Area of a circle - Wikipedia

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class="vector-toc-list"> </ul> </li> <li id="toc-Archimedes's_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Archimedes's_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Archimedes's proof</span> </div> </a> <ul id="toc-Archimedes's_proof-sublist" class="vector-toc-list"> <li id="toc-Not_greater" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Not_greater"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Not greater</span> </div> </a> <ul id="toc-Not_greater-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Not_less" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Not_less"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Not less</span> </div> </a> <ul id="toc-Not_less-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rearrangement_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rearrangement_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Rearrangement proof</span> </div> </a> <ul id="toc-Rearrangement_proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modern_proofs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Modern_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Modern proofs</span> </div> </a> <button aria-controls="toc-Modern_proofs-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 Modern proofs subsection</span> </button> <ul id="toc-Modern_proofs-sublist" class="vector-toc-list"> <li id="toc-Onion_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Onion_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Onion proof</span> </div> </a> <ul id="toc-Onion_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangle_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Triangle_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Triangle proof</span> </div> </a> <ul id="toc-Triangle_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semicircle_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semicircle_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Semicircle proof</span> </div> </a> <ul id="toc-Semicircle_proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Isoperimetric_inequality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isoperimetric_inequality"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Isoperimetric inequality</span> </div> </a> <ul id="toc-Isoperimetric_inequality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fast_approximation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fast_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Fast approximation</span> </div> </a> <button aria-controls="toc-Fast_approximation-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 Fast approximation subsection</span> </button> <ul id="toc-Fast_approximation-sublist" class="vector-toc-list"> <li id="toc-Archimedes'_doubling_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Archimedes'_doubling_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Archimedes' doubling method</span> </div> </a> <ul id="toc-Archimedes'_doubling_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Snell–Huygens_refinement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Snell–Huygens_refinement"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>The Snell–Huygens refinement</span> </div> </a> <ul id="toc-The_Snell–Huygens_refinement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_of_Archimedes'_doubling_formulae" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_of_Archimedes'_doubling_formulae"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Derivation of Archimedes' doubling formulae</span> </div> </a> <ul id="toc-Derivation_of_Archimedes'_doubling_formulae-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dart_approximation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dart_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Dart approximation</span> </div> </a> <ul id="toc-Dart_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_rearrangement" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Finite_rearrangement"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Finite rearrangement</span> </div> </a> <ul id="toc-Finite_rearrangement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-Euclidean_circles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-Euclidean_circles"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Non-Euclidean circles</span> </div> </a> <ul id="toc-Non-Euclidean_circles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</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"> 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class="mw-page-title-main">Area of a circle</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 13 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-13" 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">13 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D8%A9_%D8%A7%D9%84%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9" title="مساحة الدائرة – Arabic" lang="ar" hreflang="ar" data-title="مساحة الدائرة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80rea_del_cercle" title="Àrea del cercle – Catalan" lang="ca" hreflang="ca" data-title="Àrea del cercle" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kreis#Kreisfläche" title="Kreis – German" lang="de" hreflang="de" data-title="Kreis" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D8%AA_%D8%AF%D8%A7%DB%8C%D8%B1%D9%87" title="مساحت دایره – Persian" lang="fa" hreflang="fa" data-title="مساحت دایره" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9A%E0%A4%95%E0%A4%A4%E0%A5%80_%E0%A4%95%E0%A4%BE_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="चकती का क्षेत्रफल – Hindi" lang="hi" hreflang="hi" data-title="चकती का क्षेत्रफल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Luas_lingkaran" title="Luas lingkaran – Indonesian" lang="id" hreflang="id" data-title="Luas lingkaran" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%86%E3%81%AE%E9%9D%A2%E7%A9%8D" title="円の面積 – Japanese" lang="ja" hreflang="ja" data-title="円の面積" 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-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%80%E1%9F%92%E1%9E%9A%E1%9E%9B%E1%9E%B6%E1%9E%95%E1%9F%92%E1%9E%91%E1%9F%83%E1%9E%9A%E1%9E%84%E1%9F%92%E1%9E%9C%E1%9E%84%E1%9F%8B" title="ក្រលាផ្ទៃរង្វង់ – Khmer" lang="km" hreflang="km" data-title="ក្រលាផ្ទៃរង្វង់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0%D0%B4%D1%8C_%D0%BA%D1%80%D1%83%D0%B3%D0%B0" title="Площадь круга – Russian" lang="ru" hreflang="ru" data-title="Площадь круга" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0_%D0%BA%D1%80%D1%83%D0%B3%D0%B0" title="Площа круга – Ukrainian" lang="uk" hreflang="uk" data-title="Площа круга" 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/Di%E1%BB%87n_t%C3%ADch_h%C3%ACnh_tr%C3%B2n" title="Diện tích hình tròn – Vietnamese" lang="vi" hreflang="vi" data-title="Diện tích hình tròn" 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-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9C%93%E9%9D%A2%E7%A9%8D" title="圓面積 – Cantonese" lang="yue" hreflang="yue" data-title="圓面積" 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/%E5%9C%86%E7%9A%84%E9%9D%A2%E7%A7%AF" title="圆的面积 – Chinese" lang="zh" hreflang="zh" data-title="圆的面积" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q4115331#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li 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<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">Concept in geometry</div> <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 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.sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) 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class="sidebar-content"> <ul><li><a class="mw-selflink selflink">Area of a circle</a></li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a></li> <li><a href="/wiki/List_of_formulae_involving_%CF%80" title="List of formulae involving π">Use in other formulae</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Properties</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Proof_that_%CF%80_is_irrational" title="Proof that π is irrational">Irrationality</a></li> <li><a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Transcendence</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Value</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Less than 22/7</a></li> <li><a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">Approximations</a></li> <li><a href="/wiki/Madhava%27s_correction_term" title="Madhava's correction term">Madhava's correction term</a></li> <li><a href="/wiki/Piphilology" title="Piphilology">Memorization</a></li></ul></td> </tr><tr><th class="sidebar-heading"> People</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Method_of_exhaustion#Archimedes" title="Method of exhaustion">Archimedes</a></li> <li><a href="/wiki/Liu_Hui%27s_%CF%80_algorithm" title="Liu Hui's π algorithm">Liu Hui</a></li> <li><a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava</a></li> <li><a href="/wiki/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a></li> <li><a href="/wiki/Ludolph_van_Ceulen" title="Ludolph van Ceulen">Ludolph van Ceulen</a></li> <li><a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a></li> <li><a href="/wiki/Seki_Takakazu#Calculation_of_Pi" title="Seki Takakazu">Seki Takakazu</a></li> <li><a href="/wiki/Takebe_Kenko#Legacy" class="mw-redirect" title="Takebe Kenko"> Takebe Kenko</a></li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a></li> <li><a href="/wiki/John_Machin" title="John Machin">John Machin</a></li> <li><a href="/wiki/William_Shanks" title="William Shanks">William Shanks</a></li> <li><a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a></li> <li><a href="/wiki/John_Wrench" title="John Wrench">John Wrench</a></li> <li><a href="/wiki/Chudnovsky_brothers" title="Chudnovsky brothers">Chudnovsky brothers</a></li> <li><a href="/wiki/Yasumasa_Kanada" title="Yasumasa Kanada">Yasumasa Kanada</a></li></ul></td> </tr><tr><th class="sidebar-heading"> History</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Chronology_of_computation_of_%CF%80" title="Chronology of computation of π">Chronology</a></li> <li><i><a href="/wiki/A_History_of_Pi" title="A History of Pi">A History of Pi</a></i></li></ul></td> </tr><tr><th class="sidebar-heading"> In culture</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Indiana_pi_bill" title="Indiana pi bill">Indiana pi bill</a></li> <li><a href="/wiki/Pi_Day" title="Pi Day">Pi Day</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Related topics</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></li> <li><a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a></li> <li><a href="/wiki/Six_nines_in_pi" title="Six nines in pi">Six nines in <span class="texhtml mvar" style="font-style:italic;">π</span></a></li> <li><a href="/wiki/List_of_topics_related_to_%CF%80" title="List of topics related to π">Other topics related to <span class="texhtml mvar" 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title="Non-Euclidean geometry">Non-Euclidean</a> <ul><li><a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic</a> <ul><li><a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li> <li><a href="/wiki/Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective</a></li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine</a></li> <li><a href="/wiki/Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a> <ul><li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a> <ul><li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li> <li><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li> <li><a href="/wiki/Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li> <li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a> <ul><li><a href="/wiki/Digital_geometry" title="Digital geometry">Digital</a></li></ul></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Incidence_geometry" title="Incidence geometry">Incidence </a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a> <ul><li><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="/wiki/Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a> <ul><li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul> <ul><li><a href="/wiki/Angle" title="Angle">Angle</a></li> <li><a href="/wiki/Curve" title="Curve">Curve</a></li> <li><a href="/wiki/Diagonal" title="Diagonal">Diagonal</a></li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Perpendicular" title="Perpendicular">Perpendicular</a>)</li> <li><a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li> <li><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul> <ul><li><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">Zero-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Line_(geometry)" title="Line (geometry)">Line</a> <ul><li><a href="/wiki/Line_segment" title="Line segment">segment</a></li> <li><a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li> <li><a href="/wiki/Length" title="Length">Length</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li> <li><a href="/wiki/Area" title="Area">Area</a></li> <li><a href="/wiki/Polygon" title="Polygon">Polygon</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li> <li><a href="/wiki/Hypotenuse" title="Hypotenuse">Hypotenuse</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Diameter" title="Diameter">Diameter</a></li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a></li> <li><a class="mw-selflink selflink">Area</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Volume" title="Volume">Volume</a></li></ul> <ul><li><a