dl> <p>yang pusatnya berada di pusat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9241493be76739f2400f258f32c24f9689161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}"></span>.<sup id="cite_ref-Mamuka_13-0" class="reference"><a href="#cite_note-Mamuka-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Nilai dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> bergantung pada jumlah simpul <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> dari padatan Platonis dan sama: </p><p><b>•</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m=1,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m=1,2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee00d110de77dd283f48bbb065c528c5bc78e40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.498ex; height:2.509ex;" alt="{\textstyle m=1,2}"></span> untuk <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> reguler, </p><p><b>•</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m=1,2,3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m=1,2,3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f789fe5b205979f2ae4516ccdf9acc581fe1e8e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.694ex; height:2.509ex;" alt="{\textstyle m=1,2,3}"></span> untuk <a href="/wiki/Oktahedron" title="Oktahedron">oktahedron</a> dan <a href="/wiki/Kubus" title="Kubus">kubus</a>, </p><p><b>•</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m=1,2,3,4,5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m=1,2,3,4,5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9104225a792a1ec31c4c116ae7edd318900b5bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.087ex; height:2.509ex;" alt="{\textstyle m=1,2,3,4,5}"></span> untuk <a href="/wiki/Ikosahedron" title="Ikosahedron">ikosahedron</a> dan <a href="/wiki/Dodekahedron" title="Dodekahedron">dodekahedron</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Gambar">Gambar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=17" title="Sunting bagian: Gambar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=17" title="Sunting kode sumber bagian: Gambar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-packed" style="text-align:left"> <li class="gallerybox" style="width: 300px"> <div class="thumb" style="width: 298px;"><span typeof="mw:File"><a href="/wiki/Berkas:Einstein_gyro_gravity_probe_b.jpg" class="mw-file-description" title="Gambar salah satu bola buatan manusia yang paling akurat, karena membiaskan gambar Einstein di latar belakang. Bola ini adalah kuarsa leburan giroskop untuk percobaan Gravity Probe B, dan berbeda dalam bentuk dari bola sempurna dengan ketebalan tidak lebih dari 40 atom (kurang dari 10 nm). Diumumkan pada tanggal 1 Juli 2008 bahwa ilmuwan asal Australia telah menciptakan bidang yang lebih mendekati sempurna, akurat hingga 0,3 nm, sebagai bagian dari perburuan internasional untuk menemukan standar global baru kilogram.[14]"><img alt="Gambar salah satu bola buatan manusia yang paling akurat, karena membiaskan gambar Einstein di latar belakang. Bola ini adalah kuarsa leburan giroskop untuk percobaan Gravity Probe B, dan berbeda dalam bentuk dari bola sempurna dengan ketebalan tidak lebih dari 40 atom (kurang dari 10 nm). Diumumkan pada tanggal 1 Juli 2008 bahwa ilmuwan asal Australia telah menciptakan bidang yang lebih mendekati sempurna, akurat hingga 0,3 nm, sebagai bagian dari perburuan internasional untuk menemukan standar global baru kilogram.[14]" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Einstein_gyro_gravity_probe_b.jpg/447px-Einstein_gyro_gravity_probe_b.jpg" decoding="async" width="298" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Einstein_gyro_gravity_probe_b.jpg/671px-Einstein_gyro_gravity_probe_b.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Einstein_gyro_gravity_probe_b.jpg/894px-Einstein_gyro_gravity_probe_b.jpg 2x" data-file-width="3552" data-file-height="2384" /></a></span></div> <div class="gallerytext">Gambar salah satu bola buatan manusia yang paling akurat, karena <a href="/wiki/Refraksi" class="mw-redirect" title="Refraksi">membiaskan</a> gambar <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a> di latar belakang. Bola ini adalah <a href="/w/index.php?title=Kuarsa_leburan&action=edit&redlink=1" class="new" title="Kuarsa leburan (halaman belum tersedia)">kuarsa leburan</a> <a href="/wiki/Giroskop" title="Giroskop">giroskop</a> untuk percobaan <a href="/w/index.php?title=Gravity_Probe_B&action=edit&redlink=1" class="new" title="Gravity