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mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Enclosed_volume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Enclosed_volume"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Enclosed volume</span> </div> </a> <ul id="toc-Enclosed_volume-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surface_area" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surface_area"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Surface area</span> </div> </a> <ul id="toc-Surface_area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_geometric_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_geometric_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Other geometric properties</span> </div> </a> <ul id="toc-Other_geometric_properties-sublist" class="vector-toc-list"> <li id="toc-Pencil_of_spheres" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Pencil_of_spheres"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Pencil of spheres</span> </div> </a> <ul id="toc-Pencil_of_spheres-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties_of_the_sphere" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_of_the_sphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Properties of the sphere</span> </div> </a> <ul id="toc-Properties_of_the_sphere-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Treatment_by_area_of_mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Treatment_by_area_of_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Treatment by area of mathematics</span> </div> </a> <button aria-controls="toc-Treatment_by_area_of_mathematics-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>Toggle Treatment by area of mathematics subsection</span> </button> <ul id="toc-Treatment_by_area_of_mathematics-sublist" class="vector-toc-list"> <li id="toc-Spherical_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spherical_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Spherical geometry</span> </div> </a> <ul id="toc-Spherical_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Differential geometry</span> </div> </a> <ul id="toc-Differential_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Topology</span> </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Curves_on_a_sphere" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Curves_on_a_sphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Curves on a sphere</span> </div> </a> <button aria-controls="toc-Curves_on_a_sphere-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>Toggle Curves on a sphere subsection</span> </button> <ul id="toc-Curves_on_a_sphere-sublist" class="vector-toc-list"> <li id="toc-Circles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circles"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Circles</span> </div> </a> <ul id="toc-Circles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Loxodrome" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Loxodrome"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Loxodrome</span> </div> </a> <ul id="toc-Loxodrome-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Clelia_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clelia_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Clelia curves</span> </div> </a> <ul id="toc-Clelia_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spherical_conics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spherical_conics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Spherical conics</span> </div> </a> <ul id="toc-Spherical_conics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Intersection_of_a_sphere_with_a_more_general_surface" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intersection_of_a_sphere_with_a_more_general_surface"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Intersection of a sphere with a more general surface</span> </div> </a> <ul id="toc-Intersection_of_a_sphere_with_a_more_general_surface-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-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>Toggle Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Ellipsoids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ellipsoids"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Ellipsoids</span> </div> </a> <ul id="toc-Ellipsoids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimensionality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimensionality"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Dimensionality</span> </div> </a> <ul id="toc-Dimensionality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metric_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metric_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Metric spaces</span> </div> </a> <ul id="toc-Metric_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gallery" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Gallery"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Gallery</span> </div> </a> <ul id="toc-Gallery-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Regions"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Regions</span> </div> </a> <ul id="toc-Regions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_and_references" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes_and_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes and references</span> </div> </a> <button aria-controls="toc-Notes_and_references-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>Toggle Notes and references subsection</span> </button> <ul id="toc-Notes_and_references-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <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="Toggle the table of contents" > <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">Toggle the table of contents</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">Sphere</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="Go to an article in another language. Available in 106 languages" > <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 languages</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 – Amharic" lang="am" hreflang="am" data-title="ሉል" data-language-autonym="አማርኛ" data-language-local-name="Amharic" 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 – Arabic" lang="ar" hreflang="ar" data-title="كرة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Esfera" title="Esfera – Asturian" lang="ast" hreflang="ast" data-title="Esfera" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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-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 – Bangla" lang="bn" hreflang="bn" data-title="গোলক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</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-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 – Belarusian" lang="be" hreflang="be" data-title="Сфера" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Bulgarian" lang="bg" hreflang="bg" data-title="Сфера" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Sfera" title="Sfera – Bosnian" lang="bs" hreflang="bs" data-title="Sfera" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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 – Catalan" lang="ca" hreflang="ca" data-title="Esfera" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</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-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Sf%C3%A9ra_(matematika)" title="Sféra (matematika) – Czech" lang="cs" hreflang="cs" data-title="Sféra (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</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-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 – Danish" lang="da" hreflang="da" data-title="Kugle" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</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 – Moroccan Arabic" lang="ary" hreflang="ary" data-title="كورة" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kugel" title="Kugel – German" lang="de" hreflang="de" data-title="Kugel" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</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 – Estonian" lang="et" hreflang="et" data-title="Sfäär" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</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 – Greek" lang="el" hreflang="el" data-title="Σφαίρα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Esfera" title="Esfera – Spanish" lang="es" hreflang="es" data-title="Esfera" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</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-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) – Persian" lang="fa" hreflang="fa" data-title="کره (هندسه)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sph%C3%A8re" title="Sphère – French" lang="fr" hreflang="fr" data-title="Sphère" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sf%C3%A9ar" title="Sféar – Irish" lang="ga" hreflang="ga" data-title="Sféar" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Cruinne" title="Cruinne – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Cruinne" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" 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 – Galician" lang="gl" hreflang="gl" data-title="Esfera" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</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-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 – Gujarati" lang="gu" hreflang="gu" data-title="ગોળો" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</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) – Korean" lang="ko" hreflang="ko" data-title="구 (기하학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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 – Croatian" lang="hr" hreflang="hr" data-title="Sfera" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bola_(geometri)" title="Bola (geometri) – Indonesian" lang="id" hreflang="id" data-title="Bola (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/K%C3%BAla" title="Kúla – Icelandic" lang="is" hreflang="is" data-title="Kúla" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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 – Italian" lang="it" hreflang="it" data-title="Sfera" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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) – Hebrew" lang="he" hreflang="he" data-title="ספירה (גאומטריה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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-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 – Georgian" lang="ka" hreflang="ka" data-title="სფერო" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</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-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-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Esf%C3%A8" title="Esfè – Haitian Creole" lang="ht" hreflang="ht" data-title="Esfè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</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-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-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Sf%C4%93ra" title="Sfēra – Latvian" lang="lv" hreflang="lv" data-title="Sfēra" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sfera" title="Sfera – Lithuanian" lang="lt" hreflang="lt" data-title="Sfera" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/G%C3%B6mb" title="Gömb – Hungarian" lang="hu" hreflang="hu" data-title="Gömb" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Сфера" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</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) – Malagasy" lang="mg" hreflang="mg" data-title="Bola (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</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-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sfera" title="Sfera – Malay" lang="ms" hreflang="ms" data-title="Sfera" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</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-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 – Mongolian" lang="mn" hreflang="mn" data-title="Бөмбөлөг" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</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 – Burmese" lang="my" hreflang="my" data-title="စက်လုံး" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Vuravura_(Geometry)" title="Vuravura (Geometry) – Fijian" lang="fj" hreflang="fj" data-title="Vuravura (Geometry)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Sfeer_(wiskunde)" title="Sfeer (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Sfeer (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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 – Japanese" lang="ja" hreflang="ja" data-title="球面" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kuugel" title="Kuugel – Northern