href="/wiki/Cube" title="Cube">Cube</a> <ul><li><a href="/wiki/Cuboid" title="Cuboid">cuboid</a></li></ul></li> <li><a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li> <li><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a></li> <li><a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li> <li><a href="/wiki/Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li> <li><a href="/wiki/Sphere" title="Sphere">Sphere</a></li> <li><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">Four</a>- / other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Tesseract" title="Tesseract">Tesseract</a></li> <li><a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;"> <a href="/wiki/List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1–1400s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1400s–1700s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1700s–1900s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Present day</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_geometry" title="Template:General geometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_geometry" title="Template talk:General geometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_geometry" title="Special:EditPage/Template:General geometry"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, the <a href="/wiki/Area" title="Area">area</a> enclosed by a <a href="/wiki/Circle" title="Circle">circle</a> of <a href="/wiki/Radius" title="Radius">radius</a> <span class="texhtml mvar" style="font-style:italic;">r</span> is <span class="texhtml">π<i>r</i><sup>2</sup></span>. Here, the Greek letter <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> represents the <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> ratio of the <a href="/wiki/Circumference" title="Circumference">circumference</a> of any circle to its <a href="/wiki/Diameter" title="Diameter">diameter</a>, approximately equal to 3.14159. </p><p>One method of deriving this formula, which originated with <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a>, involves viewing the circle as the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of a sequence of <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> with an increasing number of sides. The area of a regular polygon is half its <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> multiplied by the <a href="/wiki/Apothem" title="Apothem">distance from its center to its sides</a>, and because the sequence tends to a circle, the corresponding formula–that the area is half the <a href="/wiki/Circumference" title="Circumference">circumference</a> times the radius–namely, <span class="texhtml"><i>A</i> = <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">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> × 2π<i>r</i> × <i>r</i></span>, holds for a circle. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=1" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although often referred to as the area of a circle in informal contexts, strictly speaking, the term <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a> refers to the interior region of the circle, while circle is reserved for the boundary only, which is a <a href="/wiki/Curve" title="Curve">curve</a> and covers no area itself. Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle. </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=Area_of_a_circle&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Modern mathematics can obtain the area using the methods of <a href="/wiki/Integral" title="Integral">integral calculus</a> or its more sophisticated offspring, <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>. However, the area of a disk was studied by the <a href="/wiki/Ancient_Greeks" class="mw-redirect" title="Ancient Greeks">Ancient Greeks</a>. <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a> in the fifth century B.C. had found that the area of a disk is proportional to its radius squared.<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> <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> used the tools of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> to show that the area inside a circle is equal to that of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> whose base has the length of the circle's circumference and whose height equals the circle's radius in his book <i><a href="/wiki/Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a></i>. The circumference is 2<span class="texhtml mvar" style="font-style:italic;">π</span><i>r</i>, and the area of a triangle is half the base times the height, yielding the area <span class="texhtml mvar" style="font-style:italic;">π</span> <i>r</i><sup>2</sup> for the disk. Prior to Archimedes, <a href="/wiki/Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates of Chios</a> was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the <a href="/wiki/Lune_of_Hippocrates" title="Lune of Hippocrates">lune of Hippocrates</a>,<sup id="cite_ref-heath_2-0" class="reference"><a href="#cite_note-heath-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> but did not identify the <a href="/wiki/Constant_of_proportionality" class="mw-redirect" title="Constant of proportionality">constant of proportionality</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Historical_arguments">Historical arguments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=3" title="Edit section: Historical arguments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A variety of arguments have been advanced historically to establish the 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 A=\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f7b7f93f93e7ba7bebb97efbe88e181ce332e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.276ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}}"></span> to varying degrees of mathematical rigor. The most famous of these is Archimedes' <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a>, one of the earliest uses of the mathematical concept of a <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a>, as well as the origin of <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedes' axiom</a> which remains part of the standard analytical treatment of the <a href="/wiki/Real_number_system" class="mw-redirect" title="Real number system">real number system</a>. The original proof of Archimedes is not rigorous by modern standards, because it assumes that we can compare the length of arc of a circle to the length of a secant and a tangent line, and similar statements about the area, as geometrically evident. </p> <div class="mw-heading mw-heading3"><h3 id="Using_polygons">Using polygons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=4" title="Edit section: Using polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The area of a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a> is half its perimeter times the <a href="/wiki/Apothem" title="Apothem">apothem</a>. As the number of sides of the regular polygon increases, the polygon tends to a circle, and the apothem tends to the radius. This suggests that the area of a disk is half the circumference of its bounding circle times the radius.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Archimedes's_proof"><span id="Archimedes.27s_proof"></span>Archimedes's proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=5" title="Edit section: Archimedes's proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Following Archimedes' argument in <i>The Measurement of a Circle</i> (c. 260 BCE), compare the area enclosed by a circle to a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius. If the area of the circle is not equal to that of the triangle, then it must be either greater or less. We eliminate each of these by contradiction, leaving equality as the only possibility. We use <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> in the same way. </p> <div class="mw-heading mw-heading4"><h4 id="Not_greater">Not greater</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=6" title="Edit section: Not greater"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Archimedes_circle_area_proof_-_inscribed_polygons.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Archimedes_circle_area_proof_-_inscribed_polygons.svg/220px-Archimedes_circle_area_proof_-_inscribed_polygons.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Archimedes_circle_area_proof_-_inscribed_polygons.svg/330px-Archimedes_circle_area_proof_-_inscribed_polygons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Archimedes_circle_area_proof_-_inscribed_polygons.svg/440px-Archimedes_circle_area_proof_-_inscribed_polygons.svg.png 2x" data-file-width="420" data-file-height="420" /></a><figcaption>Circle with square and octagon inscribed, showing area gap</figcaption></figure> <p>Suppose that the area <i>C</i> enclosed by the circle is greater than the area <i>T</i> = <i>cr</i>/2 of the triangle. Let <i>E</i> denote the excess amount. <a href="/wiki/Inscribe" class="mw-redirect" title="Inscribe">Inscribe</a> a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments. If the total area of those gaps, <i>G</i><sub>4</sub>, is greater than <i>E</i>, split each arc in half. This makes the inscribed square into an inscribed octagon, and produces eight segments with a smaller total gap, <i>G</i><sub>8</sub>. Continue splitting until the total gap area, <i>G<sub>n</sub></i>, is less than <i>E</i>. Now the area of the inscribed polygon, <i>P<sub>n</sub></i> = <i>C</i> − <i>G<sub>n</sub></i>, must be greater than that of the triangle. </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}E&{}=C-T\\&{}>G_{n}\\P_{n}&{}=C-G_{n}\\&{}>C-E\\P_{n}&{}>T\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>E</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>C</mi> <mo>−</mo> <mi>T</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>></mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>C</mi> <mo>−</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>></mo> <mi>C</mi> <mo>−</mo> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>></mo> <mi>T</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E&{}=C-T\\&{}>G_{n}\\P_{n}&{}=C-G_{n}\\&{}>C-E\\P_{n}&{}>T\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eea5190954993f21d627d7df6b1f2e670939704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:14.213ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}E&{}=C-T\\&{}>G_{n}\\P_{n}&{}=C-G_{n}\\&{}>C-E\\P_{n}&{}>T\end{aligned}}}"></span></dd></dl> <p>But this forces a contradiction, as follows. Draw a perpendicular from the center to the midpoint of a side of the polygon; its length, <i>h</i>, is less than the circle radius. Also, let each side of the polygon have length <i>s</i>; then the sum of the sides is <i>ns</i>, which is less than the circle circumference. The polygon area consists of <i>n</i> equal triangles with height <i>h</i> and base <i>s</i>, thus equals <i>nhs</i>/2. But since <i>h</i> < <i>r</i> and <i>ns</i> < <i>c</i>, the polygon area must be less than the triangle area, <i>cr</i>/2, a contradiction. Therefore, our supposition that <i>C</i> might be greater than <i>T</i> must be wrong. </p> <div class="mw-heading mw-heading4"><h4 id="Not_less">Not less</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=7" title="Edit section: Not less"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Archimedes_circle_area_proof_-_circumscribed_polygons.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Archimedes_circle_area_proof_-_circumscribed_polygons.svg/220px-Archimedes_circle_area_proof_-_circumscribed_polygons.