Probe B (halaman belum tersedia)">Gravity Probe B</a>, dan berbeda dalam bentuk dari bola sempurna dengan ketebalan tidak lebih dari 40 atom (kurang dari 10<span class="nowrap"> </span>nm). Diumumkan pada tanggal 1 Juli 2008 bahwa ilmuwan asal <a href="/wiki/Australia" title="Australia">Australia</a> telah menciptakan bidang yang lebih mendekati sempurna, akurat hingga 0,3<span class="nowrap"> </span>nm, sebagai bagian dari perburuan internasional untuk menemukan standar global baru <a href="/wiki/Kilogram" title="Kilogram">kilogram</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Bagian">Bagian</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=18" title="Sunting bagian: Bagian" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=18" title="Sunting kode sumber bagian: Bagian"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/wiki/Bola_(matematika)#Bagian" class="mw-redirect" title="Bola (matematika)">Bola (matematika) § Bagian</a></div> <ul><li><a href="/w/index.php?title=Tutup_bola&action=edit&redlink=1" class="new" title="Tutup bola (halaman belum tersedia)">Tutup bola</a></li> <li><a href="/w/index.php?title=Poligon_bola&action=edit&redlink=1" class="new" title="Poligon bola (halaman belum tersedia)">Poligon bola</a></li> <li><a href="/w/index.php?title=Sektor_bola&action=edit&redlink=1" class="new" title="Sektor bola (halaman belum tersedia)">Sektor bola</a></li> <li><a href="/w/index.php?title=Segmen_Bulat&action=edit&redlink=1" class="new" title="Segmen Bulat (halaman belum tersedia)">Segmen Bulat</a></li> <li><a href="/w/index.php?title=Baji_bulat&action=edit&redlink=1" class="new" title="Baji bulat (halaman belum tersedia)">Baji bulat</a></li> <li><a href="/w/index.php?title=Zona_bola&action=edit&redlink=1" class="new" title="Zona bola (halaman belum tersedia)">Zona bola</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=19" title="Sunting bagian: Lihat pula" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=19" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r18261910">.mw-parser-output .div-col{margin-top:.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width:20em;"> <ul><li><a href="/w/index.php?title=Tribola&action=edit&redlink=1" class="new" title="Tribola (halaman belum tersedia)">Tribola</a></li> <li><a href="/w/index.php?title=Bola_Affin&action=edit&redlink=1" class="new" title="Bola Affin (halaman belum tersedia)">Bola Affin</a></li> <li><a href="/w/index.php?title=Bola_bertanduk_Alexander&action=edit&redlink=1" class="new" title="Bola bertanduk Alexander (halaman belum tersedia)">Bola bertanduk Alexander</a></li> <li><a href="/w/index.php?title=Bola_kelestial&action=edit&redlink=1" class="new" title="Bola kelestial (halaman belum tersedia)">Bola kelestial</a></li> <li><a href="/wiki/Kubus" title="Kubus">Kubus</a></li> <li><a href="/wiki/Lengkungan" class="mw-redirect" title="Lengkungan">Lengkungan</a></li> <li><a href="/w/index.php?title=Statistik_arah&action=edit&redlink=1" class="new" title="Statistik arah (halaman belum tersedia)">Statistik arah</a></li> <li><a href="/w/index.php?title=Lengkungan_puncak_(matematika)&action=edit&redlink=1" class="new" title="Lengkungan puncak (matematika) (halaman belum tersedia)">Lengkungan puncak (matematika)</a></li> <li><a href="/wiki/Bola_Dyson" title="Bola Dyson">Bola Dyson</a></li> <li><a href="/w/index.php?title=Tangan_dengan_bola_refleksi&action=edit&redlink=1" class="new" title="Tangan dengan bola refleksi (halaman belum tersedia)">Tangan dengan bola refleksi</a>, <a href="/w/index.php?title=M.C._Escher&action=edit&redlink=1" class="new" title="M.C. Escher (halaman belum tersedia)">M.C. Escher</a> gambar potret diri yang menggambarkan refleksi dan sifat optik bola cermin</li> <li><a href="/w/index.php?title=Bola_Hoberman&action=edit&redlink=1" class="new" title="Bola Hoberman (halaman belum tersedia)">Bola Hoberman</a></li> <li><a href="/w/index.php?title=Bola_homologi&action=edit&redlink=1" class="new" title="Bola homologi (halaman belum tersedia)">Bola homologi</a></li> <li><a href="/w/index.php?title=Grup_bola_homotopi&action=edit&redlink=1" class="new" title="Grup bola homotopi (halaman belum tersedia)">Grup bola homotopi</a></li> <li><a href="/wiki/Hiperbola" title="Hiperbola">Hiperbola</a></li> <li><a href="/w/index.php?title=Bola_Lenart&action=edit&redlink=1" class="new" title="Bola Lenart (halaman belum tersedia)">Bola Lenart</a></li> <li><a