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vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> 1 Formal definition </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Units" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> 2 Units </div> </a> <button aria-controls="toc-Units-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Units subsection </button> <ul id="toc-Units-sublist" class="vector-toc-list"> <li id="toc-Conversions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conversions"> <div class="vector-toc-text"> 2.1 Conversions </div> </a> <ul id="toc-Conversions-sublist" class="vector-toc-list"> <li id="toc-Non-metric_units" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Non-metric_units"> <div class="vector-toc-text"> 2.1.1 Non-metric units </div> </a> <ul id="toc-Non-metric_units-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_units_including_historical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_units_including_historical"> <div class="vector-toc-text"> 2.2 Other units including historical </div> </a> <ul id="toc-Other_units_including_historical-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 3 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Circle_area" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circle_area"> <div class="vector-toc-text"> 3.1 Circle area </div> </a> <ul id="toc-Circle_area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangle_area" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Triangle_area"> <div class="vector-toc-text"> 3.2 Triangle area </div> </a> <ul id="toc-Triangle_area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quadrilateral_area" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quadrilateral_area"> <div class="vector-toc-text"> 3.3 Quadrilateral area </div> </a> <ul id="toc-Quadrilateral_area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_polygon_area" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_polygon_area"> <div class="vector-toc-text"> 3.4 General polygon area </div> </a> <ul id="toc-General_polygon_area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Areas_determined_using_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Areas_determined_using_calculus"> <div class="vector-toc-text"> 3.5 Areas determined using calculus </div> </a> <ul id="toc-Areas_determined_using_calculus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Area_formulas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Area_formulas"> <div class="vector-toc-text"> 4 Area formulas </div> </a> <button aria-controls="toc-Area_formulas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Area formulas subsection </button> <ul id="toc-Area_formulas-sublist" class="vector-toc-list"> <li id="toc-Polygon_formulas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polygon_formulas"> <div class="vector-toc-text"> 4.1 Polygon formulas </div> </a> <ul id="toc-Polygon_formulas-sublist" class="vector-toc-list"> <li id="toc-Rectangles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rectangles"> <div class="vector-toc-text"> 4.1.1 Rectangles </div> </a> <ul id="toc-Rectangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dissection,_parallelograms,_and_triangles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Dissection,_parallelograms,_and_triangles"> <div class="vector-toc-text"> 4.1.2 Dissection, parallelograms, and triangles </div> </a> <ul id="toc-Dissection,_parallelograms,_and_triangles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Area_of_curved_shapes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Area_of_curved_shapes"> <div class="vector-toc-text"> 4.2 Area of curved shapes </div> </a> <ul id="toc-Area_of_curved_shapes-sublist" class="vector-toc-list"> <li id="toc-Circles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Circles"> <div class="vector-toc-text"> 4.2.1 Circles </div> </a> <ul id="toc-Circles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ellipses" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ellipses"> <div class="vector-toc-text"> 4.2.2 Ellipses </div> </a> <ul id="toc-Ellipses-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Non-planar_surface_area" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-planar_surface_area"> <div class="vector-toc-text"> 4.3 Non-planar surface area </div> </a> <ul id="toc-Non-planar_surface_area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_formulas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_formulas"> <div class="vector-toc-text"> 4.4 General formulas </div> </a> <ul id="toc-General_formulas-sublist" class="vector-toc-list"> <li id="toc-Areas_of_2-dimensional_figures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Areas_of_2-dimensional_figures"> <div class="vector-toc-text"> 4.4.1 Areas of 2-dimensional figures </div> </a> <ul id="toc-Areas_of_2-dimensional_figures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Area_in_calculus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Area_in_calculus"> <div class="vector-toc-text"> 4.4.2 Area in calculus </div> </a> <ul