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class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Analytic_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Analytic results</span> </div> </a> <button aria-controls="toc-Analytic_results-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 Analytic results subsection</span> </button> <ul id="toc-Analytic_results-sublist" class="vector-toc-list"> <li id="toc-Circumference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circumference"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Circumference</span> </div> </a> <ul id="toc-Circumference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Area_enclosed" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Area_enclosed"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Area enclosed</span> </div> </a> <ul id="toc-Area_enclosed-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Radian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Radian"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Radian</span> </div> </a> <ul id="toc-Radian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Equations</span> </div> </a> <ul id="toc-Equations-sublist" class="vector-toc-list"> <li id="toc-Cartesian_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cartesian_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1</span> <span>Cartesian coordinates</span> </div> </a> <ul id="toc-Cartesian_coordinates-sublist" class="vector-toc-list"> <li id="toc-Equation_of_a_circle" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Equation_of_a_circle"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1.1</span> <span>Equation of a circle</span> </div> </a> <ul id="toc-Equation_of_a_circle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-One_coordinate_as_a_function_of_the_other" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#One_coordinate_as_a_function_of_the_other"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1.2</span> <span>One coordinate as a function of the other</span> </div> </a> <ul id="toc-One_coordinate_as_a_function_of_the_other-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parametric_form" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Parametric_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1.3</span> <span>Parametric form</span> </div> </a> <ul id="toc-Parametric_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-3-point_form" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#3-point_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1.4</span> <span>3-point form</span> </div> </a> <ul id="toc-3-point_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homogeneous_form" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Homogeneous_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1.5</span> <span>Homogeneous form</span> </div> </a> <ul id="toc-Homogeneous_form-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Polar_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Polar_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.2</span> <span>Polar coordinates</span> </div> </a> <ul id="toc-Polar_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_plane" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Complex_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.3</span> <span>Complex plane</span> </div> </a> <ul id="toc-Complex_plane-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tangent_lines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tangent_lines"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Tangent lines</span> </div> </a> <ul id="toc-Tangent_lines-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Chord" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chord"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Chord</span> </div> </a> <ul id="toc-Chord-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangent" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tangent"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Tangent</span> </div> </a> <ul id="toc-Tangent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theorems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Theorems</span> </div> </a> <ul id="toc-Theorems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inscribed_angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inscribed_angles"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Inscribed angles</span> </div> </a> <ul id="toc-Inscribed_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sagitta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sagitta"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Sagitta</span> </div> </a> <ul id="toc-Sagitta-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Compass_and_straightedge_constructions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Compass_and_straightedge_constructions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Compass and straightedge constructions</span> </div> </a> <button aria-controls="toc-Compass_and_straightedge_constructions-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 Compass and straightedge constructions subsection</span> </button> <ul id="toc-Compass_and_straightedge_constructions-sublist" class="vector-toc-list"> <li id="toc-Construction_with_given_diameter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_with_given_diameter"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Construction with given diameter</span> </div> </a> <ul id="toc-Construction_with_given_diameter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construction_through_three_noncollinear_points" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_through_three_noncollinear_points"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Construction through three noncollinear points</span> </div> </a> <ul id="toc-Construction_through_three_noncollinear_points-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Circle_of_Apollonius" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Circle_of_Apollonius"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Circle of Apollonius</span> </div> </a> <button aria-controls="toc-Circle_of_Apollonius-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 Circle of Apollonius subsection</span> </button> <ul id="toc-Circle_of_Apollonius-sublist" class="vector-toc-list"> <li id="toc-Cross-ratios" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cross-ratios"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Cross-ratios</span> </div> </a> <ul id="toc-Cross-ratios-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalised_circles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalised_circles"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Generalised circles</span> </div> </a> <ul id="toc-Generalised_circles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Inscription_in_or_circumscription_about_other_figures" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inscription_in_or_circumscription_about_other_figures"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Inscription in or circumscription about other figures</span> </div> </a> <ul id="toc-Inscription_in_or_circumscription_about_other_figures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limiting_case_of_other_figures" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Limiting_case_of_other_figures"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Limiting case of other figures</span> </div> </a> <ul id="toc-Limiting_case_of_other_figures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Locus_of_constant_sum" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Locus_of_constant_sum"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Locus of constant sum</span> </div> </a> <ul id="toc-Locus_of_constant_sum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Squaring_the_circle" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Squaring_the_circle"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Squaring the circle</span> </div> </a> <ul id="toc-Squaring_the_circle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalisations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalisations"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Generalisations</span> </div> </a> <button aria-controls="toc-Generalisations-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 Generalisations subsection</span> </button> <ul id="toc-Generalisations-sublist" class="vector-toc-list"> <li id="toc-In_other_p-norms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_other_p-norms"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>In other <i>p</i>-norms</span> </div> </a> <ul id="toc-In_other_p-norms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.2</span> <span>Topological definition</span> </div> </a> <ul id="toc-Topological_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Specially_named_circles" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Specially_named_circles"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Specially named circles</span> </div> </a> <button aria-controls="toc-Specially_named_circles-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 Specially named circles subsection</span> </button> <ul id="toc-Specially_named_circles-sublist" class="vector-toc-list"> <li id="toc-Of_a_triangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Of_a_triangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.1</span> <span>Of a triangle</span> </div> </a> <ul id="toc-Of_a_triangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Of_certain_quadrilaterals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Of_certain_quadrilaterals"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.2</span> <span>Of certain quadrilaterals</span> </div> </a> <ul id="toc-Of_certain_quadrilaterals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Of_a_conic_section" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Of_a_conic_section"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.3</span> <span>Of a conic section</span> </div> </a> <ul id="toc-Of_a_conic_section-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Of_a_torus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Of_a_torus"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.4</span> <span>Of a torus</span> </div> </a> <ul id="toc-Of_a_torus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">18</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">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 144 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-144" 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">144 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Sirkel" title="Sirkel – Afrikaans" lang="af" hreflang="af" data-title="Sirkel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Kreis_(Geometrie)" title="Kreis (Geometrie) – Alemannic" lang="gsw" hreflang="gsw" data-title="Kreis (Geometrie)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%AD%E1%89%A5" title="ክብ – Amharic" lang="am" hreflang="am" data-title="ክብ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9" title="دائرة – Arabic" lang="ar" hreflang="ar" data-title="دائرة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Circumferencia" title="Circumferencia – Aragonese" lang="an" hreflang="an" data-title="Circumferencia" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-roa-rup mw-list-item"><a href="https://roa-rup.wikipedia.org/wiki/%C8%9Aerc%C4%BEiu" title="Țercľiu – Aromanian" lang="rup" hreflang="rup" data-title="Țercľiu" data-language-autonym="Armãneashti" data-language-local-name="Aromanian" class="interlanguage-link-target"><span>Armãneashti</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="বৃত্ত – Assamese" lang="as" hreflang="as" data-title="বৃত্ত" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Circunferencia" title="Circunferencia – Asturian" lang="ast" hreflang="ast" data-title="Circunferencia" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Muyu" title="Muyu – Aymara" lang="ay" hreflang="ay" data-title="Muyu" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87evr%C9%99" title="Çevrə – Azerbaijani" lang="az" hreflang="az" data-title="Çevrə" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87" title="دایره – South Azerbaijani" lang="azb" hreflang="azb" data-title="دایره" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="বৃত্ত – Bangla" lang="bn" hreflang="bn" data-title="বৃত্ত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/%C3%8E%E2%81%BF-h%C3%AAng" title="Îⁿ-hêng – Minnan" lang="nan" hreflang="nan" data-title="Îⁿ-hêng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D3%98%D0%B9%D0%BB%D3%99%D0%BD%D3%99" title="Әйләнә – Bashkir" lang="ba" hreflang="ba" data-title="Әйләнә" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Акружнасць – Belarusian" lang="be" hreflang="be" data-title="Акружнасць" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%BA%D1%80%D1%83%D0%B6%D1%8B%D0%BD%D0%B0" title="Акружына – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Акружына" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Bilog" title="Bilog – Central Bikol" lang="bcl" hreflang="bcl" data-title="Bilog" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%BA%D1%80%D1%8A%D0%B6%D0%BD%D0%BE%D1%81%D1%82" title="Окръжност – Bulgarian" lang="bg" hreflang="bg" data-title="Окръжност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A6%E0%BE%92%E0%BD%BC%E0%BD%A2%E0%BC%8B%E0%BD%90%E0%BD%B2%E0%BD%82%E0%BC%8B" title="སྒོར་ཐིག་ – Tibetan" lang="bo" hreflang="bo" data-title="སྒོར་ཐིག་" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – Bosnian" lang="bs" hreflang="bs" data-title="Kružnica" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Kelc%27h" title="Kelc'h – Breton" lang="br" hreflang="br" data-title="Kelc'h" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Circumfer%C3%A8ncia" title="Circumferència – Catalan" lang="ca" hreflang="ca" data-title="Circumferència" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C3%87%D0%B0%D0%B2%D1%80%D0%B0%D0%BA%C4%83%D1%88" title="Çавракăш – Chuvash" lang="cv" hreflang="cv" data-title="Çавракăш" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kru%C5%BEnice" title="Kružnice – Czech" lang="cs" hreflang="cs" data-title="Kružnice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Denderedzwa" title="Denderedzwa – Shona" lang="sn" hreflang="sn" data-title="Denderedzwa" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cylch" title="Cylch – Welsh" lang="cy" hreflang="cy" data-title="Cylch" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Cirkel" title="Cirkel – Danish" lang="da" hreflang="da" data-title="Cirkel" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kreis" 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-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Cera_krejza" title="Cera krejza – Lower Sorbian" lang="dsb" hreflang="dsb" data-title="Cera krejza" data-language-autonym="Dolnoserbski" data-language-local-name="Lower Sorbian" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ringjoon" title="Ringjoon – Estonian" lang="et" hreflang="et" data-title="Ringjoon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%8D%CE%BA%CE%BB%CE%BF%CF%82" title="Κύκλος – Greek" lang="el" hreflang="el" data-title="Κύκλος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Ser%C4%87_(giometr%C3%ACa)" title="Serć (giometrìa) – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Serć (giometrìa)" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/C%C3%ADrculo" title="Círculo – Spanish" lang="es" hreflang="es" data-title="Círculo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Cirklo" title="Cirklo – Esperanto" lang="eo" hreflang="eo" data-title="Cirklo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zirkulu" title="Zirkulu – Basque" lang="eu" hreflang="eu" data-title="Zirkulu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87" title="دایره – Persian" lang="fa" hreflang="fa" data-title="دایره" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Circle" title="Circle – Fiji Hindi" lang="hif" hreflang="hif" data-title="Circle" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Sirkul" title="Sirkul – Faroese" lang="fo" hreflang="fo" data-title="Sirkul" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cercle" title="Cercle – French" lang="fr" hreflang="fr" data-title="Cercle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ciorcal" title="Ciorcal – Irish" lang="ga" hreflang="ga" data-title="Ciorcal" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Kiarkyl" title="Kiarkyl – Manx" lang="gv" hreflang="gv" data-title="Kiarkyl" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Cearcall" title="Cearcall – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Cearcall" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Circunferencia" title="Circunferencia – Galician" lang="gl" hreflang="gl" data-title="Circunferencia" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%9C%93%E5%BD%A2" title="圓形 – Gan" lang="gan" hreflang="gan" data-title="圓形" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B5%E0%AA%B0%E0%AB%8D%E0%AA%A4%E0%AB%81%E0%AA%B3" title="વર્તુળ – Gujarati" lang="gu" hreflang="gu" data-title="વર્તુળ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90_(%EA%B8%B0%ED%95%98%ED%95%99)" title="원 (기하학) – Korean" lang="ko" hreflang="ko" data-title="원 (기하학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%87%D6%80%D5%BB%D5%A1%D5%B6%D5%A1%D5%A3%D5%AB%D5%AE" title="Շրջանագիծ – Armenian" lang="hy" hreflang="hy" data-title="Շրջանագիծ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%83%E0%A4%A4%E0%A5%8D%E0%A4%A4" title="वृत्त – Hindi" lang="hi" hreflang="hi" data-title="वृत्त" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Kružnica" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – Croatian" lang="hr" hreflang="hr" data-title="Kružnica" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Cirklo" title="Cirklo – Ido" lang="io" hreflang="io" data-title="Cirklo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lingkaran" title="Lingkaran – Indonesian" lang="id" hreflang="id" data-title="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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Circulo" title="Circulo – Interlingua" lang="ia" hreflang="ia" data-title="Circulo" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Isazinge" title="Isazinge – Xhosa" lang="xh" hreflang="xh" data-title="Isazinge" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hringur_(r%C3%BAmfr%C3%A6%C3%B0i)" title="Hringur (rúmfræði) – Icelandic" lang="is" hreflang="is" data-title="Hringur (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Circonferenza" title="Circonferenza – Italian" lang="it" hreflang="it" data-title="Circonferenza" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%92%D7%9C" title="מעגל – Hebrew" lang="he" hreflang="he" data-title="מעגל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Bunderan" title="Bunderan – Javanese" lang="jv" hreflang="jv" data-title="Bunderan" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B3%83%E0%B2%A4%E0%B3%8D%E0%B2%A4" title="ವೃತ್ತ – Kannada" lang="kn" hreflang="kn" data-title="ವೃತ್ತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AC%E1%83%A0%E1%83%94%E1%83%AC%E1%83%98%E1%83%A0%E1%83%98" title="წრეწირი – Georgian" lang="ka" hreflang="ka" data-title="წრეწირი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A8%D0%B5%D2%A3%D0%B1%D0%B5%D1%80" title="Шеңбер – Kazakh" lang="kk" hreflang="kk" data-title="Шеңбер" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Kylgh" title="Kylgh – Cornish" lang="kw" hreflang="kw" data-title="Kylgh" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Duara" title="Duara – Swahili" lang="sw" hreflang="sw" data-title="Duara" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/S%C3%A8k_(non)" title="Sèk (non) – Haitian Creole" lang="ht" hreflang="ht" data-title="Sèk (non)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Serk" title="Serk – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Serk" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Gilover" title="Gilover – Kurdish" lang="ku" hreflang="ku" data-title="Gilover" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B9%D0%BB%D0%B0%D0%BD%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Айлана (математика) – Kyrgyz" lang="ky" hreflang="ky" data-title="Айлана (математика)" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Circulus" title="Circulus – Latin" lang="la" hreflang="la" data-title="Circulus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Ri%C5%86%C4%B7a_l%C4%ABnija" title="Riņķa līnija – Latvian" lang="lv" hreflang="lv" data-title="Riņķa līnija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Krees_(Geometrie)" title="Krees (Geometrie) – Luxembourgish" lang="lb" hreflang="lb" data-title="Krees (Geometrie)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Apskritimas" title="Apskritimas – Lithuanian" lang="lt" hreflang="lt" data-title="Apskritimas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Cirkel" title="Cirkel – Limburgish" lang="li" hreflang="li" data-title="Cirkel" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Sirculo" title="Sirculo – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Sirculo" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/S%C3%A9rcc" title="Sércc – Lombard" lang="lmo" hreflang="lmo" data-title="Sércc" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%B6r_(geometria)" title="Kör (geometria) – Hungarian" lang="hu" hreflang="hu" data-title="Kör (geometria)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0" title="Кружница – Macedonian" lang="mk" hreflang="mk" data-title="Кружница" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Faribolana" title="Faribolana – Malagasy" lang="mg" hreflang="mg" data-title="Faribolana" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%83%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%82" title="വൃത്തം – Malayalam" lang="ml" hreflang="ml" data-title="വൃത്തം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A5%81%E0%A4%B3" title="वर्तुळ – Marathi" lang="mr" hreflang="mr" data-title="वर्तुळ" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%AF%D8%A7%D9%8A%D8%B1%D9%87" title="دايره – Egyptian Arabic" lang="arz" hreflang="arz" data-title="دايره" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bulatan" title="Bulatan – Malay" lang="ms" hreflang="ms" data-title="Bulatan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Lingkaran" title="Lingkaran – Minangkabau" lang="min" hreflang="min" data-title="Lingkaran" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B9%D1%80%D0%BE%D0%B3" title="Тойрог – Mongolian" lang="mn" hreflang="mn" data-title="Тойрог" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%85%E1%80%80%E1%80%BA%E1%80%9D%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8" title="စက်ဝိုင်း – Burmese" lang="my" hreflang="my" data-title="စက်ဝိုင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Iwirini" title="Iwirini – Fijian" lang="fj" hreflang="fj" data-title="Iwirini" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Cirkel" title="Cirkel – Dutch" lang="nl" hreflang="nl" data-title="Cirkel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A5%83%E0%A4%A4" title="वृत – Nepali" lang="ne" hreflang="ne" data-title="वृत" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%9A%E0%A4%BE%E0%A4%95%E0%A4%83" title="चाकः – Newari" lang="new" hreflang="new" data-title="चाकः" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%86_(%E6%95%B0%E5%AD%A6)" title="円 (数学) – Japanese" lang="ja" hreflang="ja" data-title="円 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kreis_(Geometrii)" title="Kreis (Geometrii) – Northern Frisian" lang="frr" hreflang="frr" data-title="Kreis (Geometrii)" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sirkel" title="Sirkel – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Sirkel" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Sirkel" title="Sirkel – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Sirkel" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Cercle" title="Cercle – Occitan" lang="oc" hreflang="oc" data-title="Cercle" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9E%D2%A5%D0%B3%D0%BE" title="Оҥго – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Оҥго" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AC%E0%AD%83%E0%AC%A4%E0%AD%8D%E0%AC%A4" title="ବୃତ୍ତ – Odia" lang="or" hreflang="or" data-title="ବୃତ୍ତ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Geengoo" title="Geengoo – Oromo" lang="om" hreflang="om" data-title="Geengoo" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Aylana" title="Aylana – Uzbek" lang="uz" hreflang="uz" data-title="Aylana" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9A%E0%A9%B1%E0%A8%95%E0%A8%B0" title="ਚੱਕਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਚੱਕਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%DB%81" title="دائرہ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="دائرہ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%AF%D8%B1%D8%AF%DA%A9%D9%87" title="گردکه – Pashto" lang="ps" hreflang="ps" data-title="گردکه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Soerkl" title="Soerkl – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Soerkl" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9A%E1%9E%84%E1%9F%92%E1%9E%9C%E1%9E%84%E1%9F%8B" title="រង្វង់ – Khmer" lang="km" hreflang="km" data-title="រង្វង់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Krink" title="Krink – Low German" lang="nds" hreflang="nds" data-title="Krink" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Okr%C4%85g" title="Okrąg – Polish" lang="pl" hreflang="pl" data-title="Okrąg" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Circunfer%C3%AAncia" title="Circunferência – Portuguese" lang="pt" hreflang="pt" data-title="Circunferência" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Cerc" title="Cerc – Romanian" lang="ro" hreflang="ro" data-title="Cerc" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/P%27allta_muyu" title="P'allta muyu – Quechua" lang="qu" hreflang="qu" data-title="P'allta muyu" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B3" title="Круг – Rusyn" lang="rue" hreflang="rue" data-title="Круг" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Окружность – Russian" lang="ru" hreflang="ru" data-title="Окружность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Raing" title="Raing – Scots" lang="sco" hreflang="sco" data-title="Raing" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rrethi" title="Rrethi – Albanian" lang="sq" hreflang="sq" data-title="Rrethi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/C%C3%ACrculu_(giometr%C3%ACa)" title="Cìrculu (giometrìa) – Sicilian" lang="scn" hreflang="scn" data-title="Cìrculu (giometrìa)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Circle" title="Circle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Circle" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AF%D9%88%D9%84" title="گول – Sindhi" lang="sd" hreflang="sd" data-title="گول" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – Slovak" lang="sk" hreflang="sk" data-title="Kružnica" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kro%C5%BEnica" title="Krožnica – Slovenian" lang="sl" hreflang="sl" data-title="Krožnica" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Goobo" title="Goobo – Somali" lang="so" hreflang="so" data-title="Goobo" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%D8%A7%D8%B2%D9%86%DB%95_(%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95)" title="بازنە (ئەندازە) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بازنە (ئەندازە)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0" title="Кружница – Serbian" lang="sr" hreflang="sr" data-title="Кружница" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kružnica" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Bunderan_(%C3%A9lmu_ukur)" title="Bunderan (élmu ukur) – Sundanese" lang="su" hreflang="su" data-title="Bunderan (élmu ukur)" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ympyr%C3%A4" title="Ympyrä – Finnish" lang="fi" hreflang="fi" data-title="Ympyrä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Cirkel" title="Cirkel – Swedish" lang="sv" hreflang="sv" data-title="Cirkel" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Bilog" title="Bilog – Tagalog" lang="tl" hreflang="tl" data-title="Bilog" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="வட்டம் – Tamil" lang="ta" hreflang="ta" data-title="வட்டம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tawinest" title="Tawinest – Kabyle" lang="kab" hreflang="kab" data-title="Tawinest" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D3%98%D0%B9%D0%BB%D3%99%D0%BD%D3%99" title="Әйләнә – Tatar" lang="tt" hreflang="tt" data-title="Әйләнә" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B1%83%E0%B0%A4%E0%B1%8D%E0%B0%A4%E0%B0%AE%E0%B1%81" title="వృత్తము – Telugu" lang="te" hreflang="te" data-title="వృత్తము" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%A7%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1" title="รูปวงกลม – Thai" lang="th" hreflang="th" data-title="รูปวงกลม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87ember" title="Çember – Turkish" lang="tr" hreflang="tr" data-title="Çember" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D0%BE" title="Коло – Ukrainian" lang="uk" hreflang="uk" data-title="Коло" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%DB%81" title="دائرہ – Urdu" lang="ur" hreflang="ur" data-title="دائرہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Luenz" title="Luenz – Zhuang" lang="za" hreflang="za" data-title="Luenz" data-language-autonym="Vahcuengh" data-language-local-name="Zhuang" class="interlanguage-link-target"><span>Vahcuengh</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Sercio" title="Sercio – Venetian" lang="vec" hreflang="vec" data-title="Sercio" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_tr%C3%B2n" title="Đường tròn – Vietnamese" lang="vi" hreflang="vi" data-title="Đường 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-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Ts%C3%B5%C3%B5rjuun" title="Tsõõrjuun – Võro" lang="vro" hreflang="vro" data-title="Tsõõrjuun" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%9C%93" title="圓 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="圓" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Lidong" title="Lidong – Waray" lang="war" hreflang="war" data-title="Lidong" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9C%86" title="圆 – Wu" lang="wuu" hreflang="wuu" data-title="圆" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%A8%D7%99%D7%99%D7%96" title="קרייז – Yiddish" lang="yi" hreflang="yi" data-title="קרייז" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%92b%C3%ACr%C3%ADpo" title="Òbìrípo – Yoruba" lang="yo" hreflang="yo" data-title="Òbìrípo" data-language-autonym="Yorùbá" 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 1.5x" data-file-width="512" data-file-height="512" /></a></span></div></div> </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">Simple curve of Euclidean geometry</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the shape and mathematical concept. For other uses, see <a href="/wiki/Circle_(disambiguation)" class="mw-disambig" title="Circle (disambiguation)">Circle (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <div class="skin-invert-image"><style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3;">Circle</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Circle-withsegments.