href="/w/index.php?title=Masalah_cincin_serbet&action=edit&redlink=1" class="new" title="Masalah cincin serbet (halaman belum tersedia)">Masalah cincin serbet</a></li> <li><a href="/w/index.php?title=Orb_(optik)&action=edit&redlink=1" class="new" title="Orb (optik) (halaman belum tersedia)">Orb (optik)</a></li> <li><a href="/w/index.php?title=Pseudobola&action=edit&redlink=1" class="new" title="Pseudobola (halaman belum tersedia)">Pseudobola</a></li> <li><a href="/w/index.php?title=Bola_Riemann&action=edit&redlink=1" class="new" title="Bola Riemann (halaman belum tersedia)">Bola Riemann</a></li> <li><a href="/w/index.php?title=Sudut_padat&action=edit&redlink=1" class="new" title="Sudut padat (halaman belum tersedia)">Sudut padat</a></li> <li><a href="/w/index.php?title=Pengepakan_bola&action=edit&redlink=1" class="new" title="Pengepakan bola (halaman belum tersedia)">Pengepakan bola</a></li> <li><a href="/w/index.php?title=Koordinat_bola&action=edit&redlink=1" class="new" title="Koordinat bola (halaman belum tersedia)">Koordinat bola</a></li> <li><a href="/w/index.php?title=Bola_bumi&action=edit&redlink=1" class="new" title="Bola bumi (halaman belum tersedia)">Bola bumi</a></li> <li>Heliks bola, <a href="/w/index.php?title=Indikator_tangen&action=edit&redlink=1" class="new" title="Indikator tangen (halaman belum tersedia)">indikator tangen</a> dari kurva presesi konstan</li> <li><a href="/w/index.php?title=Kebulatan&action=edit&redlink=1" class="new" title="Kebulatan (halaman belum tersedia)">Kebulatan</a></li> <li><a href="/w/index.php?title=Teorema_bola_tenis&action=edit&redlink=1" class="new" title="Teorema bola tenis (halaman belum tersedia)">Teorema bola tenis</a></li> <li><a href="/w/index.php?title=Permukaan_Zoll&action=edit&redlink=1" class="new" title="Permukaan Zoll (halaman belum tersedia)">Bola Zoll</a></li> <li><a href="/wiki/Frustum_bola" title="Frustum bola">Frustum bola</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Catatan_dan_referensi">Catatan dan referensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=20" title="Sunting bagian: Catatan dan referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=20" title="Sunting kode sumber bagian: Catatan dan referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Catatan">Catatan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=21" title="Sunting bagian: Catatan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=21" title="Sunting kode sumber bagian: Catatan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bagian ini kosong </p> <div class="mw-heading mw-heading3"><h3 id="Referensi">Referensi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=22" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=22" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r18833634">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Albert54-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Albert54_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Albert54_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, hal. 54.</span> </li> <li id="cite_note-Woods266-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Woods266_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Woods266_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Woods266_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFWoods1961">Woods 1961</a>, p. 266.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">(<a href="#CITEREFKreyszig1972">Kreyszig 1972</a>, hlm. 342).</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 60.</span> </li> <li id="cite_note-MathWorld_Sphere-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-MathWorld_Sphere_5-0">^</a></b></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:#777; cursor:help;" title="Bahasa Inggris" lang="Inggris">(Inggris)</span> <span class="citation mathworld" id="Reference-Mathworld-Sphere"><cite id="CITEREFWeisstein" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Sphere.html">"Sphere"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Sphere&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSphere.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteinhaus1969">Steinhaus 1969</a>, p. 221.</span> </li> <li id="cite_note-delta-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-delta_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-delta_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">E.J. Borowski; J.M. Borwein (1989). <a rel="nofollow" class="external text" href="https://archive.org/details/dictionaryofmath0000boro"><i>Collins Dictionary of Mathematics</i></a>. hlm. <a rel="nofollow" class="external text" href="https://archive.org/details/dictionaryofmath0000boro/page/141">141</a>, 149. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-00-434347-1" title="Istimewa:Sumber buku/978-0-00-434347-1">978-0-00-434347-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collins+Dictionary+of+Mathematics&rft.pages=141%2C+149&rft.date=1989&rft.isbn=978-0-00-434347-1&rft.au=E.J.+Borowski&rft.au=J.M.+Borwein&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdictionaryofmath0000boro&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="Osserman" class="citation journal">Osserman, Robert (1978). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/">"The isoperimetric inequality"</a>. <i>Bulletin of the American Mathematical Society</i>. <b>84</b>: 1187<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">14 December</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=The+isoperimetric+inequality&rft.volume=84&rft.pages=1187&rft.date=1978&rft.aulast=Osserman&rft.aufirst=Robert&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1978-84-06%2FS0002-9904-1978-14553-4%2F&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 55.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, hal. 57.</span> </li> <li id="cite_note-Woods267-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Woods267_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWoods1961">Woods 1961</a>, hal. 267.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 58.</span> </li> <li id="cite_note-Mamuka-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mamuka_13-0">^</a></b></span> <span class="reference-text"><cite class="citation journal">Meskhishvili, Mamuka (2020). <a rel="nofollow" class="external text" href="https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065">"Cyclic Averages of Regular Polygons and Platonic Solids"</a>. <i>Communications in Mathematics and Applications</i>. <b>11</b>: 335–355.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematics+and+Applications&rft.atitle=Cyclic+Averages+of+Regular+Polygons+and+Platonic+Solids&rft.volume=11&rft.pages=335-355&rft.date=2020&rft.aulast=Meskhishvili&rft.aufirst=Mamuka&rft_id=https%3A%2F%2Fwww.rgnpublications.com%2Fjournals%2Findex.php%2Fcma%2Farticle%2Fview%2F1420%2F1065&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.newscientist.com/article/dn14229-roundest-objects-in-the-world-created.html">New Scientist | Technology | Roundest objects in the world created</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Bacaan_lebih_lanjut">Bacaan lebih lanjut</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=23" title="Sunting bagian: Bacaan lebih lanjut" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=23" title="Sunting kode sumber bagian: Bacaan lebih lanjut"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r23035139">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r23782729">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" class="mw-redirect" title="Wikisource">Wikisource</a> memiliki teks artikel <a href="/w/index.php?title=Ensiklopedia_Britannica_Edisi_Kesebelas&action=edit&redlink=1" class="new" title="Ensiklopedia Britannica Edisi Kesebelas (halaman belum tersedia)">Ensiklopedia Britannica 1911</a> mengenai <i><b><a href="https://id.wikisource.org/wiki/Ensiklopedia_Britannica_Edisi_Kesebelas/Sphere" class="extiw" title="s:Ensiklopedia Britannica Edisi Kesebelas/Sphere">Sphere</a></b></i>.</div></div> </div> <ul><li><cite id="CITEREFAlbert2016" class="citation">Albert, Abraham Adrian (2016) [1949], <i>Solid Analytic Geometry</i>, Dover, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-486-81026-3" title="Istimewa:Sumber buku/978-0-486-81026-3">978-0-486-81026-3</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solid+Analytic+Geometry&rft.pub=Dover&rft.date=2016&rft.isbn=978-0-486-81026-3&rft.aulast=Albert&rft.aufirst=Abraham+Adrian&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><cite class="citation book">Dunham, William (1997). <span class="plainlinks"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicaluniv00dunh"><i>The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems and Personalities</i><span style="padding-left:0.15em"><span typeof="mw:File"><span title="Akses gratis dibatasi (uji coba), biasanya perlu berlangganan"><img alt="Akses gratis dibatasi (uji coba), biasanya perlu berlangganan" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/9px-Lock-blue-alt-2.