id="toc-Area_in_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bounded_area_between_two_quadratic_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bounded_area_between_two_quadratic_functions"> <div class="vector-toc-text"> 4.4.3 Bounded area between two quadratic functions </div> </a> <ul id="toc-Bounded_area_between_two_quadratic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surface_area_of_3-dimensional_figures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Surface_area_of_3-dimensional_figures"> <div class="vector-toc-text"> 4.4.4 Surface area of 3-dimensional figures </div> </a> <ul id="toc-Surface_area_of_3-dimensional_figures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_formula_for_surface_area" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#General_formula_for_surface_area"> <div class="vector-toc-text"> 4.4.5 General formula for surface area </div> </a> <ul id="toc-General_formula_for_surface_area-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-List_of_formulas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#List_of_formulas"> <div class="vector-toc-text"> 4.5 List of formulas </div> </a> <ul id="toc-List_of_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_of_area_to_perimeter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_of_area_to_perimeter"> <div class="vector-toc-text"> 4.6 Relation of area to perimeter </div> </a> <ul id="toc-Relation_of_area_to_perimeter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fractals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fractals"> <div class="vector-toc-text"> 4.7 Fractals </div> </a> <ul id="toc-Fractals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Area_bisectors" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Area_bisectors"> <div class="vector-toc-text"> 5 Area bisectors </div> </a> <ul id="toc-Area_bisectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimization" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Optimization"> <div class="vector-toc-text"> 6 Optimization </div> </a> <ul id="toc-Optimization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 8 References </div> </a> <ul id="toc-References-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"> 9 External links </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" > <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" > Toggle the table of contents </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">Area</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 164 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-164" aria-hidden="true" > 164 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%A9%D3%80%D1%8B%D0%BF%D3%80%D1%8D_%D0%B8%D0%BD%D0%B0%D0%B3%D1%8A" title="ЩӀыпӀэ инагъ – Kabardian" lang="kbd" hreflang="kbd" data-title="ЩӀыпӀэ инагъ" data-language-autonym="Адыгэбзэ" data-language-local-name="Kabardian" class="interlanguage-link-target">Адыгэбзэ</a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Oppervlakte" title="Oppervlakte – Afrikaans" lang="af" hreflang="af" data-title="Oppervlakte" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Fl%C3%A4cheninhalt" title="Flächeninhalt – Alemannic" lang="gsw" hreflang="gsw" data-title="Flächeninhalt" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल – Angika" lang="anp" hreflang="anp" data-title="क्षेत्रफल" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target">अंगिका</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D8%A9" title="مساحة – Arabic" lang="ar" hreflang="ar" data-title="مساحة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Aria" title="Aria – Aragonese" lang="an" hreflang="an" data-title="Aria" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/%DC%AB%DC%9B%DC%9D%DC%9A%DC%98%DC%AC%DC%90" title="ܫܛܝܚܘܬܐ – Aramaic" lang="arc" hreflang="arc" data-title="ܫܛܝܚܘܬܐ" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="Aramaic" class="interlanguage-link-target">ܐܪܡܝܐ</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A7%B0%E0%A6%AB%E0%A6%B2" title="ক্ষেত্ৰফল – Assamese" lang="as" hreflang="as" data-title="ক্ষেত্ৰফল" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81rea_(xeometr%C3%ADa)" title="Área (xeometría) – Asturian" lang="ast" hreflang="ast" data-title="Área (xeometría)" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-awa mw-list-item"><a href="https://awa.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल – Awadhi" lang="awa" hreflang="awa" data-title="क्षेत्रफल" data-language-autonym="अवधी" data-language-local-name="Awadhi" class="interlanguage-link-target">अवधी</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Tendaha" title="Tendaha – Guarani" lang="gn" hreflang="gn" data-title="Tendaha" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-av mw-list-item"><a href="https://av.