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Archimedes_circle_area_proof_-_circumscribed_polygons.svg/330px-Archimedes_circle_area_proof_-_circumscribed_polygons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Archimedes_circle_area_proof_-_circumscribed_polygons.svg/440px-Archimedes_circle_area_proof_-_circumscribed_polygons.svg.png 2x" data-file-width="420" data-file-height="420" /></a><figcaption>Circle with square and octagon circumscribed, showing area gap</figcaption></figure> <p>Suppose that the area enclosed by the circle is less than the area <i>T</i> of the triangle. Let <i>D</i> denote the deficit amount. Circumscribe a square, so that the midpoint of each edge lies on the circle. If the total area gap between the square and the circle, <i>G</i><sub>4</sub>, is greater than <i>D</i>, slice off the corners with circle tangents to make a circumscribed octagon, and continue slicing until the gap area is less than <i>D</i>. The area of the polygon, <i>P<sub>n</sub></i>, must be less than <i>T</i>. </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}D&{}=T-C\\&{}>G_{n}\\P_{n}&{}=C+G_{n}\\&{}<C+D\\P_{n}&{}<T\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>D</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>></mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>C</mi> <mo>+</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo><</mo> <mi>C</mi> <mo>+</mo> <mi>D</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo><</mo> <mi>T</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D&{}=T-C\\&{}>G_{n}\\P_{n}&{}=C+G_{n}\\&{}<C+D\\P_{n}&{}<T\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7552fdb6c51f509baab5d42220ea17fcb69a7e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:14.213ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}D&{}=T-C\\&{}>G_{n}\\P_{n}&{}=C+G_{n}\\&{}<C+D\\P_{n}&{}<T\end{aligned}}}"></span></dd></dl> <p>This, too, forces a contradiction. For, a perpendicular to the midpoint of each polygon side is a radius, of length <i>r</i>. And since the total side length is greater than the circumference, the polygon consists of <i>n</i> identical triangles with total area greater than <i>T</i>. Again we have a contradiction, so our supposition that <i>C</i> might be less than <i>T</i> must be wrong as well. </p><p>Therefore, it must be the case that the area enclosed by the circle is precisely the same as the area of the triangle. This concludes the proof. </p> <div class="mw-heading mw-heading3"><h3 id="Rearrangement_proof">Rearrangement proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=8" title="Edit section: Rearrangement proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CircleArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/220px-CircleArea.svg.png" decoding="async" width="220" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/330px-CircleArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/440px-CircleArea.svg.png 2x" data-file-width="240" data-file-height="260" /></a><figcaption>Circle area by rearrangement</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Regular_polygon_side_count_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Regular_polygon_side_count_graph.svg/220px-Regular_polygon_side_count_graph.svg.png" decoding="async" width="220" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Regular_polygon_side_count_graph.svg/330px-Regular_polygon_side_count_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Regular_polygon_side_count_graph.svg/440px-Regular_polygon_side_count_graph.svg.png 2x" data-file-width="512" data-file-height="768" /></a><figcaption>Graphs of <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">side</a>, <i>s</i>; <a href="/wiki/Apothem" title="Apothem">apothem</a>, <i>a</i>; and <a href="/wiki/Area" title="Area">area</a>, <i>A</i> of <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> of <i>n</i> sides and <a href="/wiki/Circumradius" class="mw-redirect" title="Circumradius">circumradius</a> 1, with the <a href="/wiki/Base_(geometry)" title="Base (geometry)">base</a>, <i>b</i> of a <a class="mw-selflink-fragment" href="#Rearrangement_proof">rectangle with the same area</a>. The green line shows the case <span class="nowrap"><a href="/wiki/Regular_hexagon" class="mw-redirect" title="Regular hexagon"><i>n</i> = 6</a></span>.</figcaption></figure> <p>Following Satō Moshun (<a href="#CITEREFSmithMikami1914">Smith & Mikami 1914</a>, pp. 130–132), <a href="/wiki/Nicholas_of_Cusa" title="Nicholas of Cusa">Nicholas of Cusa</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> (<a href="#CITEREFBeckmann1976">Beckmann 1976</a>, p. 19), we can use inscribed regular polygons in a different way. Suppose we inscribe a <a href="/wiki/Hexagon" title="Hexagon">hexagon</a>. Cut the hexagon into six triangles by splitting it from the center. Two opposite triangles both touch two common diameters; slide them along one so the radial edges are adjacent. They now form a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>, with the hexagon sides making two opposite edges, one of which is the base, <i>s</i>. Two radial edges form slanted sides, and the height, <i>h</i> is equal to its <a href="/wiki/Apothem" title="Apothem">apothem</a> (as in the Archimedes proof). In fact, we can also assemble all the triangles into one big parallelogram by putting successive pairs next to each other. The same is true if we increase it to eight sides and so on. For a polygon with 2<i>n</i> sides, the parallelogram will have a base of length <i>ns</i>, and a height <i>h</i>. As the number of sides increases, the length of the parallelogram base approaches half the circle circumference, and its height approaches the circle radius. In the limit, the parallelogram becomes a rectangle with width <span class="texhtml mvar" style="font-style:italic;">π</span><i>r</i> and height <i>r</i>. </p> <dl><dd><table class="wikitable" frame="vsides" style="text-align:center" cellspacing="0" cellpadding="3"> <caption><i><b>Unit disk area by rearranging n polygons.</b></i> </caption> <tbody><tr> <th colspan="2">polygon </th> <td rowspan="11" style="padding:1px;"> </td> <th colspan="3">parallelogram </th></tr> <tr> <th><i>n</i></th> <th>side</th> <th>base</th> <th>height</th> <th>area </th></tr> <tr> <td align="right">4</td> <td>1.4142136</td> <td>2.8284271</td> <td>0.7071068</td> <td>2.0000000 </td></tr> <tr> <td align="right">6</td> <td>1.0000000</td> <td>3.0000000</td> <td>0.8660254</td> <td>2.5980762 </td></tr> <tr> <td align="right">8</td> <td>0.7653669</td> <td>3.0614675</td> <td>0.9238795</td> <td>2.8284271 </td></tr> <tr> <td align="right">10</td> <td>0.6180340</td> <td>3.0901699</td> <td>0.9510565</td> <td>2.9389263 </td></tr> <tr> <td align="right">12</td> <td>0.5176381</td> <td>3.1058285</td> <td>0.9659258</td> <td>3.0000000 </td></tr> <tr> <td align="right">14</td> <td>0.4450419</td> <td>3.1152931</td> <td>0.9749279</td> <td>3.0371862 </td></tr> <tr> <td align="right">16</td> <td>0.3901806</td> <td>3.1214452</td> <td>0.9807853</td> <td>3.0614675 </td></tr> <tr> <td align="right">96</td> <td>0.0654382</td> <td>3.1410320</td> <td>0.9994646</td> <td>3.1393502 </td></tr> <tr> <td>∞</td> <td>1/∞</td> <td><span class="texhtml mvar" style="font-style:italic;">π</span></td> <td>1</td> <td><span class="texhtml mvar" style="font-style:italic;">π</span> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Modern_proofs">Modern proofs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=9" title="Edit section: Modern proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are various equivalent definitions of the constant π. The conventional definition in pre-calculus geometry is the ratio of the circumference of a circle to its diameter: </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 \pi ={\frac {C}{D}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>D</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {C}{D}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b44baebd9c828a9c0f10524c60b4c86dfb069c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.838ex; height:5.343ex;" alt="{\displaystyle \pi ={\frac {C}{D}}.}"></span></dd></dl> <p>However, because the circumference of a circle is not a primitive analytical concept, this definition is not suitable in modern rigorous treatments. A standard modern definition is that <span class="texhtml mvar" style="font-style:italic;">π</span> is equal to twice the least positive root of the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> function or, equivalently, the half-period of the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> (or cosine) function. The cosine function can be defined either as a <a href="/wiki/Power_series" title="Power series">power series</a>, or as the solution of a certain <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a>. This avoids any reference to circles in the definition of <span class="texhtml mvar" style="font-style:italic;">π</span>, so that statements about the relation of <span class="texhtml mvar" style="font-style:italic;">π</span> to the circumference and area of circles are actually theorems, rather than definitions, that follow from the analytical definitions of concepts like "area" and "circumference". </p><p>The analytical definitions are seen to be equivalent, if it is agreed that the circumference of the circle is measured as a <a href="/wiki/Rectifiable_curve" class="mw-redirect" title="Rectifiable curve">rectifiable curve</a> by means of the <a href="/wiki/Integral" title="Integral">integral</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=2\int _{-R}^{R}{\frac {R\,dx}{\sqrt {R^{2}-x^{2}}}}=2R\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <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> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>R</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=2\int _{-R}^{R}{\frac {R\,dx}{\sqrt {R^{2}-x^{2}}}}=2R\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2c7ee46379feaa312f11aa0bb52b10110f3a16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.672ex; height:7.009ex;" alt="{\displaystyle C=2\int _{-R}^{R}{\frac {R\,dx}{\sqrt {R^{2}-x^{2}}}}=2R\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}"></span></dd></dl> <p>The integral appearing on the right is an <a href="/wiki/Abelian_integral" title="Abelian integral">abelian integral</a> whose value is a half-period of the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> function, equal to <span class="texhtml mvar" style="font-style:italic;">π</span>. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=2\pi R=\pi D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>R</mi> <mo>=</mo> <mi>π</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=2\pi R=\pi D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf001d870a834f75dbf9bbbe40b67b0d26b2b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.478ex; height:2.176ex;" alt="{\displaystyle C=2\pi R=\pi D}"></span> is seen to be true as a theorem. </p><p>Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula <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=\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f7b7f93f93e7ba7bebb97efbe88e181ce332e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.276ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}}"></span>, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can develop trigonometric functions and the fundamental constant <span class="texhtml mvar" style="font-style:italic;">π</span> in a way that is totally independent of their relation to geometry. We have indicated where appropriate how each of these proofs can be made totally independent of all trigonometry, but in some cases that requires more sophisticated mathematical ideas than those afforded by elementary calculus. </p> <div class="mw-heading mw-heading3"><h3 id="Onion_proof">Onion proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=10" title="Edit section: Onion proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_area_rings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Circle_area_rings.svg/220px-Circle_area_rings.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Circle_area_rings.svg/330px-Circle_area_rings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Circle_area_rings.svg/440px-Circle_area_rings.svg.png 2x" data-file-width="420" data-file-height="420" /></a><figcaption>Area of the disk via ring integration</figcaption></figure> <p>Using calculus, we can sum the area incrementally, partitioning the disk into thin concentric rings like the layers of an <a href="/wiki/Onion" title="Onion">onion</a>. This is the method of <a href="/wiki/Shell_integration" title="Shell integration">shell integration</a> in two dimensions. For an infinitesimally thin ring of the "onion" of radius <i>t</i>, the accumulated area is 2<span class="texhtml mvar" style="font-style:italic;">π</span><i>t dt</i>, the circumferential length of the ring times its infinitesimal width (one can approximate this ring by a rectangle with width=2<span class="texhtml mvar" style="font-style:italic;">π</span><i>t</i> and height=<i>dt</i>). This gives an elementary integral for a disk of radius <i>r</i>. </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}\mathrm {Area} (r)&{}=\int _{0}^{r}2\pi t\,dt\\&{}=2\pi \left[{\frac {t^{2}}{2}}\right]_{0}^{r}\\&{}=\pi r^{2}.\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mn>2</mn> <mi>π</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msubsup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\int _{0}^{r}2\pi t\,dt\\&{}=2\pi \left[{\frac {t^{2}}{2}}\right]_{0}^{r}\\&{}=\pi r^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d49256b6928ac3b3624a749b47be687a62092e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:21.159ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\int _{0}^{r}2\pi t\,dt\\&{}=2\pi \left[{\frac {t^{2}}{2}}\right]_{0}^{r}\\&{}=\pi r^{2}.\end{aligned}}}"></span></dd></dl> <p>It is rigorously justified by the <a href="/wiki/Integration_by_substitution#Substitution_for_multiple_variables" title="Integration by substitution">multivariate substitution rule</a> in polar coordinates. Namely, the area is given by a <a href="/wiki/Double_integral" class="mw-redirect" title="Double integral">double integral</a> of the constant function 1 over the disk itself. If <i>D</i> denotes the disk, then the double integral can be computed in <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a> as follows: </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}\mathrm {Area} (r)&{}=\iint _{D}1\ d(x,y)\\&{}=\iint _{D}t\ dt\ d\theta \\&{}=\int _{0}^{r}\int _{0}^{2\pi }t\ d\theta \ dt\\&{}=\int _{0}^{r}\left[t\theta \right]_{0}^{2\pi }dt\\&{}=\int _{0}^{r}2\pi t\,dt\\\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mn>1</mn> <mtext> </mtext> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mi>t</mi> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mtext> </mtext> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <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> <mi>t</mi> <mtext> </mtext> <mi>d</mi> <mi>θ</mi> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <msubsup> <mrow> <mo>[</mo> <mrow> <mi>t</mi> <mi>θ</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mn>2</mn> <mi>π</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\iint _{D}1\ d(x,y)\\&{}=\iint _{D}t\ dt\ d\theta \\&{}=\int _{0}^{r}\int _{0}^{2\pi }t\ d\theta \ dt\\&{}=\int _{0}^{r}\left[t\theta \right]_{0}^{2\pi }dt\\&{}=\int _{0}^{r}2\pi t\,dt\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a18dc84550f7b8aec1a8eabb3d29d23a66147d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.505ex; width:26.593ex; height:30.