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/14px-Lock-blue-alt-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/18px-Lock-blue-alt-2.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span></a></span>. <i>Wiley</i>. New York. hlm. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicaluniv00dunh/page/n34">28</a>, 226. <a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1994muaa.book.....D">1994muaa.book.....D</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-471-17661-9" title="Istimewa:Sumber buku/978-0-471-17661-9">978-0-471-17661-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Universe%3A+An+Alphabetical+Journey+Through+the+Great+Proofs%2C+Problems+and+Personalities&rft.place=New+York&rft.pages=28%2C+226&rft.date=1997&rft_id=info%3Abibcode%2F1994muaa.book.....D&rft.isbn=978-0-471-17661-9&rft.aulast=Dunham&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicaluniv00dunh&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFKreyszig1972" class="citation">Kreyszig, Erwin (1972), <span class="plainlinks"><a rel="nofollow" class="external text" href="https://archive.org/details/advancedengineer00krey"><i>Advanced Engineering Mathematics</i><span style="padding-left:0.15em"><span typeof="mw:File"><span title="Perlu mendaftar (gratis)"><img alt="Perlu mendaftar (gratis)" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/9px-Lock-blue-alt-2.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/14px-Lock-blue-alt-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/18px-Lock-blue-alt-2.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span></a></span> (edisi ke-3rd), New York: <a href="/wiki/John_Wiley_%26_Sons" title="John Wiley & Sons">Wiley</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-471-50728-4" title="Istimewa:Sumber buku/978-0-471-50728-4">978-0-471-50728-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Engineering+Mathematics&rft.place=New+York&rft.edition=3rd&rft.pub=Wiley&rft.date=1972&rft.isbn=978-0-471-50728-4&rft.aulast=Kreyszig&rft.aufirst=Erwin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fadvancedengineer00krey&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><cite id="CITEREFSteinhaus1969" class="citation">Steinhaus, H. (1969), <i>Mathematical Snapshots</i> (edisi ke-Third American), Oxford University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Snapshots&rft.edition=Third+American&rft.pub=Oxford+University+Press&rft.date=1969&rft.aulast=Steinhaus&rft.aufirst=H.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><cite id="CITEREFWoods1961" class="citation">Woods, Frederick S. (1961) [1922], <i>Higher Geometry / An Introduction to Advanced Methods in Analytic Geometry</i>, Dover</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Higher+Geometry+%2F+An+Introduction+to+Advanced+Methods+in+Analytic+Geometry&rft.pub=Dover&rft.date=1961&rft.aulast=Woods&rft.aufirst=Frederick+S.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Pranala_luar">Pranala luar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bola_(geometri)&veaction=edit&section=24" title="Sunting bagian: Pranala luar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bola_(geometri)&action=edit&section=24" title="Sunting kode sumber bagian: Pranala luar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="metadata plainlinks mbox-small" style="border:1px solid #aaa; background-color:#f9f9f9;padding:3px;"> <tbody><tr style="height:25px;"> <td colspan="2" style="margin: auto; text-align: center;padding-bottom:5px;"><b>Cari tahu mengenai Bola (geometri) pada proyek-proyek Wikimedia lainnya:</b> </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="//id.wiktionary.org/wiki/Special:Search/Bola_(geometri)" title="Cari di Wiktionary"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Wiktionary-logo-id.svg/21px-Wiktionary-logo-id.svg.png" decoding="async" width="21" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Wiktionary-logo-id.svg/31px-Wiktionary-logo-id.