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0%D0%B4%D1%8C" title="Площадь – Avaric" lang="av" hreflang="av" data-title="Площадь" data-language-autonym="Авар" data-language-local-name="Avaric" class="interlanguage-link-target">Авар</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Sah%C9%99_(%C3%B6l%C3%A7%C3%BC_parametri)" title="Sahə (ölçü parametri) – Azerbaijani" lang="az" hreflang="az" data-title="Sahə (ölçü parametri)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D8%AA" title="مساحت – South Azerbaijani" lang="azb" hreflang="azb" data-title="مساحت" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%AB%E0%A6%B2" title="ক্ষেত্রফল – Bangla" lang="bn" hreflang="bn" data-title="ক্ষেত্রফল" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Bi%C4%81n-chek" title="Biān-chek – Minnan" lang="nan" hreflang="nan" data-title="Biān-chek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B9%D2%99%D0%B0%D0%BD" title="Майҙан – Bashkir" lang="ba" hreflang="ba" data-title="Майҙан" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%88%D1%87%D0%B0" title="Плошча – Belarusian" lang="be" hreflang="be" data-title="Плошча" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%88%D1%87%D0%B0" title="Плошча – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Плошча" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल – Bhojpuri" lang="bh" hreflang="bh" data-title="क्षेत्रफल" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Hiwas" title="Hiwas – Central Bikol" lang="bcl" hreflang="bcl" data-title="Hiwas" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89" title="Площ – Bulgarian" lang="bg" hreflang="bg" data-title="Площ" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Fl%C3%A4chn" title="Flächn – Bavarian" lang="bar" hreflang="bar" data-title="Flächn" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Povr%C5%A1ina" title="Površina – Bosnian" lang="bs" hreflang="bs" data-title="Površina" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Gorread" title="Gorread – Breton" lang="br" hreflang="br" data-title="Gorread" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80rea" title="Àrea – Catalan" lang="ca" hreflang="ca" data-title="Àrea" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BF%D1%82%C4%83%D0%BA" title="Лаптăк – Chuvash" lang="cv" hreflang="cv" data-title="Лаптăк" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Langyab" title="Langyab – Cebuano" lang="ceb" hreflang="ceb" data-title="Langyab" data-language-autonym="Cebuano" data-language-local-name="Cebuano" class="interlanguage-link-target">Cebuano</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Obsah" title="Obsah – Czech" lang="cs" hreflang="cs" data-title="Obsah" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nharaunda" title="Nharaunda – Shona" lang="sn" hreflang="sn" data-title="Nharaunda" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Arwynebedd" title="Arwynebedd – Welsh" lang="cy" hreflang="cy" data-title="Arwynebedd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Areal" title="Areal – Danish" lang="da" hreflang="da" data-title="Areal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AA%D9%8A%D8%B3%D8%A7%D8%B9" title="تيساع – Moroccan Arabic" lang="ary" hreflang="ary" data-title="تيساع" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fl%C3%A4cheninhalt" title="Flächeninhalt – German" lang="de" hreflang="de" data-title="Flächeninhalt" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-dv mw-list-item"><a href="https://dv.wikipedia.org/wiki/%DE%87%DE%A6%DE%86%DE%A6%DE%89%DE%A8%DE%82%DE%B0" title="އަކަމިން – Divehi" lang="dv" hreflang="dv" data-title="އަކަމިން" data-language-autonym="ދިވެހިބަސް" data-language-local-name="Divehi" class="interlanguage-link-target">ދިވެހިބަސް</a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Wop%C5%9Bimje%C5%9Be_p%C5%82oni" title="Wopśimjeśe płoni – Lower Sorbian" lang="dsb" hreflang="dsb" data-title="Wopśimjeśe płoni" data-language-autonym="Dolnoserbski" data-language-local-name="Lower Sorbian" class="interlanguage-link-target">Dolnoserbski</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pindala" title="Pindala – Estonian" lang="et" hreflang="et" data-title="Pindala" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BC%CE%B2%CE%B1%CE%B4%CF%8C%CE%BD" title="Εμβαδόν – Greek" lang="el" hreflang="el" data-title="Εμβαδόν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81rea" title="Área – Spanish" lang="es" hreflang="es" data-title="Área" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Areo" title="Areo – Esperanto" lang="eo" hreflang="eo" data-title="Areo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Azalera" title="Azalera – Basque" lang="eu" hreflang="eu" data-title="Azalera" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D8%AA" title="مساحت – Persian" lang="fa" hreflang="fa" data-title="مساحت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Area" title="Area – Fiji Hindi" lang="hif" hreflang="hif" data-title="Area" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/V%C3%ADdd" title="Vídd – Faroese" lang="fo" hreflang="fo" data-title="Vídd" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Aire_(g%C3%A9om%C3%A9trie)" title="Aire (géométrie) – French" lang="fr" hreflang="fr" data-title="Aire (géométrie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Oerflak" title="Oerflak – Western Frisian" lang="fy" hreflang="fy" data-title="Oerflak" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Eaghtyr" title="Eaghtyr – Manx" lang="gv" hreflang="gv" data-title="Eaghtyr" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target">Gaelg</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Farsaingeachd" title="Farsaingeachd – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Farsaingeachd" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81rea" title="Área – Galician" lang="gl" hreflang="gl" data-title="Área" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%9D%A2%E7%A9%8D" title="面積 – Gan" lang="gan" hreflang="gan" data-title="面積" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%95%E0%AB%8D%E0%AA%B7%E0%AB%87%E0%AA%A4%E0%AB%8D%E0%AA%B0%E0%AA%AB%E0%AA%B3" title="ક્ષેત્રફળ – Gujarati" lang="gu" hreflang="gu" data-title="ક્ષેત્રફળ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Mien-chit" title="Mien-chit – Hakka Chinese" lang="hak" hreflang="hak" data-title="Mien-chit" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target">客家語 / Hak-kâ-ngî</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%84%93%EC%9D%B4" title="넓이 – Korean" lang="ko" hreflang="ko" data-title="넓이" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/%CA%BBAlea" title="ʻAlea – Hawaiian" lang="haw" hreflang="haw" data-title="ʻAlea" data-language-autonym="Hawaiʻi" data-language-local-name="Hawaiian" class="interlanguage-link-target">Hawaiʻi</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%AF%D5%A5%D6%80%D5%A5%D5%BD" title="Մակերես – Armenian" lang="hy" hreflang="hy" data-title="Մակերես" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल – Hindi" lang="hi" hreflang="hi" data-title="क्षेत्रफल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Wobsah_p%C5%99estrjenje" title="Wobsah přestrjenje – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Wobsah přestrjenje" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target">Hornjoserbsce</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Povr%C5%A1ina" title="Površina – Croatian" lang="hr" hreflang="hr" data-title="Površina" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Areo" title="Areo – Ido" lang="io" hreflang="io" data-title="Areo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Kalawa" title="Kalawa – Iloko" lang="ilo" hreflang="ilo" data-title="Kalawa" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target">Ilokano</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Luas" title="Luas – Indonesian" lang="id" hreflang="id" data-title="Luas" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Area" title="Area – Interlingua" lang="ia" hreflang="ia" data-title="Area" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%A4%C3%A6%D0%B7%D1%83%D0%B0%D1%82" title="Фæзуат – Ossetic" lang="os" hreflang="os" data-title="Фæзуат" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Flatarm%C3%A1l" title="Flatarmál – Icelandic" lang="is" hreflang="is" data-title="Flatarmál" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Area" title="Area – Italian" lang="it" hreflang="it" data-title="Area" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%98%D7%97" title="שטח – Hebrew" lang="he" hreflang="he" data-title="שטח" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Jembar" title="Jembar – Javanese" lang="jv" hreflang="jv" data-title="Jembar" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%90%E1%83%A0%E1%83%97%E1%83%9D%E1%83%91%E1%83%98" title="ფართობი – Georgian" lang="ka" hreflang="ka" data-title="ფართობი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%83%D0%B4%D0%B0%D0%BD_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Аудан (геометрия) – Kazakh" lang="kk" hreflang="kk" data-title="Аудан (геометрия)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Enep_(fysegieth)" title="Enep (fysegieth) – Cornish" lang="kw" hreflang="kw" data-title="Enep (fysegieth)" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target">Kernowek</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Eneo" title="Eneo – Swahili" lang="sw" hreflang="sw" data-title="Eneo" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/R%C3%BBerd" title="Rûerd – Kurdish" lang="ku" hreflang="ku" data-title="Rûerd" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%8F%D0%BD%D1%82" title="Аянт – Kyrgyz" lang="ky" hreflang="ky" data-title="Аянт" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%80%E0%BA%99%E0%BA%B7%E0%BB%89%E0%BA%AD%E0%BA%97%E0%BA%B5%E0%BB%88" title="ເນື້ອທີ່ – Lao" lang="lo" hreflang="lo" data-title="ເນື້ອທີ່" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Area_(geometria)" title="Area (geometria) – Latin" lang="la" hreflang="la" data-title="Area (geometria)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Laukums" title="Laukums – Latvian" lang="lv" hreflang="lv" data-title="Laukums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Fl%C3%A4ch" title="Fläch – Luxembourgish" lang="lb" hreflang="lb" data-title="Fläch" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Plotas" title="Plotas – Lithuanian" lang="lt" hreflang="lt" data-title="Plotas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Area" title="Area – Ligurian" lang="lij" hreflang="lij" data-title="Area" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target">Ligure</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Oppervlak" title="Oppervlak – Limburgish" lang="li" hreflang="li" data-title="Oppervlak" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Etando" title="Etando – Lingala" lang="ln" hreflang="ln" data-title="Etando" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target">Lingála</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Area" title="Area – Lombard" lang="lmo" hreflang="lmo" data-title="Area" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ter%C3%BClet_(matematika)" title="Terület (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Terület (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल – Maithili" lang="mai" hreflang="mai" data-title="क्षेत्रफल" data-language-autonym="मैथिली" data-language-local-name="Maithili" class="interlanguage-link-target">मैथिली</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%88%D1%82%D0%B8%D0%BD%D0%B0" title="Плоштина – Macedonian" lang="mk" hreflang="mk" data-title="Плоштина" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Velarantany" title="Velarantany – Malagasy" lang="mg" hreflang="mg" data-title="Velarantany" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%80%E0%B5%BC%E0%B4%A3%E0%B5%8D%E0%B4%A3%E0%B4%82" title="വിസ്തീർണ്ണം – Malayalam" lang="ml" hreflang="ml" data-title="വിസ്തീർണ്ണം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B3" title="क्षेत्रफळ – Marathi" lang="mr" hreflang="mr" data-title="क्षेत्रफळ" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A4%E1%83%90%E1%83%A0%E1%83%97%E1%83%9D%E1%83%91%E1%83%98" title="ფართობი – Mingrelian" lang="xmf" hreflang="xmf" data-title="ფართობი" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target">მარგალური</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D8%AD%D9%87" title="مساحه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="مساحه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%DA%AF%D8%AA%DB%8C" title="گتی – Mazanderani" lang="mzn" hreflang="mzn" data-title="گتی" data-language-autonym="مازِرونی" data-language-local-name="Mazanderani" class="interlanguage-link-target">مازِرونی</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Luas" title="Luas – Malay" lang="ms" hreflang="ms" data-title="Luas" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mni mw-list-item"><a href="https://mni.wikipedia.org/wiki/%EA%AF%84%EA%AF%A5%EA%AF%9B%EA%AF%86%EA%AF%A5%EA%AF%8E%EA%AF%95" title="ꯄꯥꯛꯆꯥꯎꯕ – Manipuri" lang="mni" hreflang="mni" data-title="ꯄꯥꯛꯆꯥꯎꯕ" data-language-autonym="ꯃꯤꯇꯩ ꯂꯣꯟ" data-language-local-name="Manipuri" class="interlanguage-link-target">ꯃꯤꯇꯩ ꯂꯣꯟ</a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/Mi%C3%AAng-c%C3%A9k" title="Miêng-cék – Mindong" lang="cdo" hreflang="cdo" data-title="Miêng-cék" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target">閩東語 / Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/%C3%81ria" title="Ária – Mirandese" lang="mwl" hreflang="mwl" data-title="Ária" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target">Mirandés</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%B0%D0%BB%D0%B1%D0%B0%D0%B9" title="Талбай – Mongolian" lang="mn" hreflang="mn" data-title="Талбай" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A7%E1%80%9B%E1%80%AD%E1%80%9A%E1%80%AC" title="ဧရိယာ – Burmese" lang="my" hreflang="my" data-title="ဧရိယာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Iwasewase" title="Iwasewase – Fijian" lang="fj" hreflang="fj" data-title="Iwasewase" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Oppervlakte" title="Oppervlakte – Dutch" lang="nl" hreflang="nl" data-title="Oppervlakte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nds-nl mw-list-item"><a href="https://nds-nl.wikipedia.org/wiki/Oppervlakte" title="Oppervlakte – Low Saxon" lang="nds-NL" hreflang="nds-NL" data-title="Oppervlakte" data-language-autonym="Nedersaksies" data-language-local-name="Low Saxon" class="interlanguage-link-target">Nedersaksies</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2" title="क्षेत्रफल – Nepali" lang="ne" hreflang="ne" data-title="क्षेत्रफल" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9D%A2%E7%A9%8D" title="面積 – Japanese" lang="ja" hreflang="ja" data-title="面積" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ce mw-list-item"><a href="https://ce.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B9%D0%B4%D0%B0" title="Майда – Chechen" lang="ce" hreflang="ce" data-title="Майда" data-language-autonym="Нохчийн" data-language-local-name="Chechen" class="interlanguage-link-target">Нохчийн</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Areaal_(Miat)" title="Areaal (Miat) – Northern Frisian" lang="frr" hreflang="frr" data-title="Areaal (Miat)" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Areal" title="Areal – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Areal" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Flatevidd" title="Flatevidd – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Flatevidd" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Aira" title="Aira – Occitan" lang="oc" hreflang="oc" data-title="Aira" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9A%D1%83%D0%BC%D0%B4%D1%8B%D0%BA" title="Кумдык – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Кумдык" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target">Олык марий</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yuza" title="Yuza – Uzbek" lang="uz" hreflang="uz" data-title="Yuza" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%96%E0%A9%87%E0%A8%A4%E0%A8%B0%E0%A8%AB%E0%A8%B2" title="ਖੇਤਰਫਲ – Punjabi" lang="pa" hreflang="pa" data-title="ਖੇਤਰਫਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Fl%C3%A4che" title="Fläche – Palatine German" lang="pfl" hreflang="pfl" data-title="Fläche" data-language-autonym="Pälzisch" data-language-local-name="Palatine German" class="interlanguage-link-target">Pälzisch</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B1%D9%82%D8%A8%DB%81" title="رقبہ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="رقبہ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Ieria" title="Ieria – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Ieria" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%80%E1%9F%92%E1%9E%9A%E1%9E%9B%E1%9E%B6%E1%9E%95%E1%9F%92%E1%9E%91%E1%9F%83" title="ក្រលាផ្ទៃ – Khmer" lang="km" hreflang="km" data-title="ក្រលាផ្ទៃ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Flach" title="Flach – Low German" lang="nds" hreflang="nds" data-title="Flach" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pole_powierzchni" title="Pole powierzchni – Polish" lang="pl" hreflang="pl" data-title="Pole powierzchni" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81rea" title="Área – Portuguese" lang="pt" hreflang="pt" data-title="Área" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Arie" title="Arie – Romanian" lang="ro" hreflang="ro" data-title="Arie" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Hallka_k%27iti_k%27anchar" title="Hallka k'iti k'anchar – Quechua" lang="qu" hreflang="qu" data-title="Hallka k'iti k'anchar" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0%D0%B4%D1%8C" title="Площадь – Russian" lang="ru" hreflang="ru" data-title="Площадь" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%98%D1%8D%D0%BD" title="Иэн – Yakut" lang="sah" hreflang="sah" data-title="Иэн" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2%E0%A4%AE%E0%A5%8D" title="क्षेत्रफलम् – Sanskrit" lang="sa" hreflang="sa" data-title="क्षेत्रफलम्" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Area" title="Area – Scots" lang="sco" hreflang="sco" data-title="Area" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sip%C3%ABrfaqja" title="Sipërfaqja – Albanian" lang="sq" hreflang="sq" data-title="Sipërfaqja" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/%C3%80ria_(supirfici)" title="Ària (supirfici) – Sicilian" lang="scn" hreflang="scn" data-title="Ària (supirfici)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Area" title="Area – Simple English" lang="en-simple" hreflang="en-simple" data-title="Area" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%A7%D9%8A%D8%B1%D8%A7%D8%B6%D9%8A" title="ايراضي – Sindhi" lang="sd" hreflang="sd" data-title="ايراضي" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Plocha_(%C3%BAtvar)" title="Plocha (útvar) – Slovak" lang="sk" hreflang="sk" data-title="Plocha (útvar)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Povr%C5%A1ina" title="Površina – Slovenian" lang="sl" hreflang="sl" data-title="Površina" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%A5" title="Пространиѥ – Church Slavic" lang="cu" hreflang="cu" data-title="Пространиѥ" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="Church Slavic" class="interlanguage-link-target">Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Plac_rozlygowa%C5%84o" title="Plac rozlygowańo – Silesian" lang="szl" hreflang="szl" data-title="Plac rozlygowańo" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Bed" title="Bed – Somali" lang="so" hreflang="so" data-title="Bed" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D9%88%D9%88%D8%A8%DB%95%D8%B1" title="ڕووبەر – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕووبەر" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D1%80%D1%88%D0%B8%D0%BD%D0%B0" title="Површина – Serbian" lang="sr" hreflang="sr" data-title="Површина" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Povr%C5%A1ina" title="Površina – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Površina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Ar%C3%A9a" title="Aréa – Sundanese" lang="su" hreflang="su" data-title="Aréa" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pinta-ala" title="Pinta-ala – Finnish" lang="fi" hreflang="fi" data-title="Pinta-ala" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Area" title="Area – Swedish" lang="sv" hreflang="sv" data-title="Area" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Sukat" title="Sukat – Tagalog" lang="tl" hreflang="tl" data-title="Sukat" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%B0%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%B3%E0%AE%B5%E0%AF%81" title="பரப்பளவு – Tamil" lang="ta" hreflang="ta" data-title="பரப்பளவு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tajumma" title="Tajumma – Kabyle" lang="kab" hreflang="kab" data-title="Tajumma" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/M%C3%A4ydan" title="Mäydan – Tatar" lang="tt" hreflang="tt" data-title="Mäydan" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B0%BF%E0%B0%B8%E0%B1%8D%E0%B0%A4%E0%B1%80%E0%B0%B0%E0%B1%8D%E0%B0%A3%E0%B0%82" title="విస్తీర్ణం – Telugu" lang="te" hreflang="te" data-title="విస్తీర్ణం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B7%E0%B9%89%E0%B8%99%E0%B8%97%E0%B8%B5%E0%B9%88" title="พื้นที่ – Thai" lang="th" hreflang="th" data-title="พื้นที่" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-ti mw-list-item"><a href="https://ti.wikipedia.org/wiki/%E1%8C%BD%E1%8D%8D%E1%88%93%E1%89%B5_%E1%88%98%E1%88%AC%E1%89%B5" title="ጽፍሓት መሬት – Tigrinya" lang="ti" hreflang="ti" data-title="ጽፍሓት መሬት" data-language-autonym="ትግርኛ" data-language-local-name="Tigrinya" class="interlanguage-link-target">ትግርኛ</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%81%D0%BE%D2%B3%D0%B0%D1%82" title="Масоҳат – Tajik" lang="tg" hreflang="tg" data-title="Масоҳат" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Alan" title="Alan – Turkish" lang="tr" hreflang="tr" data-title="Alan" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B0" title="Площа – Ukrainian" lang="uk" hreflang="uk" data-title="Площа" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%D9%82%D8%A8%DB%81" title="رقبہ – Urdu" lang="ur" hreflang="ur" data-title="رقبہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Di%E1%BB%87n_t%C3%ADch" title="Diện tích – Vietnamese" lang="vi" hreflang="vi" data-title="Diện tích" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Pindala" title="Pindala – Võro" lang="vro" hreflang="vro" data-title="Pindala" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Sitind%C3%AAye" title="Sitindêye – Walloon" lang="wa" hreflang="wa" data-title="Sitindêye" data-language-autonym="Walon" data-language-local-name="Walloon" class="interlanguage-link-target">Walon</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Ippervlak" title="Ippervlak – West Flemish" lang="vls" hreflang="vls" data-title="Ippervlak" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Kahaluag" title="Kahaluag – Waray" lang="war" hreflang="war" data-title="Kahaluag" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Yaatuwaay" title="Yaatuwaay – Wolof" lang="wo" 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Areas&redirect=no" class="mw-redirect" title="Areas">Areas</a>)</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">Size of a two-dimensional surface</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 geometric quantity. For other uses, see <a href="/wiki/Area_(disambiguation)" class="mw-disambig" title="Area (disambiguation)">Area (disambiguation)</a>.</div> <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">Area</th></tr><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:Squaring_the_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/220px-Squaring_the_circle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/330px-Squaring_the_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Squaring_the_circle.svg/440px-Squaring_the_circle.svg.png 2x" data-file-width="281" data-file-height="281" /></a><div class="infobox-caption">The areas of this square and this <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a> are the same.</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data">A or S</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Square_metre" title="Square metre">Square metre</a> [m2]</td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit">SI base units</a></th><td class="infobox-data">1 <a href="/wiki/Metre" title="Metre">m</a>2</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e315c7dc801b08df18f46f1069d4ecbb7760be35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.314ex; height:2.676ex;" alt="{\displaystyle {\mathsf {L}}^{2}}"></td></tr></tbody></table> Area is the <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a> of a <a href="/wiki/Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">region</a>'s size on a <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a>. The area of a plane region or plane area refers to the area of a <a href="/wiki/Shape" title="Shape">shape</a> or <a href="/wiki/Planar_lamina" title="Planar lamina">planar lamina</a>, while <a href="/wiki/Surface_area" title="Surface area">surface area</a> refers to the area of an <a href="/wiki/Open_surface" class="mw-redirect" title="Open surface">open surface</a> or the <a href="/wiki/Boundary_(mathematics)" class="mw-redirect" title="Boundary (mathematics)">boundary</a> of a <a href="/wiki/Solid_geometry" title="Solid geometry">three-dimensional object</a>. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of <a href="/wiki/Paint" title="Paint">paint</a> necessary to cover the surface with a single coat.