Frisian" lang="frr" hreflang="frr" data-title="Kuugel" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kule" title="Kule – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kule" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kule" title="Kule – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kule" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Esf%C3%A8ra" title="Esfèra – Occitan" lang="oc" hreflang="oc" data-title="Esfèra" data-language-autonym="Occitan" data-language-local-name="Occitan" 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-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-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-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-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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sfera" title="Sfera – Polish" lang="pl" hreflang="pl" data-title="Sfera" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Esfera" title="Esfera – Portuguese" lang="pt" hreflang="pt" data-title="Esfera" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sfer%C4%83" title="Sferă – Romanian" lang="ro" hreflang="ro" data-title="Sferă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – Russian" lang="ru" hreflang="ru" data-title="Сфера" data-language-autonym="Русский" data-language-local-name="Russian" 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 – Yakut" lang="sah" hreflang="sah" data-title="Сфера" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sfera" title="Sfera – Albanian" lang="sq" hreflang="sq" data-title="Sfera" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Sfera" title="Sfera – Sicilian" lang="scn" hreflang="scn" data-title="Sfera" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</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 – Slovenian" lang="sl" hreflang="sl" data-title="Sfera" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Kubad" title="Kubad – Somali" lang="so" hreflang="so" data-title="Kubad" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</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 – Central Kurdish" lang="ckb" hreflang="ckb" data-title="گۆ" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</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 – Serbian" lang="sr" hreflang="sr" data-title="Сфера" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Sfera" title="Sfera – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Sfera" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Buleudan" title="Buleudan – Sundanese" lang="su" hreflang="su" data-title="Buleudan" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pallo_(geometria)" title="Pallo (geometria) – Finnish" lang="fi" hreflang="fi" data-title="Pallo (geometria)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sf%C3%A4r" title="Sfär – Swedish" lang="sv" hreflang="sv" data-title="Sfär" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</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-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-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-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-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-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-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCre" title="Küre – Turkish" lang="tr" hreflang="tr" data-title="Küre" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</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 – Ukrainian" lang="uk" hreflang="uk" data-title="Сфера" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</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 – Vietnamese" lang="vi" hreflang="vi" data-title="Mặt cầu" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</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-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Espira" title="Espira – Waray" lang="war" hreflang="war" data-title="Espira" data-language-autonym="Winaray" data-language-local-name="Waray" 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" lang="wuu" hreflang="wuu" data-title="球面" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</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 – Cantonese" lang="yue" hreflang="yue" data-title="球體" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</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 – Chinese" lang="zh" 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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">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Set of points equidistant from a center</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.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}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the concept in three-dimensional geometry. For other uses, see <a href="/wiki/Sphere_(disambiguation)" class="mw-disambig" title="Sphere (disambiguation)">Sphere (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">Ball (mathematics)</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.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%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">Sphere</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File: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">A <a href="/wiki/3D_projection#Perspective_projection" title="3D projection">perspective projection</a> of a sphere</div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Smooth_surface" class="mw-redirect" title="Smooth surface">Smooth surface</a><br /><a href="/wiki/Algebraic_surface" title="Algebraic surface">Algebraic surface</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler char.</a></th><td class="infobox-data">2</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">Symmetry group</a></th><td class="infobox-data"><a href="/wiki/Orthogonal_group" title="Orthogonal group"><span class="texhtml">O(3)</span></a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Surface_area" title="Surface area">Surface area</a></th><td class="infobox-data"><span class="texhtml">4πr<sup>2</sup></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="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">4</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>πr<sup>3</sup></span></td></tr></tbody></table> <p>A <b>sphere</b> (from <a href="/wiki/Ancient_Greek" title="Ancient Greek">Greek</a> <span lang="grc"><a href="https://en.wiktionary.org/wiki/%CF%83%CF%86%CE%B1%E1%BF%96%CF%81%CE%B1#Ancient_Greek" class="extiw" title="wikt:σφαῖρα">σφαῖρα</a></span>, <span title="Ancient Greek transliteration" lang="grc-Latn"><i>sphaîra</i></span>)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a <a href="/wiki/Geometry" title="Geometry">geometrical</a> object that is a <a href="/wiki/Solid_geometry" title="Solid geometry">three-dimensional</a> analogue to a two-dimensional <a href="/wiki/Circle" title="Circle">circle</a>. Formally, a sphere is the <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">set of points</a> that are all at the same distance <span class="texhtml"><i>r</i></span> from a given point in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>.<sup id="cite_ref-Albert54_2-0" class="reference"><a href="#cite_note-Albert54-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> That given point is the <a href="/wiki/Center_(geometry)" class="mw-redirect" title="Center (geometry)">center</a> of the sphere, and <span class="texhtml"><i>r</i></span> is the sphere's <i><a href="/wiki/Radius" title="Radius">radius</a></i>. The earliest known mentions of spheres appear in the work of the <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greek mathematicians</a>. </p><p>The sphere is a fundamental surface in many fields of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. Spheres and nearly-spherical shapes also appear in nature and industry. <a href="/wiki/Bubble_(physics)" title="Bubble (physics)">Bubbles</a> such as <a href="/wiki/Soap_bubble" title="Soap bubble">soap bubbles</a> take a spherical shape in equilibrium. The Earth is <a href="/wiki/Spherical_Earth" title="Spherical Earth">often approximated as a sphere</a> in <a href="/wiki/Geography" title="Geography">geography</a>, and the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> is an important concept in <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>. Manufactured items including <a href="/wiki/Pressure_vessels" class="mw-redirect" title="Pressure vessels">pressure vessels</a> and most <a href="/wiki/Curved_mirror" title="Curved mirror">curved mirrors</a> and <a href="/wiki/Lens" title="Lens">lenses</a> are based on spheres. Spheres <a href="/wiki/Rolling" title="Rolling">roll</a> smoothly in any direction, so most <a href="/wiki/Ball" title="Ball">balls</a> used in sports and toys are spherical, as are <a href="/wiki/Ball_bearings" class="mw-redirect" title="Ball bearings">ball bearings</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_terminology">Basic terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=1" title="Edit section: Basic terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_and_Ball.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/240px-Sphere_and_Ball.png" decoding="async" width="240" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/360px-Sphere_and_Ball.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/480px-Sphere_and_Ball.png 2x" data-file-width="1548" data-file-height="1536" /></a><figcaption>Two orthogonal radii of a sphere</figcaption></figure> <p>As mentioned earlier <span class="texhtml"><i>r</i></span> is the sphere's radius; any line from the center to a point on the sphere is also called a radius. 'Radius' is used in two senses: as a line segment and also as its length.<sup id="cite_ref-EB_3-0" class="reference"><a href="#cite_note-EB-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>If a radius is extended through the center to the opposite side of the sphere, it creates a <a href="/wiki/Diameter" title="Diameter">diameter</a>. Like the radius, the length of a diameter is also called the diameter, and denoted <span class="texhtml"><i>d</i></span>. Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius, <span class="texhtml"><i>d</i> = 2<i>r</i></span>. Two points on the sphere connected by a diameter are <a href="/wiki/Antipodal_point" title="Antipodal point">antipodal points</a> of each other.<sup id="cite_ref-EB_3-1" class="reference"><a href="#cite_note-EB-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a> is a sphere with unit radius (<span class="texhtml"><i>r</i> = 1</span>). For convenience, spheres are often taken to have their center at the origin of the <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a>, and spheres in this article have their center at the origin unless a center is mentioned. </p><p><span class="anchor" id="hemisphere"></span> A <i><a href="/wiki/Great_circle" title="Great circle">great circle</a></i> on the sphere has the same center and radius as the sphere, and divides it into two equal <i><b>hemispheres</b></i>. </p><p>Although the <a href="/wiki/Figure_of_Earth" class="mw-redirect" title="Figure of Earth">figure of Earth</a> is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. A particular line passing through its center defines an <i><a href="/wiki/Axis_of_symmetry" class="mw-redirect" title="Axis of symmetry">axis</a></i> (as in Earth's <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis of rotation</a>). The sphere-axis intersection defines two antipodal <i>poles</i> (<i>north pole</i> and <i>south pole</i>). The great circle equidistant to the poles is called the <i><a href="/wiki/Equator" title="Equator">equator</a></i>. Great circles through the poles are called lines of <a href="/wiki/Longitude" title="Longitude">longitude</a> or <a href="/wiki/Meridian_(geography)" title="Meridian (geography)"><i>meridians</i></a>. Small circles on the sphere that are parallel to the equator are <a href="/wiki/Circle_of_latitude" title="Circle of latitude">circles of latitude</a> (or <i>parallels</i>). In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding.<sup id="cite_ref-EB_3-2" class="reference"><a href="#cite_note-EB-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Mathematicians consider a sphere to be a <a href="/wiki/Two-dimensional" class="mw-redirect" title="Two-dimensional">two-dimensional</a> <a href="/wiki/Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> <a href="/wiki/Embedding" title="Embedding">embedded</a> in three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. They draw a distinction between a <i>sphere</i> and a <i><a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">ball</a></i>, which is a <a href="/wiki/Solid_figure" class="mw-redirect" title="Solid figure">solid figure</a>, a three-dimensional <a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">manifold with boundary</a> that includes the volume contained by the sphere. An <i>open ball</i> excludes the sphere itself, while a <i>closed ball</i> includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> of a (closed or open) ball. The distinction between <i>ball</i> and <i>sphere</i> has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "<a href="/wiki/Circle" title="Circle">circle</a>" and "<a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a>" in the <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> is similar. </p><p>Small spheres or balls are sometimes called <i>spherules</i> (e.g., in <a href="/wiki/Martian_spherules" title="Martian spherules">Martian spherules</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Equations">Equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=2" title="Edit section: Equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, a sphere with center <span class="texhtml">(<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>, <i>z</i><sub>0</sub>)</span> and radius <span class="texhtml mvar" style="font-style:italic;">r</span> is the <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of all points <span class="texhtml">(<i>x</i>, <i>y</i>, <i>z</i>)</span> such that </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>Since it can be expressed as a quadratic polynomial, a sphere is a <a href="/wiki/Quadric_surface" class="mw-redirect" title="Quadric surface">quadric surface</a>, a type of <a href="/wiki/Algebraic_surface" title="Algebraic surface">algebraic surface</a>.<sup id="cite_ref-EB_3-3" class="reference"><a href="#cite_note-EB-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Let <span class="texhtml mvar" style="font-style:italic;">a, b, c, d, e</span> be real numbers with <span class="texhtml"><i>a</i> ≠ 0</span> and put </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>Then the equation </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>has no real points as solutions if <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> and is called the equation of an <b>imaginary sphere</b>. If <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>, the only solution of <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> is the point <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> and the equation is said to be the equation of a <b>point sphere</b>. Finally, in the case <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> is an equation of a sphere whose center is <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> and whose radius is <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_2-1" class="reference"><a href="#cite_note-Albert54-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="texhtml mvar" style="font-style:italic;">a</span> in the above equation is zero then <span class="texhtml"><i>f</i>(<i>x</i>, <i>y</i>, <i>z</i>) = 0</span> is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a>.<sup id="cite_ref-Woods266_4-0" class="reference"><a href="#cite_note-Woods266-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Parametric">Parametric</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=3" title="Edit section: Parametric"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Parametric_equation" title="Parametric equation">parametric equation</a> for the sphere with 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="{\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> and center <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> can be parameterized using <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</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 {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \\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> </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 \\z&=z_{0}+r\cos \theta \,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6eff6f3bffe842c32b34301992d7f2f765469e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:22.223ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \\z&=z_{0}+r\cos \theta \,\end{aligned}}}"></span><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The symbols used here are the same as those used in <a href="/wiki/Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a>. <span class="texhtml mvar" style="font-style:italic;">r</span> is constant, while <span class="texhtml mvar" style="font-style:italic;">θ</span> varies from 0 to <span class="texhtml mvar" style="font-style:italic;">π</span> and <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> varies from 0 to 2<span class="texhtml mvar" style="font-style:italic;">π</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Enclosed_volume">Enclosed volume<span class="anchor" id="Volume"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=5" title="Edit section: Enclosed volume"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_and_circumscribed_cylinder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Sphere_and_circumscribed_cylinder.svg/240px-Sphere_and_circumscribed_cylinder.svg.png" decoding="async" width="240" height="242" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Sphere_and_circumscribed_cylinder.svg/360px-Sphere_and_circumscribed_cylinder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Sphere_and_circumscribed_cylinder.svg/480px-Sphere_and_circumscribed_cylinder.svg.png 2x" data-file-width="963" data-file-height="973" /></a><figcaption>Sphere and circumscribed cylinder</figcaption></figure> <p>In three dimensions, the <a href="/wiki/Volume" title="Volume">volume</a> inside a sphere (that is, the volume of a <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">ball</a>, but classically referred to as the volume of a sphere) is </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}={\frac {\pi }{6}}\ d^{3}\approx 0.5236\cdot d^{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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mtext> </mtext> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>≈</mo> <mn>0.5236</mn> <mo>⋅</mo> <msup> <mi>d</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}={\frac {\pi }{6}}\ d^{3}\approx 0.5236\cdot d^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42de7dbcc7c51105e0f10e71f53c61987454ced6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.948ex; height:5.176ex;" alt="{\displaystyle V={\frac {4}{3}}\pi r^{3}={\frac {\pi }{6}}\ d^{3}\approx 0.5236\cdot d^{3}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">r</span> is the radius and <span class="texhtml mvar" style="font-style:italic;">d</span> is the diameter of the sphere. <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the <a href="/wiki/Circumscribe" class="mw-redirect" title="Circumscribe">circumscribed</a> <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a> of that sphere (having the height and diameter equal to the diameter of the sphere).<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> This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying <a href="/wiki/Cavalieri%27s_principle" title="Cavalieri's principle">Cavalieri's principle</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This formula can also be derived using <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a> (i.e., <a href="/wiki/Disk_integration" class="mw-redirect" title="Disk integration">disk integration</a>) to sum the volumes of an <a href="/wiki/Infinite_number" class="mw-redirect" title="Infinite number">infinite number</a> of <a href="/wiki/Circle#Properties" title="Circle">circular</a> disks of infinitesimally small thickness stacked side by side and centered along the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis from <span class="texhtml"><i>x</i> = −<i>r</i></span> to <span class="texhtml"><i>x</i> = <i>r</i></span>, assuming the sphere of radius <span class="texhtml mvar" style="font-style:italic;">r</span> is centered at the origin. <style data-mw-deduplicate="TemplateStyles:r1256386598">.mw-parser-output .cot-header-mainspace{background:#F0F2F5;color:inherit}.mw-parser-output .cot-header-other{background:#CCFFCC;color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-night .mw-parser-output .cot-header-other{background:#003500;color:inherit}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-os .mw-parser-output .cot-header-other{background:#003500;color:inherit}}</style> </p> <div style="margin-left:0"> <table class="mw-collapsible mw-archivedtalk mw-collapsed" style="color:inherit; background: transparent; text-align: left; border: 1px solid Silver; margin: 0.2em auto auto; width:100%; clear: both; padding: 1px;"> <tbody><tr> <th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof of sphere volume, using calculus</div> </th></tr> <tr> <td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);"> <p>At any given <span class="texhtml mvar" style="font-style:italic;">x</span>, the incremental volume (<span class="texhtml mvar" style="font-style:italic;">δV</span>) equals the product of the cross-sectional <a href="/wiki/Area_of_a_disc#Onion_proof" class="mw-redirect" title="Area of a disc">area of the disk</a> at <span class="texhtml mvar" style="font-style:italic;">x</span> and its thickness (<span class="texhtml mvar" style="font-style:italic;">δx</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>The total volume is the summation of all incremental volumes: </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>In the limit as <span class="texhtml mvar" style="font-style:italic;">δx</span> approaches zero,<sup id="cite_ref-delta_8-0" class="reference"><a href="#cite_note-delta-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> this equation becomes: </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>At any given <span class="texhtml mvar" style="font-style:italic;">x</span>, a right-angled triangle connects <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml mvar" style="font-style:italic;">r</span> to the origin; hence, applying the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> yields: </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>Using this substitution gives </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>which can be evaluated to give the result </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>An alternative formula is found using <a href="/wiki/Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a>, with <a href="/wiki/Volume_element" title="Volume element">volume element</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 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>so </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> </td></tr></tbody></table></div> <p>For most practical purposes, the volume inside a sphere <a href="/wiki/Inscribed_figure" title="Inscribed figure">inscribed</a> in a cube can be approximated as 52.4% of the volume of the cube, since <span class="texhtml"><i>V</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> <i>d</i><sup>3</sup></span>, where <span class="texhtml mvar" style="font-style:italic;">d</span> is the diameter of the sphere and also the length of a side of the cube and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1<span class="nowrap"> </span>m, or about 0.524 m<sup>3</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Surface_area">Surface area<span class="anchor" id="Area"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=6" title="Edit section: Surface area"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Surface_area" title="Surface area">surface area</a> of a sphere of radius <span class="texhtml mvar" style="font-style:italic;">r</span> is: </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 A=4\pi r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</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 A=4\pi r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95625828c519b36791d56af65d21b8448472650d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.086ex; height:2.676ex;" alt="{\displaystyle A=4\pi r^{2}.}"></span></dd></dl> <p><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> first derived this formula<sup id="cite_ref-MathWorld_Sphere_9-0" class="reference"><a href="#cite_note-MathWorld_Sphere-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> from the fact that the projection to the lateral surface of a <a href="/wiki/Circumscribe" class="mw-redirect" title="Circumscribe">circumscribed</a> cylinder is area-preserving.<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> Another approach to obtaining the formula comes from the fact that it equals the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the formula for the volume with respect to <span class="texhtml mvar" style="font-style:italic;">r</span> because the total volume inside a sphere of radius <span class="texhtml mvar" style="font-style:italic;">r</span> can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius <span class="texhtml mvar" style="font-style:italic;">r</span>. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius <span class="texhtml mvar" style="font-style:italic;">r</span> is simply the product of the surface area at radius <span class="texhtml mvar" style="font-style:italic;">r</span> and the infinitesimal thickness. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1256386598"> <div style="margin-left:0"> <table class="mw-collapsible mw-archivedtalk mw-collapsed" style="color:inherit; background: transparent; text-align: left; border: 1px solid Silver; margin: 0.2em auto auto; width:100%; clear: both; padding: 1px;"> <tbody><tr> <th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof of surface area, using calculus</div> </th></tr> <tr> <td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);"> <p>At any given radius <span class="texhtml mvar" style="font-style:italic;">r</span>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> the incremental volume (<span class="texhtml mvar" style="font-style:italic;">δV</span>) equals the product of the surface area at radius <span class="texhtml mvar" style="font-style:italic;">r</span> (<span class="texhtml"><i>A</i>(<i>r</i>)</span>) and the thickness of a shell (<span class="texhtml mvar" style="font-style:italic;">δr</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 A(r)\cdot \delta r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>V</mi> <mo>≈</mo> <mi>A</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 A(r)\cdot \delta r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd6b20b042b7b8356514cafe4bfe9323f73970e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.959ex; height:2.843ex;" alt="{\displaystyle \delta V\approx A(r)\cdot \delta r.}"></span></dd></dl> <p>The total volume is the summation of all shell volumes: </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 A(r)\cdot \delta r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>≈</mo> <mo>∑</mo> <mi>A</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 A(r)\cdot \delta r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff662cf129a7b7ae53bb28460dadef0715b60c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.652ex; height:3.843ex;" alt="{\displaystyle V\approx \sum A(r)\cdot \delta r.}"></span></dd></dl> <p>In the limit as <span class="texhtml mvar" style="font-style:italic;">δr</span> approaches zero<sup id="cite_ref-delta_8-1" class="reference"><a href="#cite_note-delta-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> this equation becomes: </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}A(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>A</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}A(r)\,dr.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde5072ef2871126b29c8eab1cc7b83ec2d365ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.61ex; height:5.843ex;" alt="{\displaystyle V=\int _{0}^{r}A(r)\,dr.}"></span></dd></dl> <p>Substitute <span class="texhtml mvar" style="font-style:italic;">V</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 {\frac {4}{3}}\pi r^{3}=\int _{0}^{r}A(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>A</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}A(r)\,dr.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/521cb0af0bc79aaa0a4e3c6d6f02d440057fb894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.256ex; height:5.843ex;" alt="{\displaystyle {\frac {4}{3}}\pi r^{3}=\int _{0}^{r}A(r)\,dr.}"></span></dd></dl> <p>Differentiating both sides of this equation with respect to <span class="texhtml mvar" style="font-style:italic;">r</span> yields <span class="texhtml mvar" style="font-style:italic;">A</span> as a function of <span class="texhtml mvar" style="font-style:italic;">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}=A(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>A</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}=A(r).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25b8b4e3cc58f767579a25d212571f323e2f740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.944ex; height:3.176ex;" alt="{\displaystyle 4\pi r^{2}=A(r).}"></span></dd></dl> <p>This is generally abbreviated as: </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 A=4\pi r^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</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 A=4\pi r^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1cbdbb8a22c7db3b3f01dad4ea19f8dfcd502b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.086ex; height:3.009ex;" alt="{\displaystyle A=4\pi r^{2},}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">r</span> is now considered to be the fixed radius of the sphere. </p><p>Alternatively, the <a href="/wiki/Area_element" class="mw-redirect" title="Area element">area element</a> on the sphere is given in <a href="/wiki/Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a> by <span class="texhtml"><i>dA</i> = <i>r</i><sup>2</sup> sin <i>θ dθ dφ</i></span>. The total area can thus be obtained by <a href="/wiki/Integral" title="Integral">integration</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 A=\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>A</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 A=\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/128b56cc8737351056a8e5fd4dfc2dd163f58bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.697ex; height:6.176ex;" alt="{\displaystyle A=\int _{0}^{2\pi }\int _{0}^{\pi }r^{2}\sin \theta \,d\theta \,d\varphi =4\pi r^{2}.}"></span></dd></dl> </td></tr></tbody></table></div> <p>The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the <a href="/wiki/Surface_tension" title="Surface tension">surface tension</a> locally minimizes surface area. </p><p>The surface area relative to the mass of a ball is called the <a href="/wiki/Specific_surface_area" title="Specific surface area">specific surface area</a> and can be expressed from the above stated equations as </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 {SSA} ={\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">S</mi> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">A</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SSA} ={\frac {A}{V\rho }}={\frac {3}{r\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a992bf548d15c31b027fcb1b6a7b6dada6587acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.437ex; height:5.843ex;" alt="{\displaystyle \mathrm {SSA} ={\frac {A}{V\rho }}={\frac {3}{r\rho }}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">ρ</span> is the <a href="/wiki/Density" title="Density">density</a> (the ratio of mass to volume). </p> <div class="mw-heading mw-heading3"><h3 id="Other_geometric_properties">Other geometric properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=7" title="Edit section: Other geometric properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A sphere can be constructed as the surface formed by rotating a <a href="/wiki/Circle" title="Circle">circle</a> one half revolution about any of its <a href="/wiki/Diameter" title="Diameter">diameters</a>; this is very similar to the traditional definition of a sphere as given in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>. Since a circle is a special type of <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>, a sphere is a special type of <a href="/wiki/Ellipsoid_of_revolution" class="mw-redirect" title="Ellipsoid of revolution">ellipsoid of revolution</a>. Replacing the circle with an ellipse rotated about its <a href="/wiki/Major_axis" class="mw-redirect" title="Major axis">major axis</a>, the shape becomes a prolate <a href="/wiki/Spheroid" title="Spheroid">spheroid</a>; rotated about the minor axis, an oblate spheroid.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>A sphere is uniquely determined by four points that are not <a href="/wiki/Coplanar" class="mw-redirect" title="Coplanar">coplanar</a>. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This property is analogous to the property that three <a href="/wiki/Collinear" class="mw-redirect" title="Collinear">non-collinear</a> points determine a unique circle in a plane. </p><p>Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle. </p><p>By examining the <a href="/wiki/Circle_of_a_sphere#Sphere-sphere_intersection" class="mw-redirect" title="Circle of a sphere">common solutions of the equations of two spheres</a>, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the <b>radical plane</b> of the intersecting spheres.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).<sup id="cite_ref-Woods267_16-0" class="reference"><a href="#cite_note-Woods267-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>The angle between two spheres at a real point of intersection is the <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> They intersect at right angles (are <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a>) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.<sup id="cite_ref-Woods266_4-1" class="reference"><a href="#cite_note-Woods266-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Pencil_of_spheres">Pencil of spheres</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=8" title="Edit section: Pencil of spheres"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Pencil_(mathematics)#Pencil_of_spheres" class="mw-redirect" title="Pencil (mathematics)">Pencil (mathematics) § Pencil of spheres</a></div> <p>If <span class="texhtml"><i>f</i>(<i>x</i>, <i>y</i>, <i>z</i>) = 0</span> and <span class="texhtml"><i>g</i>(<i>x</i>, <i>y</i>, <i>z</i>) = 0</span> are the equations of two distinct spheres then </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>is also the equation of a sphere for arbitrary values of the parameters <span class="texhtml mvar" style="font-style:italic;">s</span> and <span class="texhtml mvar" style="font-style:italic;">t</span>. The set of all spheres satisfying this equation is called a <b>pencil of spheres</b> determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.<sup id="cite_ref-Woods266_4-2" class="reference"><a href="#cite_note-Woods266-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Properties_of_the_sphere">Properties of the sphere</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=9" title="Edit section: Properties of the sphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_section.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Sphere_section.png/220px-Sphere_section.png" decoding="async" width="220" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Sphere_section.png/330px-Sphere_section.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/2a/Sphere_section.png 2x" data-file-width="384" data-file-height="410" /></a><figcaption>A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius: the radius of the sphere. This means that every point on the sphere will be an umbilical point.</figcaption></figure> <p>In their book <i>Geometry and the Imagination</i>, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="/wiki/Stephan_Cohn-Vossen" class="mw-redirect" title="Stephan Cohn-Vossen">Stephan Cohn-Vossen</a> describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Several properties hold for the <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a>, which can be thought of as a sphere with infinite radius. These properties are: </p> <ol><li><i>The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.</i> <dl><dd>The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar <a href="/wiki/Circle#Circle_of_Apollonius" title="Circle">result</a> of <a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a> for the <a href="/wiki/Circle" title="Circle">circle</a>. This second part also holds for the <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a>.</dd></dl></li> <li><i>The contours and plane sections of the sphere are circles.</i> <dl><dd>This property defines the sphere uniquely.</dd></dl></li> <li><i>The sphere has constant width and constant girth.</i> <dl><dd>The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the <a href="/wiki/Meissner_body" class="mw-redirect" title="Meissner body">Meissner body</a>. The girth of a surface is the <a href="/wiki/Circumference" title="Circumference">circumference</a> of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.</dd></dl></li> <li><i>All points of a sphere are <a href="/wiki/Umbilic" class="mw-redirect" title="Umbilic">umbilics</a>.</i> <dl><dd>At any point on a surface a <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal direction</a> is at right angles to the surface because on the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a <i>normal section,</i> and the curvature of this curve is the <i>normal curvature</i>. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the <a href="/wiki/Principal_curvature" title="Principal curvature">principal curvatures</a>. Any closed surface will have at least four points called <i><a href="/wiki/Umbilical_point" title="Umbilical point">umbilical points</a></i>. At an umbilic all the sectional curvatures are equal; in particular the <a href="/wiki/Principal_curvature" title="Principal curvature">principal curvatures</a> are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.