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/220px-Circle-withsegments.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/330px-Circle-withsegments.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/440px-Circle-withsegments.svg.png 2x" data-file-width="726" data-file-height="726" /></a></span><div class="infobox-caption">A circle <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:black solid 3px;"> </span> circumference <i>C</i></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" /><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:blue solid 2px;"> </span> diameter <i>D</i></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" /><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:red solid 2px;"> </span> radius <i>R</i></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" /><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:green solid 2px;"> </span> centre or origin <i>O</i></div></div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Conic_section" title="Conic section">Conic section</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_planar_symmetry_groups" title="List of planar symmetry groups">Symmetry group</a></th><td class="infobox-data"><a href="/wiki/Orthogonal_group" title="Orthogonal group"><span class="texhtml">O(2)</span></a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Area" title="Area">Area</a></th><td class="infobox-data"><span class="texhtml">πR<sup>2</sup></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Perimeter" title="Perimeter">Perimeter</a></th><td class="infobox-data"><span class="texhtml">C = 2πR</span></td></tr></tbody></table></div> <p>A <b>circle</b> is a <a href="/wiki/Shape" title="Shape">shape</a> consisting of all <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> in a <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> that are at a given distance from a given point, the <a href="/wiki/Centre_(geometry)" title="Centre (geometry)">centre</a>. The distance between any point of the circle and the centre is called the <a href="/wiki/Radius" title="Radius">radius</a>. The length of a line segment connecting two points on the circle and passing through the centre is called the <a href="/wiki/Diameter" title="Diameter">diameter</a>. A circle bounds a region of the plane called a <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disc</a>. </p><p>The circle has been known since before the beginning of recorded history. Natural circles are common, such as the <a href="/wiki/Full_moon" title="Full moon">full moon</a> or a slice of round fruit. The circle is the basis for the <a href="/wiki/Wheel" title="Wheel">wheel</a>, which, with related inventions such as <a href="/wiki/Gear" title="Gear">gears</a>, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> and <a href="/wiki/Calculus" title="Calculus">calculus</a>. </p> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2></div> <ul><li><a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">Annulus</a>: a ring-shaped object, the region bounded by two <a href="/wiki/Concentric" class="mw-redirect" title="Concentric">concentric</a> circles.</li> <li><a href="/wiki/Circular_arc" title="Circular arc">Arc</a>: any <a href="/wiki/Connected_space" title="Connected space">connected</a> part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.</li> <li><a href="/wiki/Centre_(geometry)" title="Centre (geometry)">Centre</a>: the point equidistant from all points on the circle.</li> <li><a href="/wiki/Chord_(geometry)" title="Chord (geometry)">Chord</a>: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.</li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a>: the <a href="/wiki/Length" title="Length">length</a> of one circuit along the circle, or the distance around the circle.</li> <li><a href="/wiki/Diameter" title="Diameter">Diameter</a>: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.</li> <li><a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">Disc</a>: the region of the plane bounded by a circle. In strict mathematical usage, a circle is only the boundary of the disc (or disk), while in everyday use the term "circle" may also refer to a disc.</li> <li><a href="/wiki/Lens_(geometry)" title="Lens (geometry)">Lens</a>: the region common to (the intersection of) two overlapping discs.</li> <li><a href="/wiki/Radius" title="Radius">Radius</a>: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> and required to be a positive number. A circle with <span class="mwe-math-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/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}" /></span> is a <a href="/wiki/Degeneracy_(mathematics)" title="Degeneracy (mathematics)">degenerate case</a> consisting of a single point.</li> <li><a href="/wiki/Circular_sector" title="Circular sector">Sector</a>: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.</li> <li><a href="/wiki/Circular_segment" title="Circular segment">Segment</a>: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term <i>segment</i> is used only for regions not containing the centre of the circle to which their arc belongs.</li> <li><a href="/wiki/Secant_line" title="Secant line">Secant</a>: an extended chord, a coplanar straight line, intersecting a circle in two points.</li> <li><a href="/wiki/Semicircle" title="Semicircle">Semicircle</a>: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.</li> <li><a href="/wiki/Tangent" title="Tangent">Tangent</a>: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point").</li></ul> <p>All of the specified regions may be considered as <i>open</i>, that is, not containing their boundaries, or as <i>closed</i>, including their respective boundaries. </p> <div style="clear:both;" class=""></div> <table class="skin-invert-image" style="float:left;" cellspacing="0" cellpadding="0"> <tbody><tr> <td><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CIRCLE_LINES-en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/220px-CIRCLE_LINES-en.svg.png" decoding="async" width="220" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/330px-CIRCLE_LINES-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/440px-CIRCLE_LINES-en.svg.png 2x" data-file-width="612" data-file-height="618" /></a><figcaption>Chord, secant, tangent, radius, and diameter</figcaption></figure> </td> <td><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_slices_(mul).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Circle_slices_%28mul%29.svg/220px-Circle_slices_%28mul%29.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Circle_slices_%28mul%29.svg/330px-Circle_slices_%28mul%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Circle_slices_%28mul%29.svg/440px-Circle_slices_%28mul%29.svg.png 2x" data-file-width="148" data-file-height="148" /></a><figcaption>Arc, sector, and segment</figcaption></figure> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2></div> <p>The word <i>circle</i> derives from the <a href="/wiki/Greek_language" title="Greek language">Greek</a> κίρκος/κύκλος (<i>kirkos/kuklos</i>), itself a <a href="/wiki/Metathesis_(linguistics)" title="Metathesis (linguistics)">metathesis</a> of the <a href="/wiki/Homeric_Greek" title="Homeric Greek">Homeric Greek</a> κρίκος (<i>krikos</i>), meaning "hoop" or "ring".<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> The origins of the words <i><a href="/wiki/Circus" title="Circus">circus</a></i> and <i><a href="https://en.wiktionary.org/wiki/circuit" class="extiw" title="wikt:circuit">circuit</a></i> are closely related. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2009_07_09_camino_cielo_paradise_137.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/2009_07_09_camino_cielo_paradise_137.jpg/220px-2009_07_09_camino_cielo_paradise_137.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/2009_07_09_camino_cielo_paradise_137.jpg/330px-2009_07_09_camino_cielo_paradise_137.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/2009_07_09_camino_cielo_paradise_137.jpg/440px-2009_07_09_camino_cielo_paradise_137.jpg 2x" data-file-width="3504" data-file-height="2336" /></a><figcaption>Circular cave paintings in <a href="/wiki/Santa_Barbara_County,_California" title="Santa Barbara County, California">Santa Barbara County, California</a></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Shatir500.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Shatir500.jpg/200px-Shatir500.jpg" decoding="async" width="200" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Shatir500.jpg/300px-Shatir500.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Shatir500.jpg/400px-Shatir500.jpg 2x" data-file-width="473" data-file-height="458" /></a><figcaption>Circles in an old <a href="/wiki/Arabic" title="Arabic">Arabic</a> <a href="/wiki/Astronomical" class="mw-redirect" title="Astronomical">astronomical</a> drawing.</figcaption></figure> <p>Prehistoric people made <a href="/wiki/Stone_circle" title="Stone circle">stone circles</a> and <a href="/wiki/Timber_circle" title="Timber circle">timber circles</a>, and circular elements are common in <a href="/wiki/Petroglyph" title="Petroglyph">petroglyphs</a> and <a href="/wiki/Cave_painting" title="Cave painting">cave paintings</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Disc-shaped prehistoric artifacts include the <a href="/wiki/Nebra_sky_disc" title="Nebra sky disc">Nebra sky disc</a> and jade discs called <a href="/wiki/Bi_(jade)" title="Bi (jade)">Bi</a>. </p><p>The Egyptian <a href="/wiki/Rhind_papyrus" class="mw-redirect" title="Rhind papyrus">Rhind papyrus</a>, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to <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">256</span><span class="sr-only">/</span><span class="den">81</span></span>⁠</span> (3.16049...) as an approximate value of <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>.<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><p>Book 3 of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a> deals with the properties of circles. Euclid's definition of a circle is: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.</p><div class="templatequotecite">— <cite><a href="/wiki/Euclid" title="Euclid">Euclid</a>, <a href="/wiki/Euclid%27s_Elements#Contents" title="Euclid's Elements">Book I</a>, <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</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><sup class="reference nowrap"><span title="Page / location: 4">: 4 </span></sup></cite></div></blockquote> <p>In <a href="/wiki/Plato" title="Plato">Plato</a>'s <a href="/wiki/Seventh_Letter" title="Seventh Letter">Seventh Letter</a> there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation. Early <a href="/wiki/Science" title="Science">science</a>, particularly <a href="/wiki/Geometry" title="Geometry">geometry</a> and <a href="/wiki/Astrology_and_astronomy" title="Astrology and astronomy">astrology and astronomy</a>, was connected to the divine for most <a href="/wiki/History_of_science_in_the_Middle_Ages" class="mw-redirect" title="History of science in the Middle Ages">medieval scholars</a>, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.<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><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><p>In 1880 CE, <a href="/wiki/Ferdinand_von_Lindemann" title="Ferdinand von Lindemann">Ferdinand von Lindemann</a> proved that <span class="texhtml mvar" style="font-style:italic;">π</span> is <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>, proving that the millennia-old problem of <a href="/wiki/Squaring_the_circle" title="Squaring the circle">squaring the circle</a> cannot be performed with straightedge and compass.<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><p>With the advent of <a href="/wiki/Abstract_art" title="Abstract art">abstract art</a> in the early 20th century, geometric objects became an artistic subject in their own right. <a href="/wiki/Wassily_Kandinsky" title="Wassily Kandinsky">Wassily Kandinsky</a> in particular often used circles as an element of his compositions.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Symbolism_and_religious_use">Symbolism and religious use</h3></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:God_the_Geometer.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/God_the_Geometer.jpg/200px-God_the_Geometer.jpg" decoding="async" width="200" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/God_the_Geometer.jpg/300px-God_the_Geometer.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/God_the_Geometer.jpg/400px-God_the_Geometer.jpg 2x" data-file-width="1244" data-file-height="1705" /></a><figcaption> The <a href="/wiki/Compass_(drafting)" class="mw-redirect" title="Compass (drafting)">compass</a> in this 13th-century manuscript is a symbol of God's act of <a href="/wiki/Creation_myth" title="Creation myth">Creation</a>. Notice also the circular shape of the <a href="/wiki/Halo_(religious_iconography)" title="Halo (religious iconography)">halo</a>.</figcaption></figure> <p>From the time of the earliest known civilisations – such as the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Greece and Rome during classical Antiquity – the circle has been used directly or indirectly in visual art to convey the artist's message and to express certain ideas. However, differences in worldview (beliefs and culture) had a great impact on artists' perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits. </p><p>The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the <a href="/wiki/Dharmachakra" title="Dharmachakra">Dharma wheel</a>, a rainbow, mandalas, rose windows and so forth.