<a href="#cite_note-MathWorld-1">[1]</a> It is the two-dimensional analogue of the <a href="/wiki/Length" title="Length">length</a> of a <a href="/wiki/Plane_curve" title="Plane curve">curve</a> (a one-dimensional concept) or the <a href="/wiki/Volume" title="Volume">volume</a> of a solid (a three-dimensional concept). Two different regions may have the same area (as in <a href="/wiki/Squaring_the_circle" title="Squaring the circle">squaring the circle</a>); by <a href="/wiki/Synecdoche" title="Synecdoche">synecdoche</a>, "area" sometimes is used to refer to the region, as in a "<a href="/wiki/Polygonal_area" class="mw-redirect" title="Polygonal area">polygonal area</a>". The area of a shape can be measured by comparing the shape to <a href="/wiki/Square" title="Square">squares</a> of a fixed size.<a href="#cite_note-AF-2">[2]</a> In the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI), the standard unit of area is the <a href="/wiki/Square_metre" title="Square metre">square metre</a> (written as m2), which is the area of a square whose sides are one <a href="/wiki/Metre" title="Metre">metre</a> long.<a href="#cite_note-B-3">[3]</a> A shape with an area of three square metres would have the same area as three such squares. In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <a href="/wiki/Unit_square" title="Unit square">unit square</a> is defined to have area one, and the area of any other shape or surface is a <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> <a href="/wiki/Real_number" title="Real number">real number</a>. There are several well-known <a href="/wiki/Formula" title="Formula">formulas</a> for the areas of simple shapes such as <a href="/wiki/Triangle" title="Triangle">triangles</a>, <a href="/wiki/Rectangle" title="Rectangle">rectangles</a>, and <a href="/wiki/Circle" title="Circle">circles</a>. Using these formulas, the area of any <a href="/wiki/Polygon" title="Polygon">polygon</a> can be found by <a href="/wiki/Polygon_triangulation" title="Polygon triangulation">dividing the polygon into triangles</a>.<a href="#cite_note-bkos-4">[4]</a> For shapes with curved boundary, <a href="/wiki/Calculus" title="Calculus">calculus</a> is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the <a href="/wiki/History_of_calculus" title="History of calculus">historical development of calculus</a>.<a href="#cite_note-5">[5]</a> For a solid shape such as a <a href="/wiki/Sphere" title="Sphere">sphere</a>, cone, or cylinder, the area of its boundary surface is called the <a href="/wiki/Surface_area" title="Surface area">surface area</a>.<a href="#cite_note-MathWorld-1">[1]</a><a href="#cite_note-MathWorldSurfaceArea-6">[6]</a><a href="#cite_note-7">[7]</a> Formulas for the surface areas of simple shapes were computed by the <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greeks</a>, but computing the surface area of a more complicated shape usually requires <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a>. Area plays an important role in modern mathematics. In addition to its obvious importance in <a href="/wiki/Geometry" title="Geometry">geometry</a> and calculus, area is related to the definition of <a href="/wiki/Determinant" title="Determinant">determinants</a> in <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, and is a basic property of surfaces in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>.<a href="#cite_note-doCarmo-8">[8]</a> In <a href="/wiki/Analysis" title="Analysis">analysis</a>, the area of a subset of the plane is defined using <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>,<a href="#cite_note-Rudin-9">[9]</a> though not every subset is measurable if one supposes the axiom of choice.<a href="#cite_note-10">[10]</a> In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.<a href="#cite_note-MathWorld-1">[1]</a> Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2>[<a href="/w/index.php?title=Area&action=edit&section=1" title="Edit section: Formal definition">edit</a>]</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/Jordan_measure" class="mw-redirect" title="Jordan measure">Jordan measure</a></div> An approach to defining what is meant by "area" is through <a href="/wiki/Axiom" title="Axiom">axioms</a>. "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:<a href="#cite_note-Apostol-11">[11]</a> <ul><li>For all S in M, a(S) ≥ 0.</li> <li>If S and T are in M then so are S ∪ T and S ∩ T, and also a(S∪T) = a(S) + a(T) − a(S ∩ T).</li> <li>If S and T are in M with S ⊆ T then T − S is in M and a(T−S) = a(T) − a(S).</li> <li>If a set S is in M and S is congruent to T then T is also in M and a(S) = a(T).</li> <li>Every rectangle R is in M. If the rectangle has length h and breadth k then a(R) = hk.</li> <li>Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. S ⊆ Q ⊆ T. If there is a unique number c such that a(S) ≤ c ≤ a(T) for all such step regions S and T, then a(Q) = c.</li></ul> It can be proved that such an area function actually exists.<a href="#cite_note-Moise-12">[12]</a> <div class="mw-heading mw-heading2"><h2 id="Units">Units</h2>[<a href="/w/index.php?title=Area&action=edit&section=2" title="Edit section: Units">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SquareMeterQuadrat.JPG" class="mw