</dd> <dd>For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.</dd></dl></li> <li><i>The sphere does not have a surface of centers.</i> <dl><dd>For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures are called the <i>focal points</i>, and the set of all such centers forms the <a href="/wiki/Focal_surface" title="Focal surface">focal surface</a>.</dd> <dd>For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special:</dd> <dd>* For <a href="/wiki/Channel_surface" title="Channel surface">channel surfaces</a> one sheet forms a curve and the other sheet is a surface</dd> <dd>* For <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cones</a>, cylinders, <a href="/wiki/Torus" title="Torus">tori</a> and <a href="/wiki/Dupin_cyclide" title="Dupin cyclide">cyclides</a> both sheets form curves.</dd> <dd>* For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere.</dd></dl></li> <li><i>All geodesics of the sphere are closed curves.</i> <dl><dd><a href="/wiki/Geodesics" class="mw-redirect" title="Geodesics">Geodesics</a> are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property.</dd></dl></li> <li><i>Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.</i> <dl><dd>It follows from <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a>. These properties define the sphere uniquely and can be seen in <a href="/wiki/Soap_bubble" title="Soap bubble">soap bubbles</a>: a soap bubble will enclose a fixed volume, and <a href="/wiki/Surface_tension" title="Surface tension">surface tension</a> minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). It can also be seen in planets and stars where gravity minimizes surface area for large celestial bodies.</dd></dl></li> <li><i>The sphere has the smallest total mean curvature among all convex solids with a given surface area.</i> <dl><dd>The <a href="/wiki/Mean_curvature" title="Mean curvature">mean curvature</a> is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.</dd></dl></li> <li><i>The sphere has constant mean curvature.</i> <dl><dd>The sphere is the only <a href="/wiki/Embedding" title="Embedding">embedded</a> surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surfaces</a> have constant mean curvature.</dd></dl></li> <li><i>The sphere has constant positive Gaussian curvature.</i> <dl><dd><a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> is the product of the two principal curvatures. It is an intrinsic property that can be determined by measuring length and angles and is independent of how the surface is <a href="/wiki/Embedding" title="Embedding">embedded</a> in space. Hence, bending a surface will not alter the Gaussian curvature, and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries, and the sphere is the only surface that lacks a boundary with constant, positive Gaussian curvature. The <a href="/wiki/Pseudosphere" title="Pseudosphere">pseudosphere</a> is an example of a surface with constant negative Gaussian curvature.</dd></dl></li> <li><i>The sphere is transformed into itself by a three-parameter family of rigid motions.</i> <dl><dd>Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a>). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the <a href="/wiki/Rotation_group_SO(3)" class="mw-redirect" title="Rotation group SO(3)">rotation group SO(3)</a>. The plane is the only other surface with a three-parameter family of transformations (translations along the <span class="texhtml mvar" style="font-style:italic;">x</span>- and <span class="texhtml mvar" style="font-style:italic;">y</span>-axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the <a href="/wiki/Surface_of_revolution" title="Surface of revolution">surfaces of revolution</a> and <a href="/wiki/Helicoid" title="Helicoid">helicoids</a> are the only surfaces with a one-parameter family.</dd></dl></li></ol> <div class="mw-heading mw-heading2"><h2 id="Treatment_by_area_of_mathematics">Treatment by area of mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=10" title="Edit section: Treatment by area of mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Spherical_geometry">Spherical geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=11" title="Edit section: Spherical geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical geometry</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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/Great_circle" title="Great circle">Great circle</a> on a sphere</figcaption></figure> <p>The basic elements of <a href="/wiki/Euclidean_plane_geometry" class="mw-redirect" title="Euclidean plane geometry">Euclidean plane geometry</a> are <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> and <a href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">lines</a>. On the sphere, points are defined in the usual sense. The analogue of the "line" is the <a href="/wiki/Geodesic" title="Geodesic">geodesic</a>, which is a <a href="/wiki/Great_circle" title="Great circle">great circle</a>; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by <a href="/wiki/Arc_length" title="Arc length">arc length</a> shows that the shortest path between two points lying on the sphere is the shorter segment of the <a href="/wiki/Great_circle" title="Great circle">great circle</a> that includes the points. </p><p>Many theorems from <a href="/wiki/Classical_geometry" class="mw-redirect" title="Classical geometry">classical geometry</a> hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's <a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulates</a>, including the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. In <a href="/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a>, <a href="/wiki/Angle" title="Angle">angles</a> are defined between great circles. Spherical trigonometry differs from ordinary <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a> in many respects. For example, the sum of the interior angles of a <a href="/wiki/Spherical_triangle" class="mw-redirect" title="Spherical triangle">spherical triangle</a> always exceeds 180 degrees. Also, any two <a href="/wiki/Similar_triangles" class="mw-redirect" title="Similar triangles">similar</a> spherical triangles are congruent. </p><p>Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e., the diameter) are called <a href="/wiki/Antipodal_point" title="Antipodal point"><i>antipodal points</i></a> – on the sphere, the distance between them is exactly half the length of the circumference.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> Any other (i.e., not antipodal) pair of distinct points on a sphere </p> <ul><li>lie on a unique great circle,</li> <li>segment it into one minor (i.e., shorter) and one major (i.e., longer) <a href="/wiki/Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">arc</a>, and</li> <li>have the minor arc's length be the <i>shortest distance</i> between them on the sphere.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup></li></ul> <p>Spherical geometry is a form of <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a>, which together with <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> makes up <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Differential_geometry">Differential geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=12" title="Edit section: Differential geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sphere is a <a href="/wiki/Smooth_surface" class="mw-redirect" title="Smooth surface">smooth surface</a> with constant <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> at each point equal to <span class="texhtml">1/<i>r</i><sup>2</sup></span>.<sup id="cite_ref-MathWorld_Sphere_9-1" class="reference"><a href="#cite_note-MathWorld_Sphere-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> As per Gauss's <a href="/wiki/Theorema_Egregium" title="Theorema Egregium">Theorema Egregium</a>, this curvature is independent of the sphere's embedding in 3-dimensional space. Also following from Gauss, a sphere cannot be mapped to a plane while maintaining both areas and angles. Therefore, any <a href="/wiki/Map_projection" title="Map projection">map projection</a> introduces some form of distortion. </p><p>A sphere of radius <span class="texhtml mvar" style="font-style:italic;">r</span> has <a href="/wiki/Area_element" class="mw-redirect" title="Area element">area element</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 dA=r^{2}\sin \theta \,d\theta \,d\varphi }"> <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> <mi>θ</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 dA=r^{2}\sin \theta \,d\theta \,d\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9129a6d7fe62e1e437d4e7d72c37d19d82ec99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.697ex; height:3.176ex;" alt="{\displaystyle dA=r^{2}\sin \theta \,d\theta \,d\varphi }"></span>. This can be found from the <a href="/wiki/Volume_element" title="Volume element">volume element</a> in <a href="/wiki/Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a> with <span class="texhtml mvar" style="font-style:italic;">r</span> held constant.<sup id="cite_ref-MathWorld_Sphere_9-2" class="reference"><a href="#cite_note-MathWorld_Sphere-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>A sphere of any radius centered at zero is an <a href="/wiki/Integral_surface" class="mw-redirect" title="Integral surface">integral surface</a> of the following <a href="/wiki/Differential_form" title="Differential form">differential form</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 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>This equation reflects that the position vector and <a href="/wiki/Tangent_plane_(geometry)" class="mw-redirect" title="Tangent plane (geometry)">tangent plane</a> at a point are always <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a> to each other. Furthermore, the outward-facing <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal vector</a> is equal to the position vector scaled by <span class="texhtml mvar" style="font-style:italic;">1/r</span>. </p><p>In <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, the <a href="/wiki/Filling_area_conjecture" title="Filling area conjecture">filling area conjecture</a> states that the hemisphere is the optimal (least area) isometric filling of the <a href="/wiki/Riemannian_circle" class="mw-redirect" title="Riemannian circle">Riemannian circle</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Topology">Topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=13" title="Edit section: Topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Remarkably, it is possible to turn an ordinary sphere inside out in a <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> with possible self-intersections but without creating any creases, in a process called <a href="/wiki/Sphere_eversion" title="Sphere eversion">sphere eversion</a>. </p><p>The antipodal quotient of the sphere is the surface called the <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a>, which can also be thought of as the <a href="/wiki/Northern_Hemisphere" title="Northern Hemisphere">Northern Hemisphere</a> with antipodal points of the equator identified. </p> <div class="mw-heading mw-heading2"><h2 id="Curves_on_a_sphere">Curves on a sphere <span class="anchor" id="Curves"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=14" title="Edit section: Curves on a sphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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>Plane section of a sphere: one circle</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kugel-zylinder-kk.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Kugel-zylinder-kk.svg/220px-Kugel-zylinder-kk.svg.png" decoding="async" width="220" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Kugel-zylinder-kk.svg/330px-Kugel-zylinder-kk.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Kugel-zylinder-kk.svg/440px-Kugel-zylinder-kk.svg.png 2x" data-file-width="441" data-file-height="288" /></a><figcaption>Coaxial intersection of a sphere and a cylinder: two circles</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Circles">Circles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=15" title="Edit section: Circles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Circle_of_a_sphere" class="mw-redirect" title="Circle of a sphere">Circle of a sphere</a></div> <p>Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles. </p><p>More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a <a href="/wiki/Surface_of_revolution" title="Surface of revolution">surface of revolution</a> whose axis contains the center of the sphere (are <i>coaxial</i>) consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty. </p> <div class="mw-heading mw-heading3"><h3 id="Loxodrome">Loxodrome</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=16" title="Edit section: Loxodrome"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Rhumb_line" title="Rhumb line">Rhumb line</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Loxodrome.