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\iint _{D}1\ d(x,y)\\&{}=\iint _{D}t\ dt\ d\theta \\&{}=\int _{0}^{r}\int _{0}^{2\pi }t\ d\theta \ dt\\&{}=\int _{0}^{r}\left[t\theta \right]_{0}^{2\pi }dt\\&{}=\int _{0}^{r}2\pi t\,dt\\\end{aligned}}}"></span></dd></dl> <p>which is the same result as obtained above. </p><p>An equivalent rigorous justification, without relying on the special coordinates of trigonometry, uses the <a href="/wiki/Coarea_formula" title="Coarea formula">coarea formula</a>. Define a function <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 :\mathbb {R} ^{2}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho :\mathbb {R} ^{2}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6a2003b3ea2e729bd9f05a5ebaae840e00c32b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.164ex; height:3.176ex;" alt="{\displaystyle \rho :\mathbb {R} ^{2}\to \mathbb {R} }"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \rho (x,y)={\sqrt {x^{2}+y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \rho (x,y)={\sqrt {x^{2}+y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814ac6fc88a67675f2eb558b67d825c48c24bd69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.392ex; height:3.509ex;" alt="{\textstyle \rho (x,y)={\sqrt {x^{2}+y^{2}}}}"></span>. Note ρ is a <a href="/wiki/Lipschitz_function" class="mw-redirect" title="Lipschitz function">Lipschitz function</a> whose <a href="/wiki/Gradient" title="Gradient">gradient</a> is a unit vector <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 |\nabla \rho |=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\nabla \rho |=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4052f4ce9686e0fe9ad45664c38a2faa0c35b4aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.692ex; height:2.843ex;" alt="{\displaystyle |\nabla \rho |=1}"></span> (<a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>). Let <i>D</i> be the disc <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 <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5f4f072a10e5f3c66c5ca6bbf24df526f0706d" 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 <1}"></span> in <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>. We will show that <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 {\mathcal {L}}^{2}(D)=\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{2}(D)=\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ff09bb5b41401f3f1992a621448cbee5024d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.822ex; height:3.176ex;" alt="{\displaystyle {\mathcal {L}}^{2}(D)=\pi }"></span>, where <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 {\mathcal {L}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e212255ffc118973a6a2a83d380004b12d46280f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.658ex; height:2.676ex;" alt="{\displaystyle {\mathcal {L}}^{2}}"></span> is the two-dimensional Lebesgue measure in <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>. We shall assume that the one-dimensional <a href="/wiki/Hausdorff_measure" title="Hausdorff measure">Hausdorff measure</a> of the circle <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 =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40db75722182575603fb982d7ddab4863bef1079" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle \rho =r}"></span> 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 2\pi r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e811131a9c6c5f45e6657e0fc506e7e2a37f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.176ex;" alt="{\displaystyle 2\pi r}"></span>, the circumference of the circle of radius <i>r</i>. (This can be taken as the definition of circumference.) Then, by the coarea formula, </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}{\mathcal {L}}^{2}(D)&=\iint _{D}|\nabla \rho |\,d{\mathcal {L}}^{2}\\&=\int _{\mathbb {R} }{\mathcal {H}}^{1}(\rho ^{-1}(r)\cap D)\,dr\\&=\int _{0}^{1}{\mathcal {H}}^{1}(\rho ^{-1}(r))\,dr\\&=\int _{0}^{1}2\pi r\,dr=\pi .\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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mo>=</mo> <mi>π</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}^{2}(D)&=\iint _{D}|\nabla \rho |\,d{\mathcal {L}}^{2}\\&=\int _{\mathbb {R} }{\mathcal {H}}^{1}(\rho ^{-1}(r)\cap D)\,dr\\&=\int _{0}^{1}{\mathcal {H}}^{1}(\rho ^{-1}(r))\,dr\\&=\int _{0}^{1}2\pi r\,dr=\pi .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9922e76813f4e7863a3cfcb78ce47ce26e7688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.466ex; margin-bottom: -0.205ex; width:31.718ex; height:24.509ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}^{2}(D)&=\iint _{D}|\nabla \rho |\,d{\mathcal {L}}^{2}\\&=\int _{\mathbb {R} }{\mathcal {H}}^{1}(\rho ^{-1}(r)\cap D)\,dr\\&=\int _{0}^{1}{\mathcal {H}}^{1}(\rho ^{-1}(r))\,dr\\&=\int _{0}^{1}2\pi r\,dr=\pi .\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Triangle_proof">Triangle proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=11" title="Edit section: Triangle proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:TriangleFromCircle.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/TriangleFromCircle.gif/220px-TriangleFromCircle.gif" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/07/TriangleFromCircle.gif 1.5x" data-file-width="300" data-file-height="200" /></a><figcaption>Circle unwrapped to form a triangle</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Area_of_circle_and_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Area_of_circle_and_triangle.svg/220px-Area_of_circle_and_triangle.svg.png" decoding="async" width="220" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Area_of_circle_and_triangle.svg/330px-Area_of_circle_and_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Area_of_circle_and_triangle.svg/440px-Area_of_circle_and_triangle.svg.png 2x" data-file-width="870" data-file-height="450" /></a><figcaption>The circle and the triangle are equal in area.</figcaption></figure> <p>Similar to the onion proof outlined above, we could exploit calculus in a different way in order to arrive at the formula for the area of a disk. Consider unwrapping the concentric circles to straight strips. This will form a right angled triangle with r as its height and 2<span class="texhtml mvar" style="font-style:italic;">π</span>r (being the outer slice of onion) as its base. </p><p>Finding the area of this triangle will give the area of the disk </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}{\text{Area}}&{}={\frac {1}{2}}\cdot {\text{base}}\cdot {\text{height}}\\[6pt]&{}={\frac {1}{2}}\cdot 2\pi r\cdot r\\[6pt]&{}=\pi r^{2}\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="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Area</mtext> </mrow> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>height</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo>⋅</mo> <mi>r</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{Area}}&{}={\frac {1}{2}}\cdot {\text{base}}\cdot {\text{height}}\\[6pt]&{}={\frac {1}{2}}\cdot 2\pi r\cdot r\\[6pt]&{}=\pi r^{2}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec19d8244481b4f7c1aae8b9d31cb2e251488a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:24.791ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}{\text{Area}}&{}={\frac {1}{2}}\cdot {\text{base}}\cdot {\text{height}}\\[6pt]&{}={\frac {1}{2}}\cdot 2\pi r\cdot r\\[6pt]&{}=\pi r^{2}\end{aligned}}}"></span></dd></dl> <p>The opposite and adjacent angles for this triangle are respectively in degrees 9.0430611..., 80.956939... and in radians 0.1578311... <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A233527" class="extiw" title="oeis:A233527">A233527</a></span>, 1.4129651...<span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A233528" class="extiw" title="oeis:A233528">A233528</a></span>. </p><p>Explicitly, we imagine dividing up a circle into triangles, each with a height equal to the circle's radius and a base that is infinitesimally small. The area of each of these triangles is equal to <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 1/2\cdot r\cdot du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>⋅</mo> <mi>r</mi> <mo>⋅</mo> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2\cdot r\cdot du}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1a066a0fa3a8379f94f0488949a50a92c8245a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.44ex; height:2.843ex;" alt="{\displaystyle 1/2\cdot r\cdot du}"></span>. By summing up (integrating) all of the areas of these triangles, we arrive at the formula for the circle's area: </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}\mathrm {Area} (r)&{}=\int _{0}^{2\pi r}{\frac {1}{2}}r\,du\\[6pt]&{}=\left[{\frac {1}{2}}ru\right]_{0}^{2\pi r}\\[6pt]&{}=\pi r^{2}.\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="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>r</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>r</mi> <mi>u</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\int _{0}^{2\pi r}{\frac {1}{2}}r\,du\\[6pt]&{}=\left[{\frac {1}{2}}ru\right]_{0}^{2\pi r}\\[6pt]&{}=\pi r^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2645ee843189090de6367ebb0d8c4a66c24c228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:23.126ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\int _{0}^{2\pi r}{\frac {1}{2}}r\,du\\[6pt]&{}=\left[{\frac {1}{2}}ru\right]_{0}^{2\pi r}\\[6pt]&{}=\pi r^{2}.\end{aligned}}}"></span></dd></dl> <p>It too can be justified by a double integral of the constant function 1 over the disk by reversing the <a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">order of integration</a> and using a change of variables in the above iterated integral: </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}\mathrm {Area} (r)&{}=\iint _{D}1\ d(x,y)\\&{}=\iint _{D}t\ dt\ d\theta \\&{}=\int _{0}^{2\pi }\int _{0}^{r}t\ dt\ d\theta \\&{}=\int _{0}^{2\pi }{\frac {1}{2}}r^{2}\ d\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mn>1</mn> <mtext> </mtext> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mi>t</mi> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mtext> </mtext> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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>r</mi> </mrow> </msubsup> <mi>t</mi> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mtext> </mtext> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\iint _{D}1\ d(x,y)\\&{}=\iint _{D}t\ dt\ d\theta \\&{}=\int _{0}^{2\pi }\int _{0}^{r}t\ dt\ d\theta \\&{}=\int _{0}^{2\pi }{\frac {1}{2}}r^{2}\ d\theta \\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de81dd427b118bc3e9d4a917d77f4406bfb1fe0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:26.593ex; height:24.509ex;" alt="{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\iint _{D}1\ d(x,y)\\&{}=\iint _{D}t\ dt\ d\theta \\&{}=\int _{0}^{2\pi }\int _{0}^{r}t\ dt\ d\theta \\&{}=\int _{0}^{2\pi }{\frac {1}{2}}r^{2}\ d\theta \\\end{aligned}}}"></span></dd></dl> <p>Making the substitution <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 u=r\theta ,\ du=r\ d\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>r</mi> <mi>θ</mi> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <mi>u</mi> <mo>=</mo> <mi>r</mi> <mtext> </mtext> <mi>d</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=r\theta ,\ du=r\ d\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b19964ea4f443c319f4aed924828774b95c5517" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.761ex; height:2.509ex;" alt="{\displaystyle u=r\theta ,\ du=r\ d\theta }"></span> converts the integral to </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 \int _{0}^{2\pi r}{\frac {1}{2}}{\frac {r^{2}}{r}}du=\int _{0}^{2\pi r}{\frac {1}{2}}r\ du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mfrac> </mrow> <mi>d</mi> <mi>u</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> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>r</mi> <mtext> </mtext> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{2\pi r}{\frac {1}{2}}{\frac {r^{2}}{r}}du=\int _{0}^{2\pi r}{\frac {1}{2}}r\ du}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7183156a76a4a5bcdb9a2289222d1d93a5e6b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.933ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{2\pi r}{\frac {1}{2}}{\frac {r^{2}}{r}}du=\int _{0}^{2\pi r}{\frac {1}{2}}r\ du}"></span></dd></dl> <p>which is the same as the above result. </p><p>The triangle proof can be reformulated as an application of <a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's theorem</a> in flux-divergence form (i.e. a two-dimensional version of the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>), in a way that avoids all mention of trigonometry and the constant <span class="texhtml mvar" style="font-style:italic;">π</span>. Consider the <a href="/wiki/Vector_field" title="Vector field">vector field</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 \mathbf {r} =x\mathbf {i} +y\mathbf {j} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d470fc9fb71b4743145c5b179c8786875bba3977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.085ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} }"></span> in the plane. So the <a href="/wiki/Divergence" title="Divergence">divergence</a> of <b>r</b> is equal to two, and hence the area of a disc <i>D</i> is equal to </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={\frac {1}{2}}\iint _{D}\operatorname {div} \mathbf {r} \,dA.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mi>div</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}\iint _{D}\operatorname {div} \mathbf {r} \,dA.