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Wiktionary-logo-id.svg/41px-Wiktionary-logo-id.svg.png 2x" data-file-width="391" data-file-height="474" /></a></span></td> <td><a href="https://id.wiktionary.org/wiki/Special:Search/Bola_(geometri)" class="extiw" title="wikt:Special:Search/Bola (geometri)">Definisi dan terjemahan</a> dari Wiktionary<br /> </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="//commons.wikimedia.org/wiki/Special:Search/Bola_(geometri)" title="Cari di Commons"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/28px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/37px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></td> <td><a href="https://commons.wikimedia.org/wiki/Special:Search/Bola_(geometri)" class="extiw" title="commons:Special:Search/Bola (geometri)">Gambar dan media</a> dari Commons<br /> </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="//id.wikinews.org/wiki/Special:Search/Bola_(geometri)" title="Cari di Wikinews"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/25px-Wikinews-logo.svg.png" decoding="async" width="25" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/38px-Wikinews-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/50px-Wikinews-logo.svg.png 2x" data-file-width="759" data-file-height="415" /></a></span></td> <td><a href="https://id.wikinews.org/wiki/Special:Search/Bola_(geometri)" class="extiw" title="n:Special:Search/Bola (geometri)">Berita</a> dari Wikinews<br /> </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="//id.wikiquote.org/wiki/Special:Search/Bola_(geometri)" title="Cari di Wikiquote"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/21px-Wikiquote-logo.svg.png" decoding="async" width="21" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/32px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/42px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span></td> <td><a href="https://id.wikiquote.org/wiki/Special:Search/Bola_(geometri)" class="extiw" title="q:Special:Search/Bola (geometri)">Kutipan</a> dari Wikiquote<br /> </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="//id.wikisource.org/wiki/Special:Search/Bola_(geometri)" title="Cari di Wikisource"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/24px-Wikisource-logo.svg.png" decoding="async" width="24" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/36px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/48px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></a></span></td> <td><a href="https://id.wikisource.org/wiki/Special:Search/Bola_(geometri)" class="extiw" title="s:Special:Search/Bola (geometri)">Teks sumber</a> dari Wikisource<br /> </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="//id.wikibooks.org/wiki/Special:Search/Bola_(geometri)" title="Cari di Wikibuku"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/25px-Wikibooks-logo.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/38px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/50px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span></td> <td><a href="https://id.wikibooks.org/wiki/Special:Search/Bola_(geometri)" class="extiw" title="b:Special:Search/Bola (geometri)">Buku</a> dari Wikibuku<br /> </td></tr> </tbody></table> <ul><li><a href="https://planetmath.org/alphabetical.html" class="extiw" title="planetmath:186">Sphere (PlanetMath.org website)</a></li> <li><span style="font-size:0.95em; font-weight:bold; color:#777; cursor:help;" title="Bahasa Inggris" lang="Inggris">(Inggris)</span> <span class="citation mathworld" id="Reference-Mathworld-Sphere"><cite id="CITEREFWeisstein" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Sphere.html">"Sphere"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Sphere&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSphere.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span></span></li> <li><a class="external text" href="https://en.wikibooks.org/wiki/Mathematica/Uniform_Spherical_Distribution">Mathematica/Uniform Spherical Distribution</a></li> <li><cite class="citation audio-visual"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070901111142/http://video.google.com/videoplay?docid=-6626464599825291409"><i>Outside In</i></a>. 2007-11-14. Diarsipkan dari <a rel="nofollow" class="external text" href="http://video.google.com/videoplay?docid=-6626464599825291409">versi asli</a> tanggal 2007-09-01<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2007-11-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Outside+In&rft.date=2007-11-14&rft_id=http%3A%2F%2Fvideo.google.com%2Fvideoplay%3Fdocid%3D-6626464599825291409&rfr_id=info%3Asid%2Fid.wikipedia.org%3ABola+%28geometri%29" class="Z3988"><span style="display:none;"> </span></span> (computer animation showing how the inside of a sphere can turn outside.)