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Magic_circle" title="Magic circle">Magic circles</a> are part of some traditions of <a href="/wiki/Western_esotericism" title="Western esotericism">Western esotericism</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Analytic_results">Analytic results</h2></div> <div class="mw-heading mw-heading3"><h3 id="Circumference">Circumference</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Circumference" title="Circumference">Circumference</a></div> <p>The ratio of a circle's circumference to its diameter is <span class="texhtml mvar" style="font-style:italic;">π</span> (pi), an <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> <a href="/wiki/Mathematical_constant" title="Mathematical constant">constant</a> approximately equal to 3.141592654. The ratio of a circle's circumference to its radius is <span class="texhtml">2<span class="texhtml mvar" style="font-style:italic;">π</span></span>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> Thus the circumference <i>C</i> is related to the radius <i>r</i> and diameter <i>d</i> by: <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 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> <mo>.</mo> </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/aa9aac4480802cc06bc0acb41c0f61ba0a3ff88f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.701ex; height:2.176ex;" alt="{\displaystyle C=2\pi r=\pi d.}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Area_enclosed">Area enclosed</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_Area.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Circle_Area.svg/220px-Circle_Area.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Circle_Area.svg/330px-Circle_Area.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Circle_Area.svg/440px-Circle_Area.svg.png 2x" data-file-width="264" data-file-height="264" /></a><figcaption>Area enclosed by a circle = <span class="texhtml mvar" style="font-style:italic;">π</span> × area of the shaded square</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Area_of_a_circle" title="Area of a circle">Area of a circle</a></div> <p>As proved by <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a>, in his <a href="/wiki/Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a>, the <a href="/wiki/Area_of_a_disk" class="mw-redirect" title="Area of a disk">area enclosed by a circle</a> is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> which comes to <span class="texhtml mvar" style="font-style:italic;">π</span> multiplied by the radius squared: <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 \mathrm {Area} =\pi r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</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 \mathrm {Area} =\pi r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ef0cf29b8b7acadd8281c97891b7937f23c447" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.03ex; height:2.676ex;" alt="{\displaystyle \mathrm {Area} =\pi r^{2}.}" /></span> </p><p>Equivalently, denoting diameter by <i>d</i>, <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 \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0.7854d^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.7854</mn> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0.7854d^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3149c663cd904c664c75336ecce4e482487cfae1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.865ex; height:5.676ex;" alt="{\displaystyle \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0.7854d^{2},}" /></span> that is, approximately 79% of the <a href="/wiki/Circumscribe" class="mw-redirect" title="Circumscribe">circumscribing</a> square (whose side is of length <i>d</i>). </p><p>The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Radian">Radian</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Radian" title="Radian">Radian</a></div> <p>If a circle of radius <span class="texhtml mvar" style="font-style:italic;">r</span> is centred at the <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a> of an <a href="/wiki/Angle" title="Angle">angle</a>, and that angle intercepts an <a href="/wiki/Circular_arc" title="Circular arc">arc of the circle</a> with an <a href="/wiki/Arc_length" title="Arc length">arc length</a> of <span class="texhtml mvar" style="font-style:italic;">s</span>, then the <a href="/wiki/Radian" title="Radian">radian</a> measure 𝜃 of the angle is the ratio of the arc length to the radius: <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 \theta ={\frac {s}{r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>r</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {s}{r}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1217fa85434c2db9527eaf06e0a9f6bced41a571" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.762ex; height:4.676ex;" alt="{\displaystyle \theta ={\frac {s}{r}}.}" /></span> </p><p>The circular arc is said to <a href="/wiki/Subtend" class="mw-redirect" title="Subtend">subtend</a> the angle, known as the <a href="/wiki/Central_angle" title="Central angle">central angle</a>, at the centre of the circle. One radian is the measure of the central angle subtended by a circular arc whose length is equal to its radius. The angle subtended by a complete circle at its centre is a <a href="/wiki/Complete_angle" class="mw-redirect" title="Complete angle">complete angle</a>, which measures <span class="texhtml">2<span class="texhtml mvar" style="font-style:italic;">π</span></span> radians, 360 <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>, or one <a href="/wiki/Turn_(angle)" title="Turn (angle)">turn</a>. </p><p>Using radians, the formula for the arc length <span class="texhtml mvar" style="font-style:italic;">s</span> of a circular arc of radius <span class="texhtml mvar" style="font-style:italic;">r</span> and subtending a central angle of measure 𝜃 is <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 s=\theta r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>θ</mi> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\theta r,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82fc3b382cf726c8084ee043cce5e0f7ac469b01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.975ex; height:2.509ex;" alt="{\displaystyle s=\theta r,}" /></span> </p><p>and the formula for the area <span class="texhtml mvar" style="font-style:italic;">A</span> of a <a href="/wiki/Circular_sector" title="Circular sector">circular sector</a> of radius <span class="texhtml mvar" style="font-style:italic;">r</span> and with central angle of measure 𝜃 is <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 A={\frac {1}{2}}\theta r^{2}.}"> <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> <mi>θ</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={\frac {1}{2}}\theta r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ae83cc06a766866484702e2a52fff8fa087b97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.68ex; height:5.176ex;" alt="{\displaystyle A={\frac {1}{2}}\theta r^{2}.}" /></span> </p><p>In the special case <span class="texhtml">𝜃 = 2<span class="texhtml mvar" style="font-style:italic;">π</span></span>, these formulae yield the circumference of a complete circle and area of a complete disc, respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Equations">Equations</h3></div> <div class="mw-heading mw-heading4"><h4 id="Cartesian_coordinates">Cartesian coordinates</h4></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_center_a_b_radius_r.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Circle_center_a_b_radius_r.svg/250px-Circle_center_a_b_radius_r.svg.png" decoding="async" width="220" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Circle_center_a_b_radius_r.svg/330px-Circle_center_a_b_radius_r.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Circle_center_a_b_radius_r.svg/440px-Circle_center_a_b_radius_r.svg.png 2x" data-file-width="288" data-file-height="260" /></a><figcaption>Circle of radius <i>r</i> = 1, centre (<i>a</i>, <i>b</i>) = (1.2, −0.5)</figcaption></figure> <div class="mw-heading mw-heading5"><h5 id="Equation_of_a_circle">Equation of a circle</h5></div> <p>In an <i>x</i>–<i>y</i> <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>, the circle with centre <a href="/wiki/Coordinate_system" title="Coordinate system">coordinates</a> (<i>a</i>, <i>b</i>) and radius <i>r</i> is the set of all points (<i>x</i>, <i>y</i>) such that <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 (x-a)^{2}+(y-b)^{2}=r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7aac23138efa42dc2ac60d99cd7b440be36f11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.809ex; height:3.176ex;" alt="{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}.}" /></span> </p><p>This <a href="/wiki/Equation" title="Equation">equation</a>, known as the <i>equation of the circle</i>, follows from the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |<i>x</i> − <i>a</i>| and |<i>y</i> − <i>b</i>|. If the circle is centred at the origin (0, 0), then the equation simplifies to <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 x^{2}+y^{2}=r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ed9d9763124224deaef61d1c3e93b8de55d911" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.287ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=r^{2}.}" /></span> </p> <div class="mw-heading mw-heading5"><h5 id="One_coordinate_as_a_function_of_the_other">One coordinate as a function of the other</h5></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_derivative.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Circle_derivative.png/220px-Circle_derivative.png" decoding="async" width="220" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Circle_derivative.png/330px-Circle_derivative.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Circle_derivative.png/440px-Circle_derivative.png 2x" data-file-width="575" data-file-height="862" /></a><figcaption> Upper semicircle with radius <span class="texhtml">1</span> and center <span class="texhtml">(0, 0)</span> and its derivative.</figcaption></figure> <p>The circle of radius <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>⁠</span> with center at <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}" /></span>⁠</span> in the <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>⁠</span>–<span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>⁠</span> plane can be broken into two semicircles each of which is the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph of a function</a>, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{+}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{+}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68be6aa418ae7745834d4896929ab018191581b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle y_{+}(x)}" /></span>⁠</span> and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{-}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{-}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/005a16deda6d722ac5b50143172fbe56f8b0183c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle y_{-}(x)}" /></span>⁠</span>, respectively: <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}y_{+}(x)=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}},\\[5mu]y_{-}(x)=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{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="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{+}(x)=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}},\\[5mu]y_{-}(x)=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/726b5709b6a33583882aa2ea2d615bdbd2231c9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:32.004ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}y_{+}(x)=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}},\\[5mu]y_{-}(x)=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}},\end{aligned}}}" /></span> for values of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>⁠</span> ranging from <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd0baa1ab62fe86d23d145acb05d12512fea8e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.273ex; height:2.343ex;" alt="{\displaystyle x_{0}-r}" /></span>⁠</span> to <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878573944b41f637f3cb5f377363f88056897e5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.273ex; height:2.343ex;" alt="{\displaystyle x_{0}+r}" /></span>⁠</span>. </p> <div class="mw-heading mw-heading5"><h5 id="Parametric_form">Parametric form</h5></div> <p>The equation can be written in <a href="/wiki/Parametric_equation" title="Parametric equation">parametric form</a> using the <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> sine and cosine as <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}x&=a+r\,\cos t,\\y&=b+r\,\sin 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>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=a+r\,\cos t,\\y&=b+r\,\sin t,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9840181971dea6e554b582db7b54e4fbd3f84003" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.057ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}x&=a+r\,\cos t,\\y&=b+r\,\sin t,\end{aligned}}}" /></span> where <i>t</i> is a <a href="/wiki/Parametric_variable" class="mw-redirect" title="Parametric variable">parametric variable</a> in the range 0 to 2<span class="texhtml mvar" style="font-style:italic;">π</span>, interpreted geometrically as the <a href="/wiki/Angle" title="Angle">angle</a> that the ray from (<i>a</i>, <i>b</i>) to (<i>x</i>, <i>y</i>) makes with the positive <i>x</i> axis. </p><p>An alternative parametrisation of the circle is <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}x&=a+r{\frac {1-t^{2}}{1+t^{2}}},\\y&=b+r{\frac {2t}{1+t^{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> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=a+r{\frac {1-t^{2}}{1+t^{2}}},\\y&=b+r{\frac {2t}{1+t^{2}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e638322f1d06b6f98538225cc5b0e96f6fe7244" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.087ex; margin-bottom: -0.251ex; width:17.678ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}x&=a+r{\frac {1-t^{2}}{1+t^{2}}},\\y&=b+r{\frac {2t}{1+t^{2}}}.\end{aligned}}}" /></span> </p><p>In this parameterisation, the ratio of <i>t</i> to <i>r</i> can be interpreted geometrically as the <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a> of the line passing through the centre parallel to the <i>x</i> axis (see <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a>). However, this parameterisation works only if <i>t</i> is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted. </p> <div class="mw-heading mw-heading5"><h5 id="3-point_form">3-point form</h5></div> <p>The equation of the circle determined by three points <span class="mwe-math-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_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201df5d993a5f9d4819a624c9d878cf81bac37c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.33ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})}" /></span> not on a line is obtained by a conversion of the <a href="/wiki/Ellipse#Circles" title="Ellipse"><i>3-point form of a circle equation</i></a>: <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 {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="green"> <mi>x</mi> </mstyle> </mrow> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="green"> <mi>x</mi> </mstyle> </mrow> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>y</mi> </mstyle> </mrow> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>y</mi> </mstyle> </mrow> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>y</mi> </mstyle> </mrow> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="green"> <mi>x</mi> </mstyle> </mrow> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>y</mi> </mstyle> </mrow> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="green"> <mi>x</mi> </mstyle> </mrow> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6670134ffaccf00af267de7ad6c78983fe975e11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:80.731ex; height:6.509ex;" alt="{\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.