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Loxodrome.png/110px-Loxodrome.png" decoding="async" width="110" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Loxodrome.png/165px-Loxodrome.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Loxodrome.png/220px-Loxodrome.png 2x" data-file-width="693" data-file-height="694" /></a><figcaption>Loxodrome</figcaption></figure> <p>In <a href="/wiki/Navigation" title="Navigation">navigation</a>, a <i>loxodrome</i> or <i>rhumb line</i> is a path whose <a href="/wiki/Bearing_(navigation)" title="Bearing (navigation)">bearing</a>, the angle between its tangent and due North, is constant. Loxodromes project to straight lines under the <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a>. Two special cases are the <a href="/wiki/Meridian_(geography)" title="Meridian (geography)">meridians</a> which are aligned directly North–South and <a href="/wiki/Circle_of_latitude" title="Circle of latitude">parallels</a> which are aligned directly East–West. For any other bearing, a loxodrome spirals infinitely around each pole. For the Earth modeled as a sphere, or for a general sphere given a <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a>, such a loxodrome is a kind of <a href="/wiki/Spherical_spiral" class="mw-redirect" title="Spherical spiral">spherical spiral</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Clelia_curves">Clelia curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=17" title="Edit section: Clelia curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cl%C3%A9lie" title="Clélie">Clélie</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kugel-spirale-1-2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/180px-Kugel-spirale-1-2.svg.png" decoding="async" width="180" height="89" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/270px-Kugel-spirale-1-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/360px-Kugel-spirale-1-2.svg.png 2x" data-file-width="759" data-file-height="377" /></a><figcaption>Clelia spiral with <span class="texhtml"><i>c</i> = 8</span></figcaption></figure> <p>Another kind of spherical spiral is the Clelia curve, for which the <a href="/wiki/Longitude" title="Longitude">longitude</a> (or azimuth) <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> and the <a href="/wiki/Colatitude" title="Colatitude">colatitude</a> (or polar angle) <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> are in a linear relationship, <span class="nowrap">⁠<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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>c</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =c\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a365cf05a4238676a4e7e910703829da0fa09d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:2.676ex;" alt="{\displaystyle \varphi =c\theta }"></span>⁠</span>. Clelia curves project to straight lines under the <a href="/wiki/Equirectangular_projection" title="Equirectangular projection">equirectangular projection</a>. <a href="/wiki/Viviani%27s_curve" title="Viviani's curve">Viviani's curve</a> (<span class="nowrap">⁠<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>⁠</span>) is a special case. Clelia curves approximate the <a href="/wiki/Ground_track" class="mw-redirect" title="Ground track">ground track</a> of satellites in <a href="/wiki/Polar_orbit" title="Polar orbit">polar orbit</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Spherical_conics">Spherical conics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=18" title="Edit section: Spherical conics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spherical_conic" title="Spherical conic">Spherical conic</a></div> <p>The analog of a <a href="/wiki/Conic_section" title="Conic section">conic section</a> on the sphere is a <a href="/wiki/Spherical_conic" title="Spherical conic">spherical conic</a>, a <a href="/wiki/Quartic_function" title="Quartic function">quartic</a> curve which can be defined in several equivalent ways. </p> <ul><li>The intersection of a sphere with a quadratic cone whose vertex is the sphere center</li> <li>The intersection of a sphere with an <a href="/wiki/Cylinder#cylindrical_surfaces" title="Cylinder">elliptic or hyperbolic cylinder</a> whose axis passes through the sphere center</li> <li>The locus of points whose sum or difference of <a href="/wiki/Great-circle_distance" title="Great-circle distance">great-circle distances</a> from a pair of <a href="/wiki/Focus_(geometry)" title="Focus (geometry)">foci</a> is a constant</li></ul> <p>Many theorems relating to planar conic sections also extend to spherical conics. </p> <div class="mw-heading mw-heading3"><h3 id="Intersection_of_a_sphere_with_a_more_general_surface">Intersection of a sphere with a more general surface</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=19" title="Edit section: Intersection of a sphere with a more general surface"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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>General intersection sphere-cylinder</figcaption></figure> <p>If a sphere is intersected by another surface, there may be more complicated spherical curves. </p> <dl><dt>Example</dt> <dd>sphere–cylinder</dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sphere%E2%80%93cylinder_intersection" title="Sphere–cylinder intersection">Sphere–cylinder intersection</a></div> <p>The intersection of the sphere with equation <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> and the cylinder with equation <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> is not just one or two circles. It is the solution of the non-linear system of equations </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>(see <a href="/wiki/Implicit_curve" title="Implicit curve">implicit curve</a> and the diagram) </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=20" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ellipsoids">Ellipsoids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=21" title="Edit section: Ellipsoids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a> is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a>. An ellipsoid bears the same relationship to the sphere that an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> does to a circle. </p> <div class="mw-heading mw-heading3"><h3 id="Dimensionality">Dimensionality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=22" title="Edit section: Dimensionality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/N-sphere" title="N-sphere">n-sphere</a></div> <p>Spheres can be generalized to spaces of any number of <a href="/wiki/Dimension" title="Dimension">dimensions</a>. For any <a href="/wiki/Natural_number" title="Natural number">natural number</a> <span class="texhtml mvar" style="font-style:italic;">n</span>, an <i><span class="texhtml mvar" style="font-style:italic;">n</span>-sphere,</i> often denoted <span class="texhtml"><i>S</i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup><i>n</i></sup></span>, is the set of points in (<span class="texhtml"><i>n</i> + 1</span>)-dimensional Euclidean space that are at a fixed distance <span class="texhtml mvar" style="font-style:italic;">r</span> from a central point of that space, where <span class="texhtml mvar" style="font-style:italic;">r</span> is, as before, a positive real number. In particular: </p> <ul><li><span class="texhtml"><i>S</i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup>0</sup></span>: a 0-sphere consists of two discrete points, <span class="texhtml">−<i>r</i></span> and <span class="texhtml"><i>r</i></span></li> <li><span class="texhtml"><i>S</i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup>1</sup></span>: a 1-sphere is a <a href="/wiki/Circle" title="Circle">circle</a> of radius <i>r</i></li> <li><span class="texhtml"><i>S</i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup>2</sup></span>: a 2-sphere is an ordinary sphere</li> <li><span class="texhtml"><i>S</i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup>3</sup></span>: a <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> is a sphere in 4-dimensional Euclidean space.</li></ul> <p>Spheres for <span class="texhtml"><i>n</i> > 2</span> are sometimes called <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">hyperspheres</a>. </p><p>The <span class="texhtml mvar" style="font-style:italic;">n</span>-sphere of unit radius centered at the origin is denoted <span class="texhtml"><i>S</i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup><i>n</i></sup></span> and is often referred to as "the" <span class="texhtml mvar" style="font-style:italic;">n</span>-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space. </p><p>In <a href="/wiki/Topology" title="Topology">topology</a>, the <span class="texhtml mvar" style="font-style:italic;">n</span>-sphere is an example of a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifold</a> without <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a>. A topological sphere need not be <a href="/wiki/Manifold#Differentiable_manifolds" title="Manifold">smooth</a>; if it is smooth, it need not be <a href="/wiki/Diffeomorphic" class="mw-redirect" title="Diffeomorphic">diffeomorphic</a> to the Euclidean sphere (an <a href="/wiki/Exotic_sphere" title="Exotic sphere">exotic sphere</a>). </p><p>The sphere is the inverse image of a one-point set under the continuous function <span class="texhtml">‖<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>‖</span>, so it is closed; <span class="texhtml"><i>S<sup>n</sup></i></span> is also bounded, so it is compact by the <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Metric_spaces">Metric spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=23" title="Edit section: Metric spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Metric_space" title="Metric space">Metric space</a></div> <p>More generally, in a <a href="/wiki/Metric_space" title="Metric space">metric space</a> <span class="texhtml">(<i>E</i>,<i>d</i>)</span>, the sphere of center <span class="texhtml mvar" style="font-style:italic;">x</span> and radius <span class="texhtml"><i>r</i> > 0</span> is the set of points <span class="texhtml mvar" style="font-style:italic;">y</span> such that <span class="texhtml"><i>d</i>(<i>x</i>,<i>y</i>) = <i>r</i></span>. </p><p>If the center is a distinguished point that is considered to be the origin of <span class="texhtml mvar" style="font-style:italic;">E</span>, as in a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">normed</a> space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a>. </p><p>Unlike a <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">ball</a>, even a large sphere may be an empty set. For example, in <span class="texhtml"><b>Z</b><sup><i>n</i></sup></span> with <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>, a sphere of radius <span class="texhtml"><i>r</i></span> is nonempty only if <span class="texhtml"><i>r</i><sup>2</sup></span> can be written as sum of <span class="texhtml"><i>n</i></span> squares of <a href="/wiki/Integer" title="Integer">integers</a>. </p><p>An <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> is a sphere in <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">taxicab geometry</a>, and a <a href="/wiki/Cube" title="Cube">cube</a> is a sphere in geometry using the <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=24" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The geometry of the sphere was studied by the Greeks. <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a></i> defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably due to <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a>. The volume and area formulas were first determined in <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a>'s <i><a href="/wiki/On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a></i> by the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a>. <a href="/wiki/Zenodorus_(mathematician)" title="Zenodorus (mathematician)">Zenodorus</a> was the first to state that, for a given surface area, the sphere is the solid of maximum volume.<sup id="cite_ref-EB_3-4" class="reference"><a href="#cite_note-EB-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it. A solution by means of the parabola and hyperbola was given by <a href="/wiki/Dionysodorus" title="Dionysodorus">Dionysodorus</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> A similar problem – to construct a segment equal in volume to a given segment, and in surface to another segment – was solved later by <a href="/wiki/Al-Quhi" class="mw-redirect" title="Al-Quhi">al-Quhi</a>.