</li> <li><a rel="nofollow" class="external text" href="http://www.start2code.com/Cresources/sphere-program-cpp.html">Program in C++ to draw a sphere using parametric equation</a></li> <li><a rel="nofollow" class="external text" href="http://mathschallenge.net/index.php?section=faq&ref=geometry/surface_sphere">Surface area of sphere proof.</a></li></ul> <p class="mw-empty-elt"> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r23782733">.mw-parser-output .hlist dl,.mw-parser-output .hlist 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class="mw-redirect" title="Ruang (geometri)">Ruang (3D)</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Besaran geometri menurut dimensi</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Panjang" title="Panjang">Panjang (1D)</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Luas" title="Luas">Luas (area) (2D)</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Volume" title="Volume">Volume (3D)</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Istilah dasar lain</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Jari-jari" title="Jari-jari">Radius (jari-jari)</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Sisi_(geometri)" title="Sisi (geometri)">Sisi (segi)</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Sudut_(geometri)" title="Sudut (geometri)">Sudut</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bangun 2 dimensi</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Belah_ketupat" title="Belah ketupat">Belah ketupat</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Elips" title="Elips">Elips</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Jajar_genjang" title="Jajar genjang">Jajar genjang</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Layang-layang_(geometri)" title="Layang-layang (geometri)">Layang-layang</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Lingkaran" title="Lingkaran">Lingkaran</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Persegi" title="Persegi">Persegi</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Persegi_panjang" title="Persegi panjang">Persegi panjang</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Poligon" title="Poligon">Poligon (segi-n)</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Segi_empat" title="Segi empat">Segi empat</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Segitiga" title="Segitiga">Segitiga</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Trapesium_(geometri)" title="Trapesium (geometri)">Trapesium</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bangun 3 dimensi</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Balok" title="Balok">Balok</a> <span style="font-weight:bold;"> ·</span>  <a class="mw-selflink selflink">Bola</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Kerucut" title="Kerucut">Kerucut</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Kubus" title="Kubus">Kubus</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Limas" title="Limas">Limas</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Polihedron" title="Polihedron">Polihedron (bidang-n)</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Prisma_(geometri)" title="Prisma (geometri)">Prisma</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Sferoid" title="Sferoid"> Sferoid (elipsoid revolusi)</a> <span style="font-weight:bold;"> ·</span>  <a href="/wiki/Tabung_(geometri)" title="Tabung (geometri)">Tabung (silinder)</a></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25847331"></div><div role="navigation" class="navbox authority-control" 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style="width:1%">Perpustakaan nasional</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119812876">Prancis</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119812876">(data)</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85126590">Amerika Serikat</a></span></li></ul> </div></td></tr></tbody></table></div> <p><br /> </p> <style data-mw-deduplicate="TemplateStyles:r18342415">.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}</style><div role="note" class="metadata plainlinks asbox stub"><table role="presentation"><tbody><tr class="noresize"><td><span typeof="mw:File"><a href="/wiki/Berkas:Dodecahedron.svg" class="mw-file-description"><img alt="Ikon rintisan" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/30px-Dodecahedron.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/45px-Dodecahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Dodecahedron.svg/60px-Dodecahedron.