}" /></span> </p> <div class="mw-heading mw-heading5"><h5 id="Homogeneous_form">Homogeneous form</h5></div> <p>In <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>, each <a href="/wiki/Conic_section" title="Conic section">conic section</a> with the equation of a circle has the form <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 x^{2}+y^{2}-2axz-2byz+cz^{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>x</mi> <mi>z</mi> <mo>−</mo> <mn>2</mn> <mi>b</mi> <mi>y</mi> <mi>z</mi> <mo>+</mo> <mi>c</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-2axz-2byz+cz^{2}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a6e471f2edc1b3f74b9c1449f2a47ea27fe1c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.233ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-2axz-2byz+cz^{2}=0.}" /></span> </p><p>It can be proven that a conic section is a circle exactly when it contains (when extended to the <a href="/wiki/Complex_projective_plane" title="Complex projective plane">complex projective plane</a>) the points <i>I</i>(1: <i>i</i>: 0) and <i>J</i>(1: −<i>i</i>: 0). These points are called the <a href="/wiki/Circular_points_at_infinity" title="Circular points at infinity">circular points at infinity</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Polar_coordinates">Polar coordinates</h4></div> <p>In <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>, the equation of a circle is <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 r^{2}-2rr_{0}\cos(\theta -\phi )+r_{0}^{2}=a^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}-2rr_{0}\cos(\theta -\phi )+r_{0}^{2}=a^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1905137531d7f246c5c5a84c931c1d7dc2ed7b95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.854ex; height:3.343ex;" alt="{\displaystyle r^{2}-2rr_{0}\cos(\theta -\phi )+r_{0}^{2}=a^{2},}" /></span> </p><p>where <i>a</i> is the radius 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 (r,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8396fdc359fb06c93722137c959e7496e47ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.982ex; height:2.843ex;" alt="{\displaystyle (r,\theta )}" /></span> are the polar coordinates of a generic point on the circle, 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 (r_{0},\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{0},\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b336f1862eef349e2c5acc16e84b8f7ea2ad848d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.332ex; height:2.843ex;" alt="{\displaystyle (r_{0},\phi )}" /></span> are the polar coordinates of the centre of the circle (i.e., <i>r</i><sub>0</sub> is the distance from the origin to the centre of the circle, and <i>φ</i> is the anticlockwise angle from the positive <i>x</i> axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e. <span class="nowrap"><i>r</i><sub>0</sub> = 0</span>, this reduces to <span class="nowrap"><i>r</i> = <i>a</i></span>. When <span class="nowrap"><i>r</i><sub>0</sub> = <i>a</i></span>, or when the origin lies on the circle, the equation becomes <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 r=2a\cos(\theta -\phi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=2a\cos(\theta -\phi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3197ba78f76fdc04355f8d6b313116cdee6de94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.81ex; height:2.843ex;" alt="{\displaystyle r=2a\cos(\theta -\phi ).}" /></span> </p><p>In the general case, the equation can be solved for <i>r</i>, giving <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 r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi )}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8bcc4b2b107cbf433eeaa44a8b4f9b1ff6c7c12" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:41.335ex; height:4.676ex;" alt="{\displaystyle r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi )}}.}" /></span> Without the ± sign, the equation would in some cases describe only half a circle. </p> <div class="mw-heading mw-heading4"><h4 id="Complex_plane">Complex plane</h4></div> <p>In the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, a circle with a centre at <i>c</i> and radius <i>r</i> has the equation <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 |z-c|=r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z-c|=r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f86aa96090f1028c9f90fcb9d29294a8999e7956" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.023ex; height:2.843ex;" alt="{\displaystyle |z-c|=r.}" /></span> </p><p>In parametric form, this can be written as <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 z=re^{it}+c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=re^{it}+c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180d3a26356d91aa889bcca534f8acebbea29c30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.206ex; height:2.843ex;" alt="{\displaystyle z=re^{it}+c.}" /></span> </p><p>The slightly generalised equation <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 pz{\overline {z}}+gz+{\overline {gz}}=q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mi>g</mi> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mi>z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/028f215ad8f29eeb316058469a7fe70c4482dc4d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.931ex; height:2.676ex;" alt="{\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}" /></span> </p><p>for real <i>p</i>, <i>q</i> and complex <i>g</i> is sometimes called a <a href="/wiki/Generalised_circle" title="Generalised circle">generalised circle</a>. This becomes the above equation for a circle with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mi>g</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mtext> </mtext> <mi>q</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e322c33ee9ecf2c3df4e4e1d36b846a49e5871aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:28.359ex; height:3.343ex;" alt="{\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}}" /></span>, 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 |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <mi>c</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>z</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2994155cf77d92fba4111d6370a8af0a1b4fccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.752ex; height:3.343ex;" alt="{\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}" /></span>. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Tangent_lines">Tangent lines</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Tangent_lines_to_circles" title="Tangent lines to circles">Tangent lines to circles</a></div> <p>The <a href="/wiki/Tangent_line" class="mw-redirect" title="Tangent line">tangent line</a> through a point <i>P</i> on the circle is perpendicular to the diameter passing through <i>P</i>. If <span class="nowrap">P = (<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>)</span> and the circle has centre (<i>a</i>, <i>b</i>) and radius <i>r</i>, then the tangent line is perpendicular to the line from (<i>a</i>, <i>b</i>) to (<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>), so it has the form <span class="nowrap">(<i>x</i><sub>1</sub> − <i>a</i>)<i>x</i> + (<i>y</i><sub>1</sub> – <i>b</i>)<i>y</i> = <i>c</i></span>. Evaluating at (<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>) determines the value of <i>c</i>, and the result is that the equation of the tangent is <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 (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87da79769ef2120d4750e7e4a5dc3ba850812f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.697ex; height:2.843ex;" alt="{\displaystyle (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},}" /></span> or <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 (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e0410363bb05276475d1cfe3a1a917e6a24f35" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.805ex; height:3.176ex;" alt="{\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}" /></span> </p><p>If <span class="nowrap"><i>y</i><sub>1</sub> ≠ <i>b</i></span>, then the slope of this line is <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 {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca37a31033bf89ed061c49ec1273f0b7b40183e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.225ex; height:5.843ex;" alt="{\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.}" /></span> </p><p>This can also be found using <a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">implicit differentiation</a>. </p><p>When the centre of the circle is at the origin, then the equation of the tangent line becomes <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 x_{1}x+y_{1}y=r^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}x+y_{1}y=r^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e42f7fab445f44d26e2eeecfba5efdf1b81269f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.751ex; height:3.009ex;" alt="{\displaystyle x_{1}x+y_{1}y=r^{2},}" /></span> and its slope is <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 {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3ecb0414f1926d49414ab6333f93282ab346f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.155ex; height:5.843ex;" alt="{\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div> <ul><li>The circle is the shape with the largest area for a given length of perimeter (see <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a>).</li> <li>The circle is a highly symmetric shape: every line through the centre forms a line of <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a>, and it has <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> around the centre for every angle. Its <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> is the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(2,<i>R</i>). The group of rotations alone is the <a href="/wiki/Circle_group" title="Circle group">circle group</a> <b>T</b>.</li> <li>All circles are <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <ul><li>A circle circumference and radius are <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a>.</li> <li>The <a href="/wiki/Area_(geometry)" class="mw-redirect" title="Area (geometry)">area</a> enclosed and the square of its radius are proportional.</li> <li>The constants of proportionality are 2<span class="texhtml mvar" style="font-style:italic;">π</span> and <span class="texhtml mvar" style="font-style:italic;">π</span> respectively.</li></ul></li> <li>The circle that is centred at the origin with radius 1 is called the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>. <ul><li>Thought of as a <a href="/wiki/Great_circle" title="Great circle">great circle</a> of the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a>, it becomes the <a href="/wiki/Riemannian_circle" class="mw-redirect" title="Riemannian circle">Riemannian circle</a>.</li></ul></li> <li>Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See <a href="/wiki/Circumcircle" title="Circumcircle">circumcircle</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Chord">Chord</h3></div> <ul><li>Chords are equidistant from the centre of a circle if and only if they are equal in length.</li> <li>The <a href="/wiki/Perpendicular_bisector" class="mw-redirect" title="Perpendicular bisector">perpendicular bisector</a> of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are: <ul><li>A perpendicular line from the centre of a circle bisects the chord.</li> <li>The <a href="/wiki/Line_segment" title="Line segment">line segment</a> through the centre bisecting a chord is <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the chord.</li></ul></li> <li>If a central angle and an <a href="/wiki/Inscribed_angle" title="Inscribed angle">inscribed angle</a> of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.</li> <li>If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.</li> <li>If two angles are inscribed on the same chord and on opposite sides of the chord, then they are <a href="/wiki/Supplementary_angles" class="mw-redirect" title="Supplementary angles">supplementary</a>. <ul><li>For a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a>, the <a href="/wiki/Exterior_angle" class="mw-redirect" title="Exterior angle">exterior angle</a> is equal to the interior opposite angle.</li></ul></li> <li>An inscribed angle subtended by a diameter is a right angle (see <a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a>).</li> <li>The diameter is the longest chord of the circle. <ul><li>Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.</li></ul></li> <li>If the <a href="/wiki/Intersecting_chords_theorem" title="Intersecting chords theorem">intersection of any two chords</a> divides one chord into lengths <i>a</i> and <i>b</i> and divides the other chord into lengths <i>c</i> and <i>d</i>, then <span class="nowrap"><i>ab</i> = <i>cd</i></span>.</li> <li>If the intersection of any two perpendicular chords divides one chord into lengths <i>a</i> and <i>b</i> and divides the other chord into lengths <i>c</i> and <i>d</i>, then <span class="nowrap"><i>a</i><sup>2</sup> + <i>b</i><sup>2</sup> + <i>c</i><sup>2</sup> + <i>d</i><sup>2</sup></span> equals the square of the diameter.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li>The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8<i>r</i><sup>2</sup> − 4<i>p</i><sup>2</sup>, where <i>r</i> is the circle radius, and <i>p</i> is the distance from the centre point to the point of intersection.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li> <li>The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.71">: p.71 </span></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Tangent">Tangent</h3></div> <ul><li>A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.</li> <li>A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.</li> <li>Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.</li> <li>If a tangent at <i>A</i> and a tangent at <i>B</i> intersect at the exterior point <i>P</i>, then denoting the centre as <i>O</i>, the angles ∠<i>BOA</i> and ∠<i>BPA</i> are supplementary.</li> <li>If <i>AD</i> is tangent to the circle at <i>A</i> and if <i>AQ</i> is a chord of the circle, then <span class="nowrap">∠<i>DAQ</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>arc(<i>AQ</i>)</span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Theorems">Theorems</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Secant-Secant_Theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Secant-Secant_Theorem.svg/220px-Secant-Secant_Theorem.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Secant-Secant_Theorem.svg/330px-Secant-Secant_Theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Secant-Secant_Theorem.svg/440px-Secant-Secant_Theorem.