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf275d2d0d2c53bdfb01ee5bad56a05480a9d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.375ex; height:5.676ex;" alt="{\displaystyle A={\frac {1}{2}}\iint _{D}\operatorname {div} \mathbf {r} \,dA.}"></span></dd></dl> <p>By Green's theorem, this is the same as the outward flux of <b>r</b> across the circle bounding <i>D</i>: </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={\frac {1}{2}}\oint _{\partial D}\mathbf {r} \cdot \mathbf {n} \,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}\oint _{\partial D}\mathbf {r} \cdot \mathbf {n} \,ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52bee127996b453f1fad60b360989356d549f525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.392ex; height:5.676ex;" alt="{\displaystyle A={\frac {1}{2}}\oint _{\partial D}\mathbf {r} \cdot \mathbf {n} \,ds}"></span></dd></dl> <p>where <b>n</b> is the unit normal and <i>ds</i> is the arc length measure. For a circle of radius <i>R</i> centered at the origin, we have <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 |\mathbf {r} |=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {r} |=R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7511197c0f4304212bd12c09e54c7fd0232ed69f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.258ex; height:2.843ex;" alt="{\displaystyle |\mathbf {r} |=R}"></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 \mathbf {n} =\mathbf {r} /R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} =\mathbf {r} /R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7476aeda8c221fe956de8e5bc3ccaf897f9c5455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.612ex; height:2.843ex;" alt="{\displaystyle \mathbf {n} =\mathbf {r} /R}"></span>, so the above equality 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={\frac {1}{2}}\oint _{\partial D}\mathbf {r} \cdot {\frac {\mathbf {r} }{R}}\,ds={\frac {R}{2}}\oint _{\partial D}\,ds.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>D</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}\oint _{\partial D}\mathbf {r} \cdot {\frac {\mathbf {r} }{R}}\,ds={\frac {R}{2}}\oint _{\partial D}\,ds.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/061fa4fae000ac530fc3e68efd538a082ee3284a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.137ex; height:5.676ex;" alt="{\displaystyle A={\frac {1}{2}}\oint _{\partial D}\mathbf {r} \cdot {\frac {\mathbf {r} }{R}}\,ds={\frac {R}{2}}\oint _{\partial D}\,ds.}"></span></dd></dl> <p>The integral of <i>ds</i> over the whole circle <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 \partial D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8fb77a08a342eb44226f19fd7c30314628075a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.242ex; height:2.176ex;" alt="{\displaystyle \partial D}"></span> is just the arc length, which is its circumference, so this shows that the area <i>A</i> enclosed by the circle is equal to <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/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ca7642176d9979f70675631339f834ef25abbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle R/2}"></span> times the circumference of the circle. </p><p>Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of <span class="texhtml"><i>d𝜃</i></span> at the centre of the circle), each with an area of <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> · <i>r</i><sup>2</sup> · <i>d𝜃</i></span> (derived from the expression for the area of a triangle: <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> · <i>a</i> · <i>b</i> · sin<i>𝜃</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> · <i>r</i> · <i>r</i> · sin(<i>d𝜃</i>) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> · <i>r</i><sup>2</sup> · <i>d𝜃</i></span>). Note that <span class="texhtml">sin(<i>d𝜃</i>) ≈ <i>d𝜃</i></span> due to <a href="/wiki/Small-angle_approximation" title="Small-angle approximation">small angle approximation</a>. Through summing the areas of the triangles, the expression for the area of the circle can therefore be found: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {Area} &{}=\int _{0}^{2\pi }{\frac {1}{2}}r^{2}\,d\theta \\&{}=\left[{\frac {1}{2}}r^{2}\theta \right]_{0}^{2\pi }\\&{}=\pi r^{2}.\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> </mrow> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {Area} &{}=\int _{0}^{2\pi }{\frac {1}{2}}r^{2}\,d\theta \\&{}=\left[{\frac {1}{2}}r^{2}\theta \right]_{0}^{2\pi }\\&{}=\pi r^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1ff8135211ae36e2d3087475420d550281b254" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; margin-top: -0.181ex; width:20.342ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {Area} &{}=\int _{0}^{2\pi }{\frac {1}{2}}r^{2}\,d\theta \\&{}=\left[{\frac {1}{2}}r^{2}\theta \right]_{0}^{2\pi }\\&{}=\pi r^{2}.\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Semicircle_proof">Semicircle proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=12" title="Edit section: Semicircle proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Note that the area of a semicircle of radius <i>r</i> can be computed by the integral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4643d34e13ebc90791bbb972bb189c4f262db79b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.32ex; height:3.843ex;" alt="{\textstyle \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx}"></span>. </p> <figure typeof="mw:File/Frame"><a href="/wiki/File:Semicircle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Semicircle.svg/198px-Semicircle.svg.png" decoding="async" width="198" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Semicircle.svg/297px-Semicircle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Semicircle.svg/396px-Semicircle.svg.png 2x" data-file-width="198" data-file-height="144" /></a><figcaption>A semicircle of radius <i>r</i></figcaption></figure> <p>By <a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric substitution</a>, we substitute <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=r\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r\sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e930a2d8e0926d2d202d0be3d0f24fc2e0fdcf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.197ex; height:2.176ex;" alt="{\displaystyle x=r\sin \theta }"></span>, hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx=r\cos \theta \,d\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx=r\cos \theta \,d\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd31939f56d3ec88039e9f9953b67268881a991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.009ex; height:2.176ex;" alt="{\displaystyle dx=r\cos \theta \,d\theta .}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx&=\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}{\sqrt {r^{2}\left(1-\sin ^{2}\theta \right)}}\cdot r\cos \theta \,d\theta \\[5pt]&=2r^{2}\int _{0}^{\frac {\pi }{2}}\cos ^{2}\theta \,d\theta \\[5pt]&={\frac {\pi r^{2}}{2}}.\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="0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>⋅</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx&=\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}{\sqrt {r^{2}\left(1-\sin ^{2}\theta \right)}}\cdot r\cos \theta \,d\theta \\[5pt]&=2r^{2}\int _{0}^{\frac {\pi }{2}}\cos ^{2}\theta \,d\theta \\[5pt]&={\frac {\pi r^{2}}{2}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4afdcf935f2a5eeb4493a4db90318db065d024" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:51.887ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx&=\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}{\sqrt {r^{2}\left(1-\sin ^{2}\theta \right)}}\cdot r\cos \theta \,d\theta \\[5pt]&=2r^{2}\int _{0}^{\frac {\pi }{2}}\cos ^{2}\theta \,d\theta \\[5pt]&={\frac {\pi r^{2}}{2}}.\end{aligned}}}"></span> </p><p>The last step follows since the trigonometric identity <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 \cos(\theta )=\sin(\pi /2-\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta )=\sin(\pi /2-\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fffb293c041aa16d6a7c93b1df65919d55bac8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.362ex; height:2.843ex;" alt="{\displaystyle \cos(\theta )=\sin(\pi /2-\theta )}"></span> implies that <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 \cos ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311894fdc65ae89f6b2e40edac3d3281a0727680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.643ex; height:2.676ex;" alt="{\displaystyle \cos ^{2}\theta }"></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 \sin ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02635acae84e17f9bae3ce887b425276ffeaa1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.387ex; height:2.676ex;" alt="{\displaystyle \sin ^{2}\theta }"></span> have equal integrals over the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\pi /2]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\pi /2]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db58baa407ae179d23402f61cb3edcfd7e4fa5b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.147ex; height:2.843ex;" alt="{\displaystyle [0,\pi /2]}"></span>, using <a href="/wiki/Integration_by_substitution" title="Integration by substitution">integration by substitution</a>. But on the other hand, since <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 \cos ^{2}\theta +\sin ^{2}\theta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9333418071b0b0662ba53f8983fe1cbb613ad005" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.132ex; height:2.843ex;" alt="{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}"></span>, the sum of the two integrals is the length of that interval, which 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 \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}"></span>. Consequently, the integral 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 \cos ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311894fdc65ae89f6b2e40edac3d3281a0727680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.643ex; height:2.676ex;" alt="{\displaystyle \cos ^{2}\theta }"></span> is equal to half the length of that interval, which 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 \pi /4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1707aa0fec2c8ef008b9e30b6045fbf95dab9e20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /4}"></span>. </p><p>Therefore, the area of a circle of radius <i>r</i>, which is twice the area of the semi-circle, is equal to <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 2\cdot {\frac {\pi r^{2}}{2}}=\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot {\frac {\pi r^{2}}{2}}=\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0d08b5a6c75c8a3bc169c815ac578ef7a5bf3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.646ex; height:5.676ex;" alt="{\displaystyle 2\cdot {\frac {\pi r^{2}}{2}}=\pi r^{2}}"></span>. </p><p>This particular proof may appear to beg the question, if the sine and cosine functions involved in the trigonometric substitution are regarded as being defined in relation to circles. However, as noted earlier, it is possible to define sine, cosine, and <span class="texhtml mvar" style="font-style:italic;">π</span> in a way that is totally independent of trigonometry, in which case the proof is valid by the <a href="/wiki/Change_of_variables_formula" class="mw-redirect" title="Change of variables formula">change of variables formula</a> and <a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's theorem</a>, assuming the basic properties of sine and cosine (which can also be proved without assuming anything about their relation to circles). </p> <div class="mw-heading mw-heading2"><h2 id="Isoperimetric_inequality">Isoperimetric inequality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=13" title="Edit section: Isoperimetric inequality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The circle is the closed curve of least perimeter that encloses the maximum area. This is known as the <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a>, which states that if a rectifiable Jordan curve in the Euclidean plane has perimeter <i>C</i> and encloses an area <i>A</i> (by the <a href="/wiki/Jordan_curve_theorem" title="Jordan curve theorem">Jordan curve theorem</a>) 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 4\pi A\leq C^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <mi>A</mi> <mo>≤</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi A\leq C^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206d5cd0fa0bfd674f559f687893f311dca9235c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.835ex; height:2.843ex;" alt="{\displaystyle 4\pi A\leq C^{2}.