<sup id="cite_ref-EB_3-5" class="reference"><a href="#cite_note-EB-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Gallery">Gallery</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=25" title="Edit section: Gallery"><span>edit</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/File:Einstein_gyro_gravity_probe_b.jpg" class="mw-file-description" title="An image of one of the most accurate human-made spheres, as it refracts the image of Einstein in the background. This sphere was a fused quartz gyroscope for the Gravity Probe B experiment, and differs in shape from a perfect sphere by no more than 40 atoms (less than 10 nm) of thickness. It was announced on 1 July 2008 that Australian scientists had created even more nearly perfect spheres, accurate to 0.3 nm, as part of an international hunt to find a new global standard kilogram.[21]"><img alt="An image of one of the most accurate human-made spheres, as it refracts the image of Einstein in the background. This sphere was a fused quartz gyroscope for the Gravity Probe B experiment, and differs in shape from a perfect sphere by no more than 40 atoms (less than 10 nm) of thickness. It was announced on 1 July 2008 that Australian scientists had created even more nearly perfect spheres, accurate to 0.3 nm, as part of an international hunt to find a new global standard kilogram.[21]" 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">An image of one of the most accurate human-made spheres, as it <a href="/wiki/Refraction" title="Refraction">refracts</a> the image of <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a> in the background. This sphere was a <a href="/wiki/Fused_quartz" title="Fused quartz">fused quartz</a> <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a> for the <a href="/wiki/Gravity_Probe_B" title="Gravity Probe B">Gravity Probe B</a> experiment, and differs in shape from a perfect sphere by no more than 40 atoms (less than 10<span class="nowrap"> </span>nm) of thickness. It was announced on 1 July 2008 that <a href="/wiki/Australia" title="Australia">Australian</a> scientists had created even more nearly perfect spheres, accurate to 0.3<span class="nowrap"> </span>nm, as part of an international hunt <a href="/wiki/Alternative_approaches_to_redefining_the_kilogram#Alternative_approaches_to_redefining_the_kilogram" title="Alternative approaches to redefining the kilogram">to find a new global standard kilogram</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></div> </li> <li class="gallerybox" style="width: 132.66666666667px"> <div class="thumb" style="width: 130.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:King_of_spades-_spheres.jpg" class="mw-file-description" title="Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres"><img alt="Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/King_of_spades-_spheres.jpg/196px-King_of_spades-_spheres.jpg" decoding="async" width="131" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/King_of_spades-_spheres.jpg/294px-King_of_spades-_spheres.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/King_of_spades-_spheres.jpg/392px-King_of_spades-_spheres.jpg 2x" data-file-width="497" data-file-height="760" /></a></span></div> <div class="gallerytext">Deck of playing cards illustrating engineering instruments, England, 1702. <a href="/wiki/King_of_spades" class="mw-redirect" title="King of spades">King of spades</a>: Spheres</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Regions">Regions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=26" title="Edit section: Regions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Ball_(mathematics)#Regions" title="Ball (mathematics)">Ball (mathematics) § Regions</a></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;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>Hemisphere</li> <li><a href="/wiki/Spherical_cap" title="Spherical cap">Spherical cap</a></li> <li><a href="/wiki/Spherical_lune" title="Spherical lune">Spherical lune</a></li> <li><a href="/wiki/Spherical_polygon" class="mw-redirect" title="Spherical polygon">Spherical polygon</a></li> <li><a href="/wiki/Spherical_sector" title="Spherical sector">Spherical sector</a></li> <li><a href="/wiki/Spherical_segment" title="Spherical segment">Spherical segment</a></li> <li><a href="/wiki/Spherical_wedge" title="Spherical wedge">Spherical wedge</a></li> <li><a href="/wiki/Spherical_zone" class="mw-redirect" title="Spherical zone">Spherical zone</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=27" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/3-sphere" title="3-sphere">3-sphere</a></li> <li><a href="/wiki/Affine_sphere" title="Affine sphere">Affine sphere</a></li> <li><a href="/wiki/Alexander_horned_sphere" title="Alexander horned sphere">Alexander horned sphere</a></li> <li><a href="/wiki/Celestial_spheres" title="Celestial spheres">Celestial spheres</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Directional_statistics" title="Directional statistics">Directional statistics</a></li> <li><a href="/wiki/Dyson_sphere" title="Dyson sphere">Dyson sphere</a></li> <li><a href="/wiki/Gauss_map" title="Gauss map">Gauss map</a></li> <li><a href="/wiki/Hand_with_Reflecting_Sphere" title="Hand with Reflecting Sphere">Hand with Reflecting Sphere</a>, <a href="/wiki/M.C._Escher" class="mw-redirect" title="M.C. Escher">M.C. Escher</a> self-portrait drawing illustrating reflection and the optical properties of a mirror sphere</li> <li><a href="/wiki/Hoberman_sphere" title="Hoberman sphere">Hoberman sphere</a></li> <li><a href="/wiki/Homology_sphere" title="Homology sphere">Homology sphere</a></li> <li><a href="/wiki/Homotopy_groups_of_spheres" title="Homotopy groups of spheres">Homotopy groups of spheres</a></li> <li><a href="/wiki/Homotopy_sphere" title="Homotopy sphere">Homotopy sphere</a></li> <li><a href="/wiki/Lenart_Sphere" class="mw-redirect" title="Lenart Sphere">Lenart Sphere</a></li> <li><a href="/wiki/Napkin_ring_problem" title="Napkin ring problem">Napkin ring problem</a></li> <li><a href="/wiki/Orb_(optics)" class="mw-redirect" title="Orb (optics)">Orb (optics)</a></li> <li><a href="/wiki/Pseudosphere" title="Pseudosphere">Pseudosphere</a></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li> <li><a href="/wiki/Solid_angle" title="Solid angle">Solid angle</a></li> <li><a href="/wiki/Sphere_packing" title="Sphere packing">Sphere packing</a></li> <li><a href="/wiki/Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">Spherical coordinates</a></li> <li><a href="/wiki/Spherical_cow" title="Spherical cow">Spherical cow</a></li> <li>Spherical helix, <a href="/wiki/Tangent_indicatrix" title="Tangent indicatrix">tangent indicatrix</a> of a curve of constant precession</li> <li><a href="/wiki/Spherical_polyhedron" title="Spherical polyhedron">Spherical polyhedron</a></li> <li><a href="/wiki/Sphericity" title="Sphericity">Sphericity</a></li> <li><a href="/wiki/Tennis_ball_theorem" title="Tennis ball theorem">Tennis ball theorem</a></li> <li><a href="/wiki/Volume-equivalent_radius" class="mw-redirect" title="Volume-equivalent radius">Volume-equivalent radius</a></li> <li><a href="/wiki/Zoll_surface" title="Zoll surface">Zoll sphere</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=28" title="Edit section: Notes and references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=29" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.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"><ol class="references"> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><span class="texhtml mvar" style="font-style:italic;">r</span> is being considered as a variable in this computation.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">It does not matter which direction is chosen, the distance is the sphere's radius × <i>π</i>.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">The distance between two non-distinct points (i.e., a point and itself) on the sphere is zero.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="References">References</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dsfai%3Dra^">σφαῖρα</a>, Henry George Liddell, Robert Scott, <i>A Greek-English Lexicon</i>, on Perseus.</span> </li> <li id="cite_note-Albert54-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Albert54_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Albert54_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 54.</span> </li> <li id="cite_note-EB-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-EB_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-EB_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-EB_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-EB_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-EB_3-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-EB_3-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFChisholm1911" class="citation encyclopaedia cs1"><a href="/wiki/Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a>, ed. (1911). <span class="cs1-ws-icon" title="s:1911 Encyclopædia Britannica/Sphere"><a class="external text" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Sphere">"Sphere" </a></span>. <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i>. Vol. 25 (11th ed.). Cambridge University Press. pp. <span class="nowrap">647–</span>648.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Sphere&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.pages=%3Cspan+class%3D%22nowrap%22%3E647-%3C%2Fspan%3E648&rft.edition=11th&rft.pub=Cambridge+University+Press&rft.date=1911&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span> </li> <li id="cite_note-Woods266-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Woods266_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Woods266_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Woods266_4-2"><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-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFKreyszig1972">Kreyszig (1972</a>, p. 342).</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. 223.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://mathcentral.uregina.ca/QQ/database/QQ.09.01/rahul1.html">"The volume of a sphere – Math Central"</a>. <i>mathcentral.uregina.ca</i><span class="reference-accessdate">. Retrieved <span class="nowrap">10 June</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathcentral.uregina.ca&rft.atitle=The+volume+of+a+sphere+%E2%80%93+Math+Central&rft_id=http%3A%2F%2Fmathcentral.uregina.ca%2FQQ%2Fdatabase%2FQQ.09.01%2Frahul1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span> </li> <li id="cite_note-delta-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-delta_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-delta_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE.J._BorowskiJ.M._Borwein1989" class="citation book cs1">E.J. Borowski; J.M. Borwein (1989). <i>Collins Dictionary of Mathematics</i>. Collins. pp. 141, 149. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-00-434347-1" title="Special:BookSources/978-0-00-434347-1"><bdi>978-0-00-434347-1</bdi></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.pub=Collins&rft.date=1989&rft.isbn=978-0-00-434347-1&rft.au=E.J.+Borowski&rft.au=J.M.+Borwein&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span> </li> <li id="cite_note-MathWorld_Sphere-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-MathWorld_Sphere_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MathWorld_Sphere_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-MathWorld_Sphere_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Sphere"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><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%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span></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="#CITEREFSteinhaus1969">Steinhaus 1969</a>, p. 221.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="Osserman" class="citation journal cs1">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> (6): 1187. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1978-14553-4">10.1090/S0002-9904-1978-14553-4</a></span><span class="reference-accessdate">. Retrieved <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.issue=6&rft.pages=1187&rft.date=1978&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1978-14553-4&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%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 60.</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 href="#CITEREFAlbert2016">Albert 2016</a>, p. 55.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 57.</span> </li> <li id="cite_note-Woods267-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Woods267_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWoods1961">Woods 1961</a>, p. 267.