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>Secant–secant theorem</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Power_of_a_point" title="Power of a point">Power of a point</a></div> <ul><li>The chord theorem states that if two chords, <i>CD</i> and <i>EB</i>, intersect at <i>A</i>, then <span class="nowrap"><i>AC</i> × <i>AD</i> = <i>AB</i> × <i>AE</i></span>.</li> <li>If two secants, <i>AE</i> and <i>AD</i>, also cut the circle at <i>B</i> and <i>C</i> respectively, then <span class="nowrap"><i>AC</i> × <i>AD</i> = <i>AB</i> × <i>AE</i></span> (corollary of the chord theorem).</li> <li><span class="anchor" id="Tangent-secant_theorem"></span>A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point <i>A</i> meets the circle at <i>F</i> and a secant from the external point <i>A</i> meets the circle at <i>C</i> and <i>D</i> respectively, then <span class="nowrap"><i>AF</i><sup>2</sup> = <i>AC</i> × <i>AD</i></span> (tangent–secant theorem).</li> <li>The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle).</li> <li>If the angle subtended by the chord at the centre is 90<a href="/wiki/Degree_(angle)" title="Degree (angle)">°</a>, then <span class="nowrap"><i>ℓ</i> = <i>r</i> √2</span>, where <i>ℓ</i> is the length of the chord, and <i>r</i> is the radius of the circle.</li> <li><span class="anchor" id="Secant-secant_theorem"></span>If two secants are inscribed in the circle as shown at right, then the measurement of angle <i>A</i> is equal to one half the difference of the measurements of the enclosed arcs (<span class="mwe-math-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 {\overset {\frown }{DE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>D</mi> <mi>E</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{DE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ccf660369a3a705bff5c9fd879707b0d404a56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.7ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{DE}}}" /></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 {\overset {\frown }{BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d5cffe92c89482266f356039a49ba3d4333277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{BC}}}" /></span>). That 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\angle {CAB}=\angle {DOE}-\angle {BOC}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> <mi>B</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>O</mi> <mi>E</mi> </mrow> <mo>−</mo> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>O</mi> <mi>C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b56445a07a4e59a22a790a21a2b2fbbb08b2a569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.186ex; height:2.343ex;" alt="{\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}}" /></span>, where <i>O</i> is the centre of the circle (secant–secant theorem).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Inscribed_angles">Inscribed angles</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Inscribed_angle_theorem" class="mw-redirect" title="Inscribed angle theorem">Inscribed angle theorem</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Inscribed_angle_theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Inscribed_angle_theorem.svg/250px-Inscribed_angle_theorem.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Inscribed_angle_theorem.svg/330px-Inscribed_angle_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Inscribed_angle_theorem.svg/400px-Inscribed_angle_theorem.svg.png 2x" data-file-width="470" data-file-height="470" /></a><figcaption>Inscribed-angle theorem</figcaption></figure> <p>An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding <a href="/wiki/Central_angle" title="Central angle">central angle</a> (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a <a href="/wiki/Right_angle" title="Right angle">right angle</a> (since the central angle is 180°). </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Sagitta">Sagitta</h3></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_Sagitta.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Circle_Sagitta.svg/277px-Circle_Sagitta.svg.png" decoding="async" width="277" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Circle_Sagitta.svg/416px-Circle_Sagitta.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Circle_Sagitta.svg/554px-Circle_Sagitta.svg.png 2x" data-file-width="540" data-file-height="195" /></a><figcaption>The sagitta is the vertical segment.</figcaption></figure> <p>The <a href="/wiki/Sagitta_(geometry)" title="Sagitta (geometry)">sagitta</a> (also known as the <a href="/wiki/Versine" title="Versine">versine</a>) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle. </p><p>Given the length <i>y</i> of a chord and the length <i>x</i> of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines: <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 r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>8</mn> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd79ddf39cdc709f20f2ce73a0c08747f976af24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.128ex; height:5.676ex;" alt="{\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.}" /></span> </p><p>Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of length <i>y</i> and with sagitta of length <i>x</i>, since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is (<span class="nowrap">2<i>r</i> − <i>x</i></span>) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (<span class="nowrap">2<i>r</i> − <i>x</i>)<i>x</i> = (<i>y</i> / 2)<sup>2</sup></span>. Solving for <i>r</i>, we find the required result. </p> <div class="mw-heading mw-heading2"><h2 id="Compass_and_straightedge_constructions">Compass and straightedge constructions</h2></div> <p>There are many <a href="/wiki/Compass-and-straightedge_construction" class="mw-redirect" title="Compass-and-straightedge construction">compass-and-straightedge constructions</a> resulting in circles. </p><p>The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the <a href="/wiki/Compass_(drawing_tool)" title="Compass (drawing tool)">compass</a> on the centre point, the movable leg on the point on the circle and rotate the compass. </p> <div class="mw-heading mw-heading3"><h3 id="Construction_with_given_diameter">Construction with given diameter</h3></div> <ul><li>Construct the <a href="/wiki/Midpoint" title="Midpoint">midpoint</a> <span class="texhtml"><b>M</b></span> of the diameter.</li> <li>Construct the circle with centre <span class="texhtml"><b>M</b></span> passing through one of the endpoints of the diameter (it will also pass through the other endpoint).</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Circunferencia_10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Circunferencia_10.svg/220px-Circunferencia_10.svg.png" decoding="async" width="220" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Circunferencia_10.svg/330px-Circunferencia_10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Circunferencia_10.svg/440px-Circunferencia_10.svg.png 2x" data-file-width="900" data-file-height="700" /></a><figcaption>Construct a circle through points A, B and C by finding the perpendicular bisectors (red) of the sides of the triangle (blue). Only two of the three bisectors are needed to find the centre.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Construction_through_three_noncollinear_points">Construction through three noncollinear points</h3></div> <ul><li>Name the points <span class="texhtml"><b>P</b></span>, <span class="texhtml"><b>Q</b></span> and <span class="texhtml"><b>R</b></span>,</li> <li>Construct the <a href="/wiki/Perpendicular_bisector" class="mw-redirect" title="Perpendicular bisector">perpendicular bisector</a> of the segment <span class="texhtml"><span style="text-decoration:overline;"><b>PQ</b></span></span>.</li> <li>Construct the <a href="/wiki/Perpendicular_bisector" class="mw-redirect" title="Perpendicular bisector">perpendicular bisector</a> of the segment <span class="texhtml"><span style="text-decoration:overline;"><b>PR</b></span></span>.</li> <li>Label the point of intersection of these two perpendicular bisectors <span class="texhtml"><b>M</b></span>. (They meet because the points are not <a href="/wiki/Collinear" class="mw-redirect" title="Collinear">collinear</a>).</li> <li>Construct the circle with centre <span class="texhtml"><b>M</b></span> passing through one of the points <span class="texhtml"><b>P</b></span>, <span class="texhtml"><b>Q</b></span> or <span class="texhtml"><b>R</b></span> (it will also pass through the other two points).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Circle_of_Apollonius">Circle of Apollonius</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Circles_of_Apollonius" title="Circles of Apollonius">Circles of Apollonius</a></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Apollonius_circle_definition_labels.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Apollonius_circle_definition_labels.svg/250px-Apollonius_circle_definition_labels.svg.png" decoding="async" width="250" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Apollonius_circle_definition_labels.svg/375px-Apollonius_circle_definition_labels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Apollonius_circle_definition_labels.svg/500px-Apollonius_circle_definition_labels.svg.png 2x" data-file-width="228" data-file-height="153" /></a><figcaption>Apollonius' definition of a circle: <span class="nowrap"><i>d</i><sub>1</sub>/<i>d</i><sub>2</sub></span> constant</figcaption></figure> <p><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a> showed that a circle may also be defined as the set of points in a plane having a constant <i>ratio</i> (other than 1) of distances to two fixed foci, <i>A</i> and <i>B</i>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> (The set of points where the distances are equal is the perpendicular bisector of segment <i>AB</i>, a line.) That circle is sometimes said to be drawn <i>about</i> two points. </p><p>The proof is in two parts. First, one must prove that, given two foci <i>A</i> and <i>B</i> and a ratio of distances, any point <i>P</i> satisfying the ratio of distances must fall on a particular circle. Let <i>C</i> be another point, also satisfying the ratio and lying on segment <i>AB</i>. By the <a href="/wiki/Angle_bisector_theorem" title="Angle bisector theorem">angle bisector theorem</a> the line segment <i>PC</i> will bisect the <a href="/wiki/Interior_angle" class="mw-redirect" title="Interior angle">interior angle</a> <i>APB</i>, since the segments are similar: <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 {\frac {AP}{BP}}={\frac {AC}{BC}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>P</mi> </mrow> <mrow> <mi>B</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {AP}{BP}}={\frac {AC}{BC}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4508f34276933287249e21f514504bc4e9429f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.457ex; height:5.509ex;" alt="{\displaystyle {\frac {AP}{BP}}={\frac {AC}{BC}}.}" /></span> </p><p>Analogously, a line segment <i>PD</i> through some point <i>D</i> on <i>AB</i> extended bisects the corresponding exterior angle <i>BPQ</i> where <i>Q</i> is on <i>AP</i> extended. Since the interior and exterior angles sum to 180 degrees, the angle <i>CPD</i> is exactly 90 degrees; that is, a right angle. The set of points <i>P</i> such that angle <i>CPD</i> is a right angle forms a circle, of which <i>CD</i> is a diameter. </p><p>Second, see<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 15">: 15 </span></sup> for a proof that every point on the indicated circle satisfies the given ratio. </p> <div class="mw-heading mw-heading3"><h3 id="Cross-ratios">Cross-ratios</h3></div> <p>A closely related property of circles involves the geometry of the <a href="/wiki/Cross-ratio" title="Cross-ratio">cross-ratio</a> of points in the complex plane. If <i>A</i>, <i>B</i>, and <i>C</i> are as above, then the circle of Apollonius for these three points is the collection of points <i>P</i> for which the absolute value of the cross-ratio is equal to one: <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 {\bigl |}[A,B;C,P]{\bigr |}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>;</mo> <mi>C</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl |}[A,B;C,P]{\bigr |}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d29d2d228c14547650d9f23bc20322d363157303" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.616ex; height:3.176ex;" alt="{\displaystyle {\bigl |}[A,B;C,P]{\bigr |}=1.}" /></span> </p><p>Stated another way, <i>P</i> is a point on the circle of Apollonius if and only if the cross-ratio <span class="nowrap">[<i>A</i>, <i>B</i>; <i>C</i>, <i>P</i>]</span> is on the unit circle in the complex plane. </p> <div class="mw-heading mw-heading3"><h3 id="Generalised_circles"><span class="anchor" id="Generalized_circles"></span> Generalised circles</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Generalised_circle" title="Generalised circle">Generalised circle</a></div> <p>If <i>C</i> is the midpoint of the segment <i>AB</i>, then the collection of points <i>P</i> satisfying the Apollonius condition <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 {\frac {|AP|}{|BP|}}={\frac {|AC|}{|BC|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AC|}{|BC|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e86b9ad901d924803c59778e69fd49c2617bab0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.398ex; height:6.509ex;" alt="{\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AC|}{|BC|}}}" /></span> is not a circle, but rather a line. </p><p>Thus, if <i>A</i>, <i>B</i>, and <i>C</i> are given distinct points in the plane, then the <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of points <i>P</i> satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius. </p> <div class="mw-heading mw-heading2"><h2 id="Inscription_in_or_circumscription_about_other_figures">Inscription in or circumscription about other figures</h2></div> <p>In every <a href="/wiki/Triangle" title="Triangle">triangle</a> a unique circle, called the <a href="/wiki/Incircle_and_excircles_of_a_triangle" class="mw-redirect" title="Incircle and excircles of a triangle">incircle</a>, can be inscribed such that it is tangent to each of the three sides of the triangle.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Tangential_polygon" title="Tangential polygon">tangential polygon</a>, such as a <a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">tangential quadrilateral</a>, is any <a href="/wiki/Convex_polygon" title="Convex polygon">convex polygon</a> within which a <a href="/wiki/Inscribed_circle" class="mw-redirect" title="Inscribed circle">circle can be inscribed</a> that is tangent to each side of the polygon.