}"></span></dd></dl> <p>Moreover, equality holds in this inequality if and only if the curve is a circle, in which 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 A=\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f7b7f93f93e7ba7bebb97efbe88e181ce332e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.276ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}}"></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 C=2\pi r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=2\pi r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398305eb631c365e21449ec4e9c9d7dca3f8c788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.408ex; height:2.176ex;" alt="{\displaystyle C=2\pi r}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Fast_approximation">Fast approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=14" title="Edit section: Fast approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The calculations Archimedes used to approximate the area numerically were laborious, and he stopped with a polygon of 96 sides. A faster method uses ideas of <a href="/wiki/Willebrord_Snell" class="mw-redirect" title="Willebrord Snell">Willebrord Snell</a> (<i>Cyclometricus</i>, 1621), further developed by <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> (<i>De Circuli Magnitudine Inventa</i>, 1654), described in <a href="#CITEREFGerretsenVerdenduin1983">Gerretsen & Verdenduin (1983</a>, pp. 243–250). </p> <div class="mw-heading mw-heading3"><h3 id="Archimedes'_doubling_method"><span id="Archimedes.27_doubling_method"></span>Archimedes' doubling method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=15" title="Edit section: Archimedes' doubling method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a circle, let <i>u<sub>n</sub></i> be the <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> of an inscribed regular <i>n-</i>gon, and let <i>U<sub>n</sub></i> be the perimeter of a circumscribed regular <i>n-</i>gon. Then <i>u<sub>n</sub></i> and <i>U<sub>n</sub></i> are lower and upper bounds for the circumference of the circle that become sharper and sharper as <i>n</i> increases, and their average (<i>u<sub>n</sub></i> + <i>U<sub>n</sub></i>)/2 is an especially good approximation to the circumference. To compute <i>u<sub>n</sub></i> and <i>U<sub>n</sub></i> for large <i>n</i>, Archimedes derived the following doubling formulae: </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 u_{2n}={\sqrt {U_{2n}u_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2n}={\sqrt {U_{2n}u_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d6b0413585820942fe8d0934bf48f02c0a33a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.968ex; height:3.509ex;" alt="{\displaystyle u_{2n}={\sqrt {U_{2n}u_{n}}}}"></span>   (<a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a>), and</dd></dl> <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 U_{2n}={\frac {2U_{n}u_{n}}{U_{n}+u_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2n}={\frac {2U_{n}u_{n}}{U_{n}+u_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bde8040dd0b47695d703e8cc93d4cedfd36e77a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.757ex; height:5.676ex;" alt="{\displaystyle U_{2n}={\frac {2U_{n}u_{n}}{U_{n}+u_{n}}}}"></span>    (<a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a>).</dd></dl> <p>Starting from a hexagon, Archimedes doubled <i>n</i> four times to get a 96-gon, which gave him a good approximation to the circumference of the circle. </p><p>In modern notation, we can reproduce his computation (and go further) as follows. For a unit circle, an inscribed hexagon has <i>u</i><sub>6</sub> = 6, and a circumscribed hexagon has <i>U</i><sub>6</sub> = 4<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>. Doubling seven times yields </p> <dl><dd><table class="wikitable" frame="vsides" style="text-align:center" cellspacing="0" cellpadding="3"> <caption><b>Archimedes doubling seven times; <i>n</i> = 6 × 2<sup><i>k</i></sup>.</b> </caption> <tbody><tr style="background-color:#eeeeee"> <th><i>k</i></th> <th><i>n</i></th> <th><i>u<sub>n</sub></i></th> <th><i>U<sub>n</sub></i></th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>u<sub>n</sub></i> + <i>U<sub>n</sub></i></span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> </th></tr> <tr> <td>0</td> <td>6</td> <td>6.0000000</td> <td>6.9282032</td> <td>3.2320508 </td></tr> <tr> <td>1</td> <td>12</td> <td>6.2116571</td> <td>6.4307806</td> <td>3.1606094 </td></tr> <tr> <td>2</td> <td>24</td> <td>6.2652572</td> <td>6.3193199</td> <td>3.1461443 </td></tr> <tr> <td>3</td> <td>48</td> <td>6.2787004</td> <td>6.2921724</td> <td>3.1427182 </td></tr> <tr> <td>4</td> <td>96</td> <td>6.2820639</td> <td>6.2854292</td> <td>3.1418733 </td></tr> <tr> <td>5</td> <td>192</td> <td>6.2829049</td> <td>6.2837461</td> <td>3.1416628 </td></tr> <tr> <td>6</td> <td>384</td> <td>6.2831152</td> <td>6.2833255</td> <td>3.1416102 </td></tr> <tr> <td>7</td> <td>768</td> <td>6.2831678</td> <td>6.2832204</td> <td>3.1415970 </td></tr></tbody></table></dd></dl> <p>(Here <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>u<sub>n</sub></i> + <i>U<sub>n</sub></i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> approximates the circumference of the unit circle, which is 2<span class="texhtml mvar" style="font-style:italic;">π</span>, so <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>u<sub>n</sub></i> + <i>U<sub>n</sub></i></span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> approximates <span class="texhtml mvar" style="font-style:italic;">π</span>.) </p><p>The last entry of the table has <sup>355</sup>⁄<sub>113</sub> as one of its <a href="/wiki/Continued_fraction#Best_rational_approximation" title="Continued fraction">best rational approximations</a>; i.e., there is no better approximation among rational numbers with denominator up to 113. The number <sup>355</sup>⁄<sub>113</sub> is also an excellent approximation to <span class="texhtml mvar" style="font-style:italic;">π</span>, attributed to Chinese mathematician <a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a>, who named it <a href="/wiki/Mil%C3%BC" title="Milü">Milü</a>.<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> This approximation is better than any other rational number with denominator less than 16,604.<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> </p> <div class="mw-heading mw-heading3"><h3 id="The_Snell–Huygens_refinement"><span id="The_Snell.E2.80.93Huygens_refinement"></span>The Snell–Huygens refinement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=16" title="Edit section: The Snell–Huygens refinement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Snell proposed (and Huygens proved) a tighter bound than Archimedes': </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 n{\frac {3\sin {\frac {\pi }{n}}}{2+\cos {\frac {\pi }{n}}}}<\pi <n\left(2\sin {\frac {\pi }{3n}}+\tan {\frac {\pi }{3n}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> </mrow> <mrow> <mn>2</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo><</mo> <mi>π</mi> <mo><</mo> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>3</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>3</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n{\frac {3\sin {\frac {\pi }{n}}}{2+\cos {\frac {\pi }{n}}}}<\pi <n\left(2\sin {\frac {\pi }{3n}}+\tan {\frac {\pi }{3n}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86536f9d37cbcfb29eb6a427d3041661a8310542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:42.84ex; height:6.843ex;" alt="{\displaystyle n{\frac {3\sin {\frac {\pi }{n}}}{2+\cos {\frac {\pi }{n}}}}<\pi <n\left(2\sin {\frac {\pi }{3n}}+\tan {\frac {\pi }{3n}}\right).}"></span></dd></dl> <p>This for <i>n</i> = 48 gives a better approximation (about 3.14159292) than Archimedes' method for <i>n</i> = 768. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_of_Archimedes'_doubling_formulae"><span id="Derivation_of_Archimedes.27_doubling_formulae"></span>Derivation of Archimedes' doubling formulae</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=17" title="Edit section: Derivation of Archimedes' doubling formulae"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Huygens_%2B_Snell_%2B_van_Ceulen_-_regular_polygon_doubling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Huygens_%2B_Snell_%2B_van_Ceulen_-_regular_polygon_doubling.svg/220px-Huygens_%2B_Snell_%2B_van_Ceulen_-_regular_polygon_doubling.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Huygens_%2B_Snell_%2B_van_Ceulen_-_regular_polygon_doubling.svg/330px-Huygens_%2B_Snell_%2B_van_Ceulen_-_regular_polygon_doubling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Huygens_%2B_Snell_%2B_van_Ceulen_-_regular_polygon_doubling.svg/440px-Huygens_%2B_Snell_%2B_van_Ceulen_-_regular_polygon_doubling.svg.png 2x" data-file-width="420" data-file-height="420" /></a><figcaption>Circle with similar triangles: circumscribed side, inscribed side and complement, inscribed split side and complement</figcaption></figure> <p>Let one side of an inscribed regular <i>n-</i>gon have length <i>s<sub>n</sub></i> and touch the circle at points A and B. Let A′ be the point opposite A on the circle, so that A′A is a diameter, and A′AB is an inscribed triangle on a diameter. By <a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a>, this is a right triangle with right angle at B. Let the length of A′B be <i>c<sub>n</sub></i>, which we call the complement of <i>s<sub>n</sub></i>; thus <i>c<sub>n</sub></i><sup>2</sup>+<i>s<sub>n</sub></i><sup>2</sup> = (2<i>r</i>)<sup>2</sup>. Let C bisect the arc from A to B, and let C′ be the point opposite C on the circle. Thus the length of CA is <i>s</i><sub>2<i>n</i></sub>, the length of C′A is <i>c</i><sub>2<i>n</i></sub>, and C′CA is itself a right triangle on diameter C′C. Because C bisects the arc from A to B, C′C perpendicularly bisects the chord from A to B, say at P. Triangle C′AP is thus a right triangle, and is <a href="/wiki/Similarity_(geometry)#Similar_triangles" title="Similarity (geometry)">similar</a> to C′CA since they share the angle at C′. Thus all three corresponding sides are in the same proportion; in particular, we have C′A : C′C = C′P : C′A and AP : C′A = CA : C′C. The center of the circle, O, bisects A′A, so we also have triangle OAP similar to A′AB, with OP half the length of A′B. In terms of side lengths, this gives us </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}c_{2n}^{2}&{}=\left(r+{\frac {1}{2}}c_{n}\right)2r\\c_{2n}&{}={\frac {s_{n}}{s_{2n}}}.\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> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> <mi>r</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c_{2n}^{2}&{}=\left(r+{\frac {1}{2}}c_{n}\right)2r\\c_{2n}&{}={\frac {s_{n}}{s_{2n}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec280ac13e1450d5a24fd526952d400b682dfe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:21.03ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}c_{2n}^{2}&{}=\left(r+{\frac {1}{2}}c_{n}\right)2r\\c_{2n}&{}={\frac {s_{n}}{s_{2n}}}.\end{aligned}}}"></span></dd></dl> <p>In the first equation C′P is C′O+OP, length <i>r</i> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>c<sub>n</sub></i>, and C′C is the diameter, 2<i>r</i>. For a unit circle we have the famous doubling equation of <a href="/wiki/Ludolph_van_Ceulen" title="Ludolph van Ceulen">Ludolph van Ceulen</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 c_{2n}={\sqrt {2+c_{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{2n}={\sqrt {2+c_{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cef43c0b37b00567ab8546d4d7390549f2a0679e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.345ex; height:3.509ex;" alt="{\displaystyle c_{2n}={\sqrt {2+c_{n}}}.}"></span></dd></dl> <p>If we now circumscribe a regular <i>n-</i>gon, with side A″B″ parallel to AB, then OAB and OA″B″ are similar triangles, with A″B″ : AB = OC : OP. Call the circumscribed side <i>S<sub>n</sub></i>; then this is <i>S<sub>n</sub></i> : <i>s<sub>n</sub></i> = 1 : <sup>1</sup>⁄<sub>2</sub><i>c<sub>n</sub></i>. (We have again used that OP is half the length of A′B.) Thus we obtain </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 c_{n}=2{\frac {s_{n}}{S_{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}=2{\frac {s_{n}}{S_{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b061aef26814208a0f9fe1ea53fb309ac7b879ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.613ex; height:5.176ex;" alt="{\displaystyle c_{n}=2{\frac {s_{n}}{S_{n}}}.}"></span></dd></dl> <p>Call the inscribed perimeter <i>u<sub>n</sub></i> = <i>ns<sub>n</sub></i>, and the circumscribed perimeter <i>U<sub>n</sub></i> = <i>nS<sub>n</sub></i>. Then combining equations, we have </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 c_{2n}={\frac {s_{n}}{s_{2n}}}=2{\frac {s_{2n}}{S_{2n}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{2n}={\frac {s_{n}}{s_{2n}}}=2{\frac {s_{2n}}{S_{2n}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07eb7dfab4dd77c486b80b2ce9bdbdbdc007fbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.322ex; height:5.176ex;" alt="{\displaystyle c_{2n}={\frac {s_{n}}{s_{2n}}}=2{\frac {s_{2n}}{S_{2n}}},}"></span></dd></dl> <p>so 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 u_{2n}^{2}=u_{n}U_{2n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2n}^{2}=u_{n}U_{2n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b922e528b38ccf8a06fac1703d2b96075b8e7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.292ex; height:3.176ex;" alt="{\displaystyle u_{2n}^{2}=u_{n}U_{2n}.}"></span></dd></dl> <p>This gives a <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> equation. </p><p>We can also deduce </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 2{\frac {s_{2n}}{S_{2n}}}{\frac {s_{n}}{s_{2n}}}=2+2{\frac {s_{n}}{S_{n}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\frac {s_{2n}}{S_{2n}}}{\frac {s_{n}}{s_{2n}}}=2+2{\frac {s_{n}}{S_{n}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e55724fd390dc5e2926ecf9638f5c5d9fc7be2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.821ex; height:5.176ex;" alt="{\displaystyle 2{\frac {s_{2n}}{S_{2n}}}{\frac {s_{n}}{s_{2n}}}=2+2{\frac {s_{n}}{S_{n}}},}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{U_{2n}}}={\frac {1}{u_{n}}}+{\frac {1}{U_{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{U_{2n}}}={\frac {1}{u_{n}}}+{\frac {1}{U_{n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cbcda2b6cfd07af45818ee1cc270795417fc609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.076ex; height:5.676ex;" alt="{\displaystyle {\frac {2}{U_{2n}}}={\frac {1}{u_{n}}}+{\frac {1}{U_{n}}}.