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 58.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert,_DavidCohn-Vossen,_Stephan1952" class="citation book cs1"><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a>; Cohn-Vossen, Stephan (1952). "Eleven properties of the sphere". <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/geometryimaginat00davi_0"><i>Geometry and the Imagination</i></a></span> (2nd ed.). Chelsea. pp. <span class="nowrap">215–</span>231. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8284-1087-8" title="Special:BookSources/978-0-8284-1087-8"><bdi>978-0-8284-1087-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Eleven+properties+of+the+sphere&rft.btitle=Geometry+and+the+Imagination&rft.pages=%3Cspan+class%3D%22nowrap%22%3E215-%3C%2Fspan%3E231&rft.edition=2nd&rft.pub=Chelsea&rft.date=1952&rft.isbn=978-0-8284-1087-8&rft.au=Hilbert%2C+David&rft.au=Cohn-Vossen%2C+Stephan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryimaginat00davi_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Spheric_section"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><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/SphericSection.html">"Spheric section"</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=Spheric+section&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSphericSection.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Loxodrome.html">"Loxodrome"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Loxodrome&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLoxodrome.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFried2019" class="citation web cs1">Fried, Michael N. (25 February 2019). <a rel="nofollow" class="external text" href="https://oxfordre.com/view/10.1093/acrefore/9780199381135.001.0001/acrefore-9780199381135-e-8161">"conic sections"</a>. <i>Oxford Research Encyclopedia of Classics</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Facrefore%2F9780199381135.013.8161">10.1093/acrefore/9780199381135.013.8161</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-938113-5" title="Special:BookSources/978-0-19-938113-5"><bdi>978-0-19-938113-5</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">4 November</span> 2022</span>. <q>More significantly, Vitruvius (On Architecture, Vitr. 9.8) associated conical sundials with Dionysodorus (early 2nd century bce), and Dionysodorus, according to Eutocius of Ascalon (c. 480–540 ce), used conic sections to complete a solution for Archimedes' problem of cutting a sphere by a plane so that the ratio of the resulting volumes would be the same as a given ratio.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Oxford+Research+Encyclopedia+of+Classics&rft.atitle=conic+sections&rft.date=2019-02-25&rft_id=info%3Adoi%2F10.1093%2Facrefore%2F9780199381135.013.8161&rft.isbn=978-0-19-938113-5&rft.aulast=Fried&rft.aufirst=Michael+N.&rft_id=https%3A%2F%2Foxfordre.com%2Fview%2F10.1093%2Facrefore%2F9780199381135.001.0001%2Facrefore-9780199381135-e-8161&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</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="Further_reading">Further reading</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=31" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.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:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.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;min-width:0}}@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><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox 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data-file-height="430" /></a></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has the text of the <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">1911 <i>Encyclopædia Britannica</i></a> article "<span style="font-weight:bold;"><a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Sphere" class="extiw" title="wikisource:1911 Encyclopædia Britannica/Sphere">Sphere</a></span>".</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><style data-mw-deduplicate="TemplateStyles:r1250146164">.mw-parser-output .sister-box .side-box-abovebelow{padding:0.75em 0;text-align:center}.mw-parser-output .sister-box .side-box-abovebelow>b{display:block}.mw-parser-output .sister-box .side-box-text>ul{border-top:1px solid #aaa;padding:0.75em 0;width:217px;margin:0 auto}.mw-parser-output .sister-box .side-box-text>ul>li{min-height:31px}.mw-parser-output .sister-logo{display:inline-block;width:31px;line-height:31px;vertical-align:middle;text-align:center}.mw-parser-output .sister-link{display:inline-block;margin-left:4px;width:182px;vertical-align:middle}@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}</style><div role="navigation" aria-labelledby="sister-projects" class="side-box metadata side-box-right sister-box sistersitebox plainlinks"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-abovebelow"> <b>Sphere</b> at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects"><span id="sister-projects">sister projects</span></a></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/27px-Wiktionary-logo-v2.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/41px-Wiktionary-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/54px-Wiktionary-logo-v2.svg.png 2x" data-file-width="391" data-file-height="391" /></span></span></span><span class="sister-link"><a href="https://en.wiktionary.org/wiki/Special:Search/Sphere" class="extiw" title="wikt:Special:Search/Sphere">Definitions</a> from Wiktionary</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></span><span class="sister-link"><a href="https://commons.wikimedia.org/wiki/Sphere" class="extiw" title="c:Sphere">Media</a> from Commons</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/27px-Wikinews-logo.svg.png" decoding="async" width="27" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/41px-Wikinews-logo.svg.png 1.5x, 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decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/41px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/54px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span></span><span class="sister-link"><a href="https://en.wikibooks.org/wiki/Special:Search/Sphere" class="extiw" title="b:Special:Search/Sphere">Textbooks</a> from Wikibooks</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" decoding="async" width="27" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/41px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/54px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></span><span class="sister-link"><a href="https://en.wikiversity.org/wiki/Special:Search/Sphere" class="extiw" title="v:Special:Search/Sphere">Resources</a> from Wikiversity</span></li></ul></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlbert2016" class="citation cs2">Albert, Abraham Adrian (2016) [1949], <i>Solid Analytic Geometry</i>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-81026-3" title="Special:BookSources/978-0-486-81026-3"><bdi>978-0-486-81026-3</bdi></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%2Fen.wikipedia.org%3ASphere" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunham1997" class="citation book cs1">Dunham, William (1997). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><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></a></span>. New York: Wiley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicaluniv00dunh/page/n34">28</a>, 226. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994muaa.book.....D">1994muaa.book.....D</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-17661-9" title="Special:BookSources/978-0-471-17661-9"><bdi>978-0-471-17661-9</bdi></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.pub=Wiley&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%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKreyszig1972" class="citation cs2">Kreyszig, Erwin (1972), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/advancedengineer00krey"><i>Advanced Engineering Mathematics</i></a></span> (3rd ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-50728-4" title="Special:BookSources/978-0-471-50728-4"><bdi>978-0-471-50728-4</bdi></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%2Fen.wikipedia.org%3ASphere" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinhaus1969" class="citation cs2">Steinhaus, H. (1969), <i>Mathematical Snapshots</i> (Third American ed.), 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%2Fen.wikipedia.org%3ASphere" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoods1961" class="citation cs2">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%2Fen.wikipedia.org%3ASphere" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_C._Polking1999" class="citation web cs1">John C. Polking (15 April 1999). <a rel="nofollow" class="external text" href="https://www.math.csi.cuny.edu/~ikofman/Polking/sphere.html#basic">"The Geometry of the Sphere"</a>. <i>www.math.csi.cuny.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">21 January</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.math.csi.cuny.edu&rft.atitle=The+Geometry+of+the+Sphere&rft.date=1999-04-15&rft.au=John+C.+Polking&rft_id=https%3A%2F%2Fwww.math.csi.cuny.edu%2F~ikofman%2FPolking%2Fsphere.html%23basic&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASphere" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sphere&action=edit&section=32" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="https://en.wikibooks.org/wiki/Mathematica/Uniform_Spherical_Distribution" class="extiw" title="b:Mathematica/Uniform Spherical Distribution">Mathematica/Uniform Spherical Distribution</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> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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href="/wiki/Special:EditPage/Template:Compact_topological_surfaces" title="Special:EditPage/Template:Compact topological surfaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Compact_topological_surfaces_and_their_immersions_in_3D102" style="font-size:114%;margin:0 4em">Compact topological surfaces and their immersions in 3D</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Without boundary</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Orientable</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Sphere</a> (genus 0)</li> <li><a href="/wiki/Torus" title="Torus">Torus</a> (genus 1)</li> <li>Number 8 (genus 2)</li> <li>Pretzel (genus 3) ...</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Non-orientable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Real_projective_plane" title="Real projective plane">Real projective plane</a> <ul><li>genus 1; <a href="/wiki/Boy%27s_surface" title="Boy's surface">Boy's surface</a></li> <li><a href="/wiki/Roman_surface" title="Roman surface">Roman surface</a></li></ul></li> <li><a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> (genus 2)</li> <li><a href="/wiki/Dyck%27s_surface" class="mw-redirect" title="Dyck's surface">Dyck's surface</a> (genus 3) ...</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With boundary</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">Disk</a> <ul><li>Semisphere</li></ul></li> <li>Ribbon <ul><li><a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">Annulus</a></li> <li><a href="/wiki/Cylinder" title="Cylinder">Cylinder</a></li></ul></li> <li><a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> <ul><li><a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">Cross-cap</a></li></ul></li> <li><a href="/wiki/Pair_of_pants_(mathematics)" title="Pair of pants (mathematics)">Sphere with three holes</a> ...</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />notions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Connected_space" title="Connected space">Connectedness</a></li> <li><a href="/wiki/Compact_space" title="Compact space">Compactness</a></li> <li><a href="/wiki/Triangulation_(topology)" title="Triangulation (topology)">Triangulatedness</a> or <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">smoothness</a></li> <li><a href="/wiki/Orientability" title="Orientability">Orientability</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Characteristics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Number of <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> components</li> <li><a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">Genus</a></li> <li><a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Connected_sum" title="Connected sum">Connected sum</a></li> <li>Making a hole</li> <li>Gluing a <a href="/wiki/Handle_decomposition" title="Handle decomposition">handle</a></li> <li>Gluing a <a href="/wiki/Cross-cap" class="mw-redirect" title="Cross-cap">cross-cap</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style 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