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Every <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a> and every triangle is a tangential polygon. </p><p>A <a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">cyclic polygon</a> is any convex polygon about which a <a href="/wiki/Circumcircle" title="Circumcircle">circle can be circumscribed</a>, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a <a href="/wiki/Bicentric_polygon" title="Bicentric polygon">bicentric polygon</a>. </p><p>A <a href="/wiki/Hypocycloid" title="Hypocycloid">hypocycloid</a> is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle. </p> <div class="mw-heading mw-heading2"><h2 id="Limiting_case_of_other_figures">Limiting case of other figures</h2></div> <p>The circle can be viewed as a <a href="/wiki/Limiting_case_(mathematics)" title="Limiting case (mathematics)">limiting case</a> of various other figures: </p> <ul><li>The series of <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> with <i>n</i> sides has the circle as its limit as <i>n</i> approaches infinity. This fact was applied by <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> to <a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">approximate π</a>.</li> <li>A <a href="/wiki/Cartesian_oval" title="Cartesian oval">Cartesian oval</a> is a set of points such that a <a href="/wiki/Weighted_sum" class="mw-redirect" title="Weighted sum">weighted sum</a> of the distances from any of its points to two fixed points (foci) is a constant. An <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero.</li> <li>A <a href="/wiki/Superellipse" title="Superellipse">superellipse</a> has an equation of the form <span class="mwe-math-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 \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="negativethinmathspace"></mspace> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>b</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="negativethinmathspace"></mspace> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7278d088496cad8d362fa0d40d71076bd39d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.509ex; height:5.176ex;" alt="{\displaystyle \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=1}" /></span> for positive <i>a</i>, <i>b</i>, and <i>n</i>. A supercircle has <span class="nowrap"><i>b</i> = <i>a</i></span>. A circle is the special case of a supercircle in which <span class="nowrap"><i>n</i> = 2</span>.</li> <li>A <a href="/wiki/Cassini_oval" title="Cassini oval">Cassini oval</a> is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results.</li> <li>A <a href="/wiki/Curve_of_constant_width" title="Curve of constant width">curve of constant width</a> is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Locus_of_constant_sum">Locus of constant sum</h2></div> <p>Consider a finite set 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> points in the plane. The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose centre is at the centroid of the given points.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> A generalisation for higher powers of distances is obtained if, instead 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> points, the vertices of the regular polygon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5949c8b1de44005a1af3a11188361f2a830842d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.711ex; height:2.509ex;" alt="{\displaystyle P_{n}}" /></span> are taken.<sup id="cite_ref-Mamuka_24-0" class="reference"><a href="#cite_note-Mamuka-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The locus of points such that the sum of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e100f32f96dc84bf0591df4f5c5bd40d71189f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle 2m}" /></span>-th power of distances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}" /></span> to the vertices of a given regular polygon with circumradius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> is constant is a circle, if <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 \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m},\quad {\text{ where }}~m=1,2,\dots ,n-1;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msubsup> <mo>></mo> <mi>n</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <mtext> </mtext> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m},\quad {\text{ where }}~m=1,2,\dots ,n-1;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e440f93cf80c77be882ba593f8ce43dba5aa065d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.977ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m},\quad {\text{ where }}~m=1,2,\dots ,n-1;}" /></span> whose centre is the centroid of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5949c8b1de44005a1af3a11188361f2a830842d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.711ex; height:2.509ex;" alt="{\displaystyle P_{n}}" /></span>. </p><p>In the case of the <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>, the loci of the constant sums of the second and fourth powers are circles, whereas for the square, the loci are circles for the constant sums of the second, fourth, and sixth powers. For the <a href="/wiki/Regular_pentagon" class="mw-redirect" title="Regular pentagon">regular pentagon</a> the constant sum of the eighth powers of the distances will be added and so forth. </p> <div class="mw-heading mw-heading2"><h2 id="Squaring_the_circle">Squaring the circle</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></div> <p>Squaring the circle is the problem, proposed by <a href="/wiki/Classical_antiquity" title="Classical antiquity">ancient</a> <a href="/wiki/Geometers" class="mw-redirect" title="Geometers">geometers</a>, of constructing a square with the same area as a given circle by using only a finite number of steps with <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a>. </p><p>In 1882, the task was proven to be impossible, as a consequence of the <a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a>, which proves that pi (<span class="texhtml mvar" style="font-style:italic;">π</span>) is a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental number</a>, rather than an <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic irrational number</a>; that is, it is not the <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> of any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with <a href="/wiki/Rational_number" title="Rational number">rational</a> coefficients. Despite the impossibility, this topic continues to be of interest for <a href="/wiki/Pseudomath" class="mw-redirect" title="Pseudomath">pseudomath</a> enthusiasts. </p> <div class="mw-heading mw-heading2"><h2 id="Generalisations">Generalisations</h2></div> <div class="mw-heading mw-heading3"><h3 id="In_other_p-norms">In other <i>p</i>-norms</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vector-p-Norms_qtl1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Vector-p-Norms_qtl1.svg/220px-Vector-p-Norms_qtl1.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Vector-p-Norms_qtl1.svg/330px-Vector-p-Norms_qtl1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Vector-p-Norms_qtl1.svg/440px-Vector-p-Norms_qtl1.svg.png 2x" data-file-width="410" data-file-height="410" /></a><figcaption>Illustrations of unit circles (see also <a href="/wiki/Superellipse" title="Superellipse">superellipse</a>) in different <span class="texhtml"><i>p</i></span>-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding <span class="texhtml"><i>p</i></span>).</figcaption></figure> <p>Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In <a href="/wiki/P-norm" class="mw-redirect" title="P-norm"><i>p</i>-norm</a>, distance is determined by <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 \left\|x\right\|_{p}=\left(\left|x_{1}\right|^{p}+\left|x_{2}\right|^{p}+\dotsb +\left|x_{n}\right|^{p}\right)^{1/p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo symmetric="true">‖</mo> <mi>x</mi> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|x\right\|_{p}=\left(\left|x_{1}\right|^{p}+\left|x_{2}\right|^{p}+\dotsb +\left|x_{n}\right|^{p}\right)^{1/p}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b1c7d79b79edd195edec9623a7e2e3a80df4ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.59ex; height:3.843ex;" alt="{\displaystyle \left\|x\right\|_{p}=\left(\left|x_{1}\right|^{p}+\left|x_{2}\right|^{p}+\dotsb +\left|x_{n}\right|^{p}\right)^{1/p}.}" /></span> In Euclidean geometry, <i>p</i> = 2, giving the familiar <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 \left\|x\right\|_{2}={\sqrt {\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}+\dotsb +\left|x_{n}\right|^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo symmetric="true">‖</mo> <mi>x</mi> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|x\right\|_{2}={\sqrt {\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}+\dotsb +\left|x_{n}\right|^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923b8c205fc0561ae767f83ffebd6fcf3695d4b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:36.382ex; height:4.843ex;" alt="{\displaystyle \left\|x\right\|_{2}={\sqrt {\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}+\dotsb +\left|x_{n}\right|^{2}}}.}" /></span> </p><p>In <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">taxicab geometry</a>, <i>p</i> = 1. Taxicab circles are <a href="/wiki/Square" title="Square">squares</a> with sides oriented at a 45° angle to the coordinate axes. While each side would have length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4f35f5d92df4b263980763817a1a89756b413f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.147ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}r}" /></span> using a <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>, where <i>r</i> is the circle's radius, its length in taxicab geometry is 2<i>r</i>. Thus, a circle's circumference is 8<i>r</i>. Thus, the value of a geometric analog 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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span> is 4 in this geometry. The formula for the unit circle in taxicab geometry 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 |x|+|y|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|+|y|=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09db8ee8cbfafa057da38d82c651787ecbef8f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.174ex; height:2.843ex;" alt="{\displaystyle |x|+|y|=1}" /></span> in Cartesian coordinates and <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 r={\frac {1}{\left|\sin \theta \right|+\left|\cos \theta \right|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow> <mo>|</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {1}{\left|\sin \theta \right|+\left|\cos \theta \right|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0981b4d405ef1165e94c0cbfac64694a5008e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.333ex; height:6.009ex;" alt="{\displaystyle r={\frac {1}{\left|\sin \theta \right|+\left|\cos \theta \right|}}}" /></span> in polar coordinates. </p><p>A circle of radius 1 (using this distance) is the <a href="/wiki/Von_Neumann_neighborhood" title="Von Neumann neighborhood">von Neumann neighborhood</a> of its centre. </p><p>A circle of radius <i>r</i> for the <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a> (<a href="/wiki/Lp_space" title="Lp space"><i>L</i><sub>∞</sub> metric</a>) on a plane is also a square with side length 2<i>r</i> parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between <i>L</i><sub>1</sub> and <i>L</i><sub>∞</sub> metrics does not generalise to higher dimensions. </p> <div class="mw-heading mw-heading3"><h3 id="Topological_definition">Topological definition</h3></div> <p>The circle is the <a href="/wiki/One-dimensional" class="mw-redirect" title="One-dimensional">one-dimensional</a> <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">hypersphere</a> (the 1-sphere). </p><p>In <a href="/wiki/Topology" title="Topology">topology</a>, a circle is not limited to the geometric concept, but to all of its <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a>. Two topological circles are equivalent if one can be transformed into the other via a deformation of <a href="/wiki/Real_coordinate_space" title="Real coordinate space"><b>R</b><sup>3</sup></a> upon itself (known as an <a href="/wiki/Ambient_isotopy" title="Ambient isotopy">ambient isotopy</a>).<sup id="cite_ref-gamelin_25-0" class="reference"><a href="#cite_note-gamelin-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Specially_named_circles">Specially named circles</h2></div> <style data-mw-deduplicate="TemplateStyles:r1216972533">.mw-parser-output .col-begin{border-collapse:collapse;padding:0;color:inherit;width:100%;border:0;margin:0}.mw-parser-output .col-begin-small{font-size:90%}.mw-parser-output .col-break{vertical-align:top;text-align:left}.mw-parser-output .col-break-2{width:50%}.mw-parser-output .col-break-3{width:33.3%}.mw-parser-output .col-break-4{width:25%}.mw-parser-output .col-break-5{width:20%}@media(max-width:720px){.mw-parser-output .col-begin,.mw-parser-output .col-begin>tbody,.mw-parser-output .col-begin>tbody>tr,.mw-parser-output .col-begin>tbody>tr>td{display:block!important;width:100%!important}.mw-parser-output .col-break{padding-left:0!important}}</style><div> <table class="col-begin" role="presentation"> <tbody><tr> <td class="col-break"> <ul><li><a href="/wiki/Apollonian_circles" title="Apollonian circles">Apollonian circles</a></li> <li><a href="/wiki/Archimedean_circle" title="Archimedean circle">Archimedean circle</a></li> <li><a href="/wiki/Archimedes%27_twin_circles" class="mw-redirect" title="Archimedes' twin circles">Archimedes' twin circles</a></li> <li><a href="/wiki/Bankoff_circle" title="Bankoff circle">Bankoff circle</a></li> <li><a href="/wiki/Carlyle_circle" title="Carlyle circle">Carlyle circle</a></li> <li><a href="/wiki/Chromatic_circle" title="Chromatic circle">Chromatic circle</a></li> <li><a href="/wiki/Circle_of_antisimilitude" title="Circle of antisimilitude">Circle of antisimilitude</a></li> <li><a href="/wiki/Ford_circle" title="Ford circle">Ford circle</a></li> <li><a href="/wiki/Geodesic_circle" title="Geodesic circle">Geodesic circle</a></li> <li><a href="/wiki/Johnson_circles" title="Johnson circles">Johnson circles</a></li> <li><a href="/wiki/Schoch_circles" title="Schoch circles">Schoch circles</a></li> <li><a href="/wiki/Woo_circles" title="Woo circles">Woo circles</a></li></ul> <p><br /> </p> </td> <td class="col-break"> <div class="mw-heading mw-heading3"><h3 id="Of_a_triangle">Of a triangle</h3></div> <ul><li><a href="/wiki/Incircle_and_excircles_of_a_triangle#Other_excircle_properties" class="mw-redirect" title="Incircle and excircles of a triangle">Apollonius circle of the excircles</a></li> <li><a href="/wiki/Brocard_circle" title="Brocard circle">Brocard circle</a></li> <li><a href="/wiki/Excircle" class="mw-redirect" title="Excircle">Excircle</a></li> <li><a href="/wiki/Incircle" class="mw-redirect" title="Incircle">Incircle</a></li> <li><a href="/wiki/Lemoine_circle" class="mw-redirect" title="Lemoine circle">Lemoine circle</a></li> <li><a href="/wiki/Lester_circle" class="mw-redirect" title="Lester circle">Lester circle</a></li> <li><a href="/wiki/Malfatti_circles" title="Malfatti circles">Malfatti circles</a></li> <li><a href="/wiki/Mandart_circle" class="mw-redirect" title="Mandart circle">Mandart circle</a></li> <li><a href="/wiki/Nine-point_circle" title="Nine-point circle">Nine-point circle</a></li> <li><a href="/wiki/Orthocentroidal_circle" title="Orthocentroidal circle">Orthocentroidal circle</a></li> <li><a