}"></span></dd></dl> <p>This gives a <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> equation. </p> <div class="mw-heading mw-heading2"><h2 id="Dart_approximation">Dart approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=18" title="Edit section: Dart approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_area_Monte_Carlo_integration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Circle_area_Monte_Carlo_integration.svg/220px-Circle_area_Monte_Carlo_integration.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Circle_area_Monte_Carlo_integration.svg/330px-Circle_area_Monte_Carlo_integration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Circle_area_Monte_Carlo_integration.svg/440px-Circle_area_Monte_Carlo_integration.svg.png 2x" data-file-width="420" data-file-height="420" /></a><figcaption>Unit circle area Monte Carlo integration. Estimate by these 900 samples is 4×<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">709</span><span class="sr-only">/</span><span class="den">900</span></span>⁠</span> = 3.15111...</figcaption></figure> <p>When more efficient methods of finding areas are not available, we can resort to "throwing darts". This <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a> uses the fact that if random samples are taken uniformly scattered across the surface of a square in which a disk resides, the proportion of samples that hit the disk approximates the ratio of the area of the disk to the area of the square. This should be considered a method of last resort for computing the area of a disk (or any shape), as it requires an enormous number of samples to get useful accuracy; an estimate good to 10<sup>−<i>n</i></sup> requires about 100<sup><i>n</i></sup> random samples (<a href="#CITEREFThijssen2006">Thijssen 2006</a>, p. 273). </p> <div class="mw-heading mw-heading2"><h2 id="Finite_rearrangement">Finite rearrangement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=19" title="Edit section: Finite rearrangement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We have seen that by partitioning the disk into an infinite number of pieces we can reassemble the pieces into a rectangle. A remarkable fact discovered relatively recently (<a href="#CITEREFLaczkovich1990">Laczkovich 1990</a>) is that we can dissect the disk into a large but <i>finite</i> number of pieces and then reassemble the pieces into a square of equal area. This is called <a href="/wiki/Tarski%27s_circle-squaring_problem" title="Tarski's circle-squaring problem">Tarski's circle-squaring problem</a>. The nature of Laczkovich's proof is such that it proves the existence of such a partition (in fact, of many such partitions) but does not exhibit any particular partition. </p> <div class="mw-heading mw-heading2"><h2 id="Non-Euclidean_circles">Non-Euclidean circles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=20" title="Edit section: Non-Euclidean circles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Circles can be defined in <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, and in particular in the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic</a> and <a href="/wiki/Elliptic_plane" class="mw-redirect" title="Elliptic plane">elliptic</a> planes. </p><p>For example, the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</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 S^{2}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4636bdf770b534f693150ebb7fbadfb784487b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.548ex; height:3.176ex;" alt="{\displaystyle S^{2}(1)}"></span> is a model for the two-dimensional elliptic plane. It carries an <a href="/wiki/Intrinsic_metric" title="Intrinsic metric">intrinsic metric</a> that arises by measuring <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> length. The geodesic circles are the parallels in a <a href="/wiki/Latitude" title="Latitude">geodesic coordinate system</a>. </p><p>More precisely, fix a 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 \mathbf {z} \in S^{2}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>∈</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {z} \in S^{2}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1b6091f6bf3c3ed153525e1f5225c8e033d8e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.576ex; height:3.176ex;" alt="{\displaystyle \mathbf {z} \in S^{2}(1)}"></span> that we place at the zenith. Associated to that zenith is a geodesic polar coordinate system <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 ,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varphi ,\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a110fb79b1172f7ed4035655811f0d8d223efb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.454ex; height:2.843ex;" alt="{\displaystyle (\varphi ,\theta )}"></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 0\leq \varphi \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \varphi \leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd53d652e90ac4b674bdfd8a764a1f10f831cebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.212ex; height:2.676ex;" alt="{\displaystyle 0\leq \varphi \leq \pi }"></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 0\leq \theta <2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \theta <2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6186f8c500d093d06941c6508a3ad10570f851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.944ex; height:2.343ex;" alt="{\displaystyle 0\leq \theta <2\pi }"></span>, where <b>z</b> 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 \varphi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.676ex;" alt="{\displaystyle \varphi =0}"></span>. In these coordinates, the geodesic distance from <b>z</b> to any other 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 \mathbf {x} \in S^{2}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in S^{2}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a45e6fb5eb4517d48f7e1ed9683d7c8dce6366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.799ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} \in S^{2}(1)}"></span> having coordinates <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 ,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varphi ,\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a110fb79b1172f7ed4035655811f0d8d223efb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.454ex; height:2.843ex;" alt="{\displaystyle (\varphi ,\theta )}"></span> is the value 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 \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> at <b>x</b>. A spherical circle is the set of points a geodesic distance <i>R</i> from the zenith point <b>z</b>. Equivalently, with a fixed embedding into <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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>, the spherical circle of 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\leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c173c39db61e8959cc2dd99a37336c4faa0021b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.194ex; height:2.343ex;" alt="{\displaystyle R\leq \pi }"></span> centered at <b>z</b> is the set of <b>x</b> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4636bdf770b534f693150ebb7fbadfb784487b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.548ex; height:3.176ex;" alt="{\displaystyle S^{2}(1)}"></span> such that <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 \mathbf {x} \cdot \mathbf {z} =\cos R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \cdot \mathbf {z} =\cos R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be3f540f11aac9c7169444b99b2fb09cf0974af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.639ex; height:2.176ex;" alt="{\displaystyle \mathbf {x} \cdot \mathbf {z} =\cos R}"></span>. </p><p>We can also measure the area of the spherical disk enclosed within a spherical circle, using the intrinsic surface area measure on the sphere. The area of the disk of radius <i>R</i> is then given by </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}^{R}\sin(\varphi )\,d\varphi \,d\theta =2\pi (1-\cos R).}"> <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>R</mi> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\int _{0}^{2\pi }\int _{0}^{R}\sin(\varphi )\,d\varphi \,d\theta =2\pi (1-\cos R).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd65f4e910ad34c58c8820c81f8b3c21dc9c25d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.335ex; height:6.176ex;" alt="{\displaystyle A=\int _{0}^{2\pi }\int _{0}^{R}\sin(\varphi )\,d\varphi \,d\theta =2\pi (1-\cos R).}"></span></dd></dl> <p>More generally, if a sphere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}(\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}(\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17dbbacbebb0fc0065e70780ed3ed351823db228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.587ex; height:3.176ex;" alt="{\displaystyle S^{2}(\rho )}"></span> has radius of curvature <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>, then the area of the disk of radius <i>R</i> is given by </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=2\pi \rho ^{2}(1-\cos(R/\rho )).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi \rho ^{2}(1-\cos(R/\rho )).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5911dc61f7dbc9eb176f47b8893f8f1d7c371a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.1ex; height:3.176ex;" alt="{\displaystyle A=2\pi \rho ^{2}(1-\cos(R/\rho )).}"></span></dd></dl> <p>Observe that, as an application of <a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a>, this tends to the Euclidean area <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 \pi R^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi R^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4bdad3187c4001ed7079d18509f3b52d18479c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.15ex; height:2.676ex;" alt="{\displaystyle \pi R^{2}}"></span> in the flat limit <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 \to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/330b19cae37f63c358115adb2510a4dc75f00f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.14ex; height:2.343ex;" alt="{\displaystyle \rho \to \infty }"></span>. </p><p>The hyperbolic case is similar, with the area of a disk of intrinsic radius <i>R</i> in the (constant curvature <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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span>) hyperbolic plane given by </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=2\pi (1-\cosh R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi (1-\cosh R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b4cbe978b64a6621d81b2a8d8a47c311af3c16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.703ex; height:2.843ex;" alt="{\displaystyle A=2\pi (1-\cosh R)}"></span></dd></dl> <p>where cosh is the <a href="/wiki/Hyperbolic_cosine" class="mw-redirect" title="Hyperbolic cosine">hyperbolic cosine</a>. More generally, for the constant curvature <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 -k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641790be442610fecf0307a8cc06a19adb14ba7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.019ex; height:2.343ex;" alt="{\displaystyle -k}"></span> hyperbolic plane, the answer 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=2\pi k^{-2}(1-\cosh(kR)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi k^{-2}(1-\cosh(kR)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e14b32b92404532db1e62765c41bb7ecb5605f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.527ex; height:3.176ex;" alt="{\displaystyle A=2\pi k^{-2}(1-\cosh(kR)).}"></span></dd></dl> <p>These identities are important for comparison inequalities in geometry. For example, the area enclosed by a circle of radius <i>R</i> in a flat space is always greater than the area of a spherical circle and smaller than a hyperbolic circle, provided all three circles have the same (intrinsic) radius. That 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 2\pi (1-\cos R)<\pi R^{2}<2\pi (1-\cosh R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi (1-\cos R)<\pi R^{2}<2\pi (1-\cosh R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7983b5208abcd07971ca2d97cde9ab09ad6b1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.777ex; height:3.176ex;" alt="{\displaystyle 2\pi (1-\cos R)<\pi R^{2}<2\pi (1-\cosh R)}"></span></dd></dl> <p>for all <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/57914127d03a5cea02c60a32cfbb22f34904f00d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle R>0}"></span>. Intuitively, this is because the sphere tends to curve back on itself, yielding circles of smaller area than those in the plane, whilst the hyperbolic plane, when immersed into space, develops fringes that produce additional area. It is more generally true that the area of the circle of a fixed radius <i>R</i> is a strictly decreasing function of the curvature. </p><p>In all cases, 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is the curvature (constant, positive or negative), then the <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a> for a domain with area <i>A</i> and perimeter <i>L</i> 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 L^{2}\geq 4\pi A-kA^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <mn>4</mn> <mi>π</mi> <mi>A</mi> <mo>−</mo> <mi>k</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}\geq 4\pi A-kA^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2773d6b658e70bfafa898a1eeb404ee35162d500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.822ex; height:2.843ex;" alt="{\displaystyle L^{2}\geq 4\pi A-kA^{2}}"></span></dd></dl> <p>where equality is achieved precisely for the circle.