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span></td><td><p class="asbox-body">Artikel bertopik geometri ini adalah sebuah <a href="/wiki/Wikipedia:Rintisan" class="mw-redirect" title="Wikipedia:Rintisan">rintisan</a>. Anda dapat membantu Wikipedia dengan <a class="external text" href="https://id.wikipedia.org/w/index.php?title=Bola_(geometri)&action=edit">mengembangkannya</a>.</p></td></tr></tbody></table><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18590415"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-lihat"><a href="/wiki/Templat:Geometri-stub" class="mw-redirect" title="Templat:Geometri-stub"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/w/index.php?title=Pembicaraan_Templat:Geometri-stub&action=edit&redlink=1" class="new" title="Pembicaraan Templat:Geometri-stub (halaman belum tersedia)"><abbr title="Diskusikan templat ini">b</abbr></a></li><li class="nv-sunting"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Templat:Geometri-stub&action=edit"><abbr title="Sunting templat ini">s</abbr></a></li></ul></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Diperoleh dari "<a dir="ltr" href="https://id.wikipedia.org/w/index.php?title=Bola_(geometri)&oldid=25994388">https://id.wikipedia.org/w/index.php?title=Bola_(geometri)&oldid=25994388</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Istimewa:Daftar_kategori" title="Istimewa:Daftar kategori">Kategori</a>: <ul><li><a href="/wiki/Kategori:Geometri" title="Kategori:Geometri">Geometri</a></li><li><a href="/wiki/Kategori:Bentuk" title="Kategori:Bentuk">Bentuk</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Kategori tersembunyi: <ul><li><a href="/wiki/Kategori:Missing_redirects" title="Kategori:Missing redirects">Missing redirects</a></li><li><a href="/wiki/Kategori:Artikel_yang_dimintakan_pemeriksaan_atas_penerjemahannya" title="Kategori:Artikel yang dimintakan pemeriksaan atas penerjemahannya">Artikel yang dimintakan pemeriksaan atas penerjemahannya</a></li><li><a href="/wiki/Kategori:Artikel_yang_perlu_diperiksa_terjemahannya_Juli_2024" title="Kategori:Artikel yang perlu diperiksa terjemahannya Juli 2024">Artikel yang perlu diperiksa terjemahannya Juli 2024</a></li><li><a href="/wiki/Kategori:Artikel_yang_diterjemahkan_secara_kasar" title="Kategori:Artikel yang diterjemahkan secara kasar">Artikel yang diterjemahkan secara kasar</a></li><li><a href="/wiki/Kategori:Semua_artikel_yang_perlu_dirapikan" title="Kategori:Semua artikel yang perlu dirapikan">Semua artikel yang perlu dirapikan</a></li><li><a href="/wiki/Kategori:Artikel_yang_belum_dirapikan_Juli_2024" title="Kategori:Artikel yang belum dirapikan Juli 2024">Artikel yang belum dirapikan Juli 2024</a></li><li><a href="/wiki/Kategori:Use_dmy_dates_from_March_2011" title="Kategori:Use dmy dates from March 2011">Use dmy dates from March 2011</a></li><li><a href="/wiki/Kategori:Artikel_Wikipedia_dengan_penanda_GND" title="Kategori:Artikel Wikipedia dengan penanda GND">Artikel Wikipedia dengan penanda GND</a></li><li><a href="/wiki/Kategori:Artikel_Wikipedia_dengan_penanda_BNF" title="Kategori:Artikel Wikipedia dengan penanda BNF">Artikel Wikipedia dengan penanda BNF</a></li><li><a href="/wiki/Kategori:Artikel_Wikipedia_dengan_penanda_LCCN" title="Kategori:Artikel Wikipedia dengan penanda LCCN">Artikel Wikipedia dengan penanda LCCN</a></li><li><a href="/wiki/Kategori:Semua_artikel_rintisan" title="Kategori:Semua artikel rintisan">Semua artikel rintisan</a></li><li><a href="/wiki/Kategori:Rintisan_bertopik_geometri" title="Kategori:Rintisan bertopik geometri">Rintisan bertopik geometri</a></li><li><a href="/wiki/Kategori:Semua_artikel_rintisan_Juli_2024" title="Kategori:Semua artikel rintisan Juli 2024">Semua artikel rintisan Juli 2024</a></li><li><a href="/wiki/Kategori:Halaman_yang_menggunakan_format_tag_matematika_usang" title="Kategori:Halaman yang menggunakan format tag matematika usang">Halaman yang menggunakan format tag matematika usang</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Halaman ini terakhir diubah pada 4 Juli 2024, pukul 15.37.</li> <li id="footer-info-copyright">Teks tersedia di bawah <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Lisensi Atribusi-BerbagiSerupa Creative Commons</a>; ketentuan tambahan mungkin berlaku. 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Bola (geometri) - Wikipedia bahasa Indonesia, ensiklopedia bebas