href="/wiki/Parry_circle" class="mw-redirect" title="Parry circle">Parry circle</a></li> <li><a href="/wiki/Polar_circle_(geometry)" title="Polar circle (geometry)">Polar circle (geometry)</a></li> <li><a href="/wiki/Spieker_circle" title="Spieker circle">Spieker circle</a></li> <li><a href="/wiki/Van_Lamoen_circle" title="Van Lamoen circle">Van Lamoen circle</a></li></ul> <p><br /> </p> </td> <td class="col-break"> <div class="mw-heading mw-heading3"><h3 id="Of_certain_quadrilaterals">Of certain quadrilaterals</h3></div> <ul><li><a href="/wiki/Eight-point_circle" class="mw-redirect" title="Eight-point circle">Eight-point circle</a> of an orthodiagonal quadrilateral</li></ul> <div class="mw-heading mw-heading3"><h3 id="Of_a_conic_section">Of a conic section</h3></div> <ul><li><a href="/wiki/Director_circle" title="Director circle">Director circle</a></li> <li><a href="/wiki/Directrix_circle" class="mw-redirect" title="Directrix circle">Directrix circle</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Of_a_torus">Of a torus</h3></div> <ul><li><a href="/wiki/Villarceau_circles" title="Villarceau circles">Villarceau circles</a></li></ul> <p>  </p> </td></tr></tbody></table></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Affine_sphere" title="Affine sphere">Affine sphere</a> – Mathematical concept</li> <li><a href="/wiki/Apeirogon" title="Apeirogon">Apeirogon</a> – Polygon with an infinite number of sides</li> <li><a href="/wiki/Circle_fitting" class="mw-redirect" title="Circle fitting">Circle fitting</a> – Process of constructing a curve that has the best fit to a series of data points<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Distance" title="Distance">Distance</a> – Separation between two points</li> <li><a href="/wiki/Gauss_circle_problem" title="Gauss circle problem">Gauss circle problem</a> – How many integer lattice points there are in a circle</li> <li><a href="/wiki/Inversion_in_a_circle" class="mw-redirect" title="Inversion in a circle">Inversion in a circle</a> – Study of angle-preserving transformations<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Line%E2%80%93circle_intersection" class="mw-redirect" title="Line–circle intersection">Line–circle intersection</a></li> <li><a href="/wiki/List_of_circle_topics" title="List of circle topics">List of circle topics</a></li> <li><a href="/wiki/Sphere" title="Sphere">Sphere</a> – Set of points equidistant from a center</li> <li><a href="/wiki/Three_points_determine_a_circle" class="mw-redirect" title="Three points determine a circle">Three points determine a circle</a> – Number of points needed to determine an algebraic curve<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Translation_of_axes" title="Translation of axes">Translation of axes</a> – Transformation of coordinates that moves the origin</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Also known as <a href="/wiki/Tau_(mathematical_constant)" class="mw-redirect" title="Tau (mathematical constant)">𝜏 (tau)</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dkri%2Fkos">krikos</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131106164504/http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dkri%2Fkos">Archived</a> 2013-11-06 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Henry George Liddell, Robert Scott, <i>A Greek-English Lexicon</i>, on Perseus</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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="CITEREFSimekCresslerHerrmannSherwood2013" class="citation journal cs1">Simek, Jan F.; Cressler, Alan; Herrmann, Nicholas P.; Sherwood, Sarah C. (1 June 2013). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/product/identifier/S0003598X00049048/type/journal_article">"Sacred landscapes of the south-eastern USA: prehistoric rock and cave art in Tennessee"</a>. <i>Antiquity</i>. <b>87</b> (336): <span class="nowrap">430–</span>446. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0003598X00049048">10.1017/S0003598X00049048</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-598X">0003-598X</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:130296519">130296519</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Antiquity&rft.atitle=Sacred+landscapes+of+the+south-eastern+USA%3A+prehistoric+rock+and+cave+art+in+Tennessee&rft.volume=87&rft.issue=336&rft.pages=%3Cspan+class%3D%22nowrap%22%3E430-%3C%2Fspan%3E446&rft.date=2013-06-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A130296519%23id-name%3DS2CID&rft.issn=0003-598X&rft_id=info%3Adoi%2F10.1017%2FS0003598X00049048&rft.aulast=Simek&rft.aufirst=Jan+F.&rft.au=Cressler%2C+Alan&rft.au=Herrmann%2C+Nicholas+P.&rft.au=Sherwood%2C+Sarah+C.&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fproduct%2Fidentifier%2FS0003598X00049048%2Ftype%2Fjournal_article&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" 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"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/Chronology/30000BC_500BC.html#1700BC">Chronology for 30000 BC to 500 BC</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080322085509/http://www-history.mcs.st-andrews.ac.uk/history/Chronology/30000BC_500BC.html">Archived</a> 2008-03-22 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. 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Retrieved on 2012-05-03.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL7227282M">7227282M</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="/wiki/Arthur_Koestler" title="Arthur Koestler">Arthur Koestler</a>, <i><a href="/wiki/The_Sleepwalkers_(Koestler_book)" class="mw-redirect" title="The Sleepwalkers (Koestler book)">The Sleepwalkers</a>: A History of Man's Changing Vision of the Universe</i> (1959)</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="/wiki/Proclus" title="Proclus">Proclus</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=E1HYAAAAMAAJ"><i>The Six Books of Proclus, the Platonic Successor, on the Theology of Plato</i></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170123072440/https://books.google.com/books?id=E1HYAAAAMAAJ">Archived</a> 2017-01-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Tr. Thomas Taylor (1816) Vol. 2, Ch. 2, "Of Plato"</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"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Squaring_the_circle.html">Squaring the circle</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080624144640/http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Squaring_the_circle.html">Archived</a> 2008-06-24 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://philamuseum.org/collection/object/51019">"Circles in a Circle"</a>. <i>Philadelphia Museum of Art</i><span class="reference-accessdate">. Retrieved <span class="nowrap">28 December</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Philadelphia+Museum+of+Art&rft.atitle=Circles+in+a+Circle&rft_id=https%3A%2F%2Fphilamuseum.org%2Fcollection%2Fobject%2F51019&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLesso2022" class="citation web cs1">Lesso, Rosie (15 June 2022). <a rel="nofollow" class="external text" href="https://www.thecollector.com/why-did-wassily-kandinsky-paint-circles/">"Why Did Wassily Kandinsky Paint Circles?"</a>. <i>TheCollector</i><span class="reference-accessdate">. Retrieved <span class="nowrap">28 December</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=TheCollector&rft.atitle=Why+Did+Wassily+Kandinsky+Paint+Circles%3F&rft.date=2022-06-15&rft.aulast=Lesso&rft.aufirst=Rosie&rft_id=https%3A%2F%2Fwww.thecollector.com%2Fwhy-did-wassily-kandinsky-paint-circles%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAbdullahi2019" class="citation encyclopaedia cs1">Abdullahi, Yahya (29 October 2019). "The Circle from East to West". In Charnier, Jean-François (ed.). <i>The Louvre Abu Dhabi: A World Vision of Art</i>. Rizzoli International Publications, Incorporated. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9782370741004" title="Special:BookSources/9782370741004"><bdi>9782370741004</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+Circle+from+East+to+West&rft.btitle=The+Louvre+Abu+Dhabi%3A+A+World+Vision+of+Art&rft.pub=Rizzoli+International+Publications%2C+Incorporated&rft.date=2019-10-29&rft.isbn=9782370741004&rft.aulast=Abdullahi&rft.aufirst=Yahya&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKatz1998" class="citation cs1">Katz, Victor J. (1998). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00katz/page/108"><i>A History of Mathematics / An Introduction</i></a></span> (2nd ed.). Addison Wesley Longman. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00katz/page/108">108</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-01618-8" title="Special:BookSources/978-0-321-01618-8"><bdi>978-0-321-01618-8</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+Mathematics+%2F+An+Introduction&rft.pages=108&rft.edition=2nd&rft.pub=Addison+Wesley+Longman&rft.date=1998&rft.isbn=978-0-321-01618-8&rft.aulast=Katz&rft.aufirst=Victor+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00katz%2Fpage%2F108&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRicheson2015" class="citation journal cs1">Richeson, David (2015). 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Stanley</a>, <i>Excursions in Geometry</i>, Dover, 1969, 14–17.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Altshiller-Court, Nathan, <i>College Geometry</i>, Dover, 2007 (orig. 1952).</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Incircle.html">Incircle – from Wolfram MathWorld</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120121111333/http://mathworld.wolfram.com/Incircle.html">Archived</a> 2012-01-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Circumcircle.html">Circumcircle – from Wolfram MathWorld</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120120120814/http://mathworld.wolfram.com/Circumcircle.html">Archived</a> 2012-01-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/TangentialPolygon.html">Tangential Polygon – from Wolfram MathWorld</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130903051014/http://mathworld.wolfram.com/TangentialPolygon.html">Archived</a> 2013-09-03 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFApostolMnatsakanian2003" class="citation journal cs1">Apostol, Tom; Mnatsakanian, Mamikon (2003). "Sums of squares of distances in m-space". <i>American Mathematical Monthly</i>. <b>110</b> (6): <span class="nowrap">516–</span>526. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2003.11919989">10.1080/00029890.2003.11919989</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:12641658">12641658</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Sums+of+squares+of+distances+in+m-space&rft.volume=110&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E516-%3C%2Fspan%3E526&rft.date=2003&rft_id=info%3Adoi%2F10.1080%2F00029890.2003.11919989&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12641658%23id-name%3DS2CID&rft.aulast=Apostol&rft.aufirst=Tom&rft.au=Mnatsakanian%2C+Mamikon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></span> </li> <li id="cite_note-Mamuka-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mamuka_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMeskhishvili2020" class="citation journal cs1">Meskhishvili, Mamuka (2020). <a rel="nofollow" class="external text" href="https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065">"Cyclic Averages of Regular Polygons and Platonic Solids"</a>. <i>Communications in Mathematics and Applications</i>. <b>11</b>: <span class="nowrap">335–</span>355. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2010.12340">2010.12340</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.26713%2Fcma.v11i3.1420">10.26713/cma.v11i3.1420</a> (inactive 1 November 2024). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210422211229/https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065">Archived</a> from the original on 22 April 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">17 May</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematics+and+Applications&rft.atitle=Cyclic+Averages+of+Regular+Polygons+and+Platonic+Solids&rft.volume=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E335-%3C%2Fspan%3E355&rft.date=2020&rft_id=info%3Aarxiv%2F2010.12340&rft_id=info%3Adoi%2F10.26713%2Fcma.v11i3.1420&rft.aulast=Meskhishvili&rft.aufirst=Mamuka&rft_id=https%3A%2F%2Fwww.rgnpublications.com%2Fjournals%2Findex.php%2Fcma%2Farticle%2Fview%2F1420%2F1065&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: DOI inactive as of November 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">link</a>)</span></span> </li> <li id="cite_note-gamelin-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-gamelin_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGamelin1999" class="citation book cs1">Gamelin, Theodore (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoto00game"><i>Introduction to topology</i></a></span>. Mineola, N.Y: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0486406806" title="Special:BookSources/0486406806"><bdi>0486406806</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+topology&rft.place=Mineola%2C+N.Y&rft.pub=Dover+Publications&rft.date=1999&rft.isbn=0486406806&rft.aulast=Gamelin&rft.aufirst=Theodore&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoto00game&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPedoe,_Dan1988" class="citation book cs1">Pedoe, Dan (1988). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/geometrycomprehe0000pedo"><i>Geometry: a comprehensive course</i></a></span>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486658124" title="Special:BookSources/9780486658124"><bdi>9780486658124</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+a+comprehensive+course&rft.pub=Dover&rft.date=1988&rft.isbn=9780486658124&rft.au=Pedoe%2C+Dan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometrycomprehe0000pedo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a> has media related to:<br /> <a href="https://commons.wikimedia.org/wiki/Circles" class="extiw" title="commons:Circles"><span style="font-style:italic; 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Circle.html">"Circle"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Circle&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCircle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathopenref.com/tocs/circlestoc.html">"Interactive Java applets"</a>. <q>for the properties of and elementary constructions involving circles</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Interactive+Java+applets&rft_id=http%3A%2F%2Fwww.mathopenref.com%2Ftocs%2Fcirclestoc.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathwarehouse.com/geometry/circle/interactive-circle-equation.php">"Interactive Standard Form Equation of Circle"</a>. <q>Click and drag points to see standard form equation in action</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Interactive+Standard+Form+Equation+of+Circle&rft_id=http%3A%2F%2Fwww.mathwarehouse.com%2Fgeometry%2Fcircle%2Finteractive-circle-equation.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircle" class="Z3988"></span></li> <li><link 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