<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> </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=Area_of_a_circle&action=edit&section=21" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We can stretch a disk to form an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>. Because this stretch is a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> of the plane, it has a distortion factor which will change the area but preserve <i>ratios</i> of areas. This observation can be used to compute the area of an arbitrary ellipse from the area of a unit circle. </p><p>Consider the unit circle circumscribed by a square of side length 2. The transformation sends the circle to an ellipse by stretching or shrinking the horizontal and vertical diameters to the major and minor axes of the ellipse. The square gets sent to a rectangle circumscribing the ellipse. The ratio of the area of the circle to the square is <span class="texhtml mvar" style="font-style:italic;">π</span>/4, which means the ratio of the ellipse to the rectangle is also <span class="texhtml mvar" style="font-style:italic;">π</span>/4. Suppose <i>a</i> and <i>b</i> are the lengths of the major and minor axes of the ellipse. Since the area of the rectangle is <i>ab</i>, the area of the ellipse is <span class="texhtml mvar" style="font-style:italic;">π</span><i>ab</i>/4. </p><p>We can also consider analogous measurements in higher dimensions. For example, we may wish to find the volume inside a sphere. When we have a formula for the surface area, we can use the same kind of "onion" approach we used for the disk. </p> <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=Area_of_a_circle&action=edit&section=22" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Area-equivalent_radius" class="mw-redirect" title="Area-equivalent radius">Area-equivalent radius</a></li> <li><a href="/wiki/Area_of_a_triangle" title="Area of a triangle">Area of a triangle</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=23" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><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"><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="CITEREFStewart2003" class="citation book cs1">Stewart, James (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/singlevariableca00stew/page/3"><i>Single variable calculus early transcendentals</i></a></span> (5th. ed.). Toronto ON: Brook/Cole. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/singlevariableca00stew/page/3">3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-39330-6" title="Special:BookSources/0-534-39330-6"><bdi>0-534-39330-6</bdi></a>. <q>However, by indirect reasoning, Eudoxus (fifth century B.C.) used exhaustion to prove the familiar formula for the area of a disk: <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=\pi r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <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=\pi r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecdb193f0c084cce66161e383a7e8acb1eea80f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.923ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}.}"></span></q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Single+variable+calculus+early+transcendentals.&rft.place=Toronto+ON&rft.pages=3&rft.edition=5th.&rft.pub=Brook%2FCole&rft.date=2003&rft.isbn=0-534-39330-6&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsinglevariableca00stew%2Fpage%2F3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span> </span> </li> <li id="cite_note-heath-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-heath_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath2003" class="citation cs2"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Thomas L.</a> (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA121"><i>A Manual of Greek Mathematics</i></a>, Courier Dover Publications, pp. 121–132, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-43231-9" title="Special:BookSources/0-486-43231-9"><bdi>0-486-43231-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Manual+of+Greek+Mathematics&rft.pages=121-132&rft.pub=Courier+Dover+Publications&rft.date=2003&rft.isbn=0-486-43231-9&rft.aulast=Heath&rft.aufirst=Thomas+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_HZNr_mGFzQC%26pg%3DPA121&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Hill, George. <i><a rel="nofollow" class="external text" href="https://archive.org/details/lessonsingeomet04hillgoog/page/n136">Lessons in Geometry: For the Use of Beginners</a></i>, page 124 (1894).</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClegg2012" class="citation book cs1">Clegg, Brian (2012). <i>Introducing Infinity</i>. Icon Books. p. 69. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84831-406-1" title="Special:BookSources/978-1-84831-406-1"><bdi>978-1-84831-406-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introducing+Infinity&rft.pages=69&rft.pub=Icon+Books&rft.date=2012&rft.isbn=978-1-84831-406-1&rft.aulast=Clegg&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartzloff2006" class="citation book cs1">Martzloff, Jean-Claude (2006). <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/historychinesema00mart_058"><i>A History of Chinese Mathematics</i></a></span>. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historychinesema00mart_058/page/n298">281</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540337829" title="Special:BookSources/9783540337829"><bdi>9783540337829</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Chinese+Mathematics&rft.pages=281&rft.pub=Springer&rft.date=2006&rft.isbn=9783540337829&rft.aulast=Martzloff&rft.aufirst=Jean-Claude&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistorychinesema00mart_058&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-approximations-are-the-convergents-of-the-continued-fraction/">Not all best rational approximations are the convergents of the continued fraction!</a></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 id="CITEREFIsaac_Chavel2001" class="citation cs2">Isaac Chavel (2001), <i>Isoperimetric inequalities</i>, Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Isoperimetric+inequalities&rft.pub=Cambridge+University+Press&rft.date=2001&rft.au=Isaac+Chavel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Area_of_a_circle&action=edit&section=24" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArchimedesc._260_BCE" class="citation cs2"><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (1897), <a rel="nofollow" class="external text" href="https://archive.org/stream/worksofarchimede029517mbp#page/n279/mode/2up">"Measurement of a circle"</a>, in <a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, T. L.</a> (ed.), <a rel="nofollow" class="external text" href="https://www.archive.org/details/worksofarchimede029517mbp"><i>The Works of Archimedes</i></a>, Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Measurement+of+a+circle&rft.btitle=The+Works+of+Archimedes&rft.pub=Cambridge+University+Press&rft.date=1897&rft.au=Archimedes&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fworksofarchimede029517mbp%23page%2Fn279%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span><br />(Originally published by <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, 1897, based on J. L. Heiberg's Greek version.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeckmann1976" class="citation cs2"><a href="/wiki/Petr_Beckmann" title="Petr Beckmann">Beckmann, Petr</a> (1976), <i>A History of Pi</i>, <a href="/wiki/St._Martin%27s_Press" title="St. Martin's Press">St. Martin's Griffin</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-312-38185-1" title="Special:BookSources/978-0-312-38185-1"><bdi>978-0-312-38185-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Pi&rft.pub=St.+Martin%27s+Griffin&rft.date=1976&rft.isbn=978-0-312-38185-1&rft.aulast=Beckmann&rft.aufirst=Petr&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGerretsenVerdenduin1983" class="citation cs2">Gerretsen, J.; Verdenduin, P. (1983), "Chapter 8: Polygons and Polyhedra", in H. Behnke; F. Bachmann; K. Fladt; H. Kunle (eds.), <i>Fundamentals of Mathematics, Volume II: Geometry</i>, translated by S. H. Gould, <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>, pp. 243–250, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-52094-2" title="Special:BookSources/978-0-262-52094-2"><bdi>978-0-262-52094-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+8%3A+Polygons+and+Polyhedra&rft.btitle=Fundamentals+of+Mathematics%2C+Volume+II%3A+Geometry&rft.pages=243-250&rft.pub=MIT+Press&rft.date=1983&rft.isbn=978-0-262-52094-2&rft.aulast=Gerretsen&rft.aufirst=J.&rft.au=Verdenduin%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span><br />(Originally <i>Grundzüge der Mathematik</i>, Vandenhoeck & Ruprecht, Göttingen, 1971.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaczkovich1990" class="citation cs2"><a href="/wiki/Mikl%C3%B3s_Laczkovich" title="Miklós Laczkovich">Laczkovich, Miklós</a> (1990), <a rel="nofollow" class="external text" href="https://eudml.org/doc/153197">"Equidecomposability and discrepancy: A solution to Tarski's circle squaring problem"</a>, <i><a href="/wiki/Crelle%27s_Journal" title="Crelle's Journal">Journal für die reine und angewandte Mathematik</a></i>, <b>1990</b> (404): 77–117, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1990.404.77">10.1515/crll.1990.404.77</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1037431">1037431</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117762563">117762563</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=Equidecomposability+and+discrepancy%3A+A+solution+to+Tarski%27s+circle+squaring+problem&rft.volume=1990&rft.issue=404&rft.pages=77-117&rft.date=1990&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1037431%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117762563%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1515%2Fcrll.1990.404.77&rft.aulast=Laczkovich&rft.aufirst=Mikl%C3%B3s&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F153197&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1985" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1985), "The length of the circle", <i>Math! : Encounters with High School Students</i>, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96129-3" title="Special:BookSources/978-0-387-96129-3"><bdi>978-0-387-96129-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+length+of+the+circle&rft.btitle=Math%21+%3A+Encounters+with+High+School+Students&rft.pub=Springer-Verlag&rft.date=1985&rft.isbn=978-0-387-96129-3&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithMikami1914" class="citation cs2"><a href="/wiki/David_Eugene_Smith" title="David Eugene Smith">Smith, David Eugene</a>; Mikami, Yoshio (1914), <a rel="nofollow" class="external text" href="https://archive.org/details/historyofjapanes00smituoft"><i>A history of Japanese mathematics</i></a>, Chicago: <a href="/wiki/Open_Court_Publishing_Company" title="Open Court Publishing Company">Open Court Publishing</a>, pp. 130–132, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-87548-170-8" title="Special:BookSources/978-0-87548-170-8"><bdi>978-0-87548-170-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+Japanese+mathematics&rft.place=Chicago&rft.pages=130-132&rft.pub=Open+Court+Publishing&rft.date=1914&rft.isbn=978-0-87548-170-8&rft.aulast=Smith&rft.aufirst=David+Eugene&rft.au=Mikami%2C+Yoshio&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofjapanes00smituoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThijssen2006" class="citation cs2">Thijssen, J. M. (2006), <i>Computational Physics</i>, Cambridge University Press, p. 273, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57588-1" title="Special:BookSources/978-0-521-57588-1"><bdi>978-0-521-57588-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Physics&rft.pages=273&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-57588-1&rft.aulast=Thijssen&rft.aufirst=J.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea+of+a+circle" 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=Area_of_a_circle&action=edit&section=25" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.sciencenews.org/articles/20041030/mathtrek.asp">Science News on Tarski problem</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080413114409/http://www.sciencenews.org/articles/20041030/mathtrek.asp">Archived</a> 2008-04-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Area_of_a_circle&oldid=1258021794">https://en.wikipedia.org/w/index.php?title=Area_of_a_circle&oldid=1258021794</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Area" title="Category:Area">Area</a></li><li><a href="/wiki/Category:Circles" title="Category:Circles">Circles</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Pages_using_sidebar_with_the_child_parameter" title="Category:Pages using sidebar with the child parameter">Pages using sidebar with the child parameter</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_containing_proofs" title="Category:Articles containing proofs">Articles containing proofs</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 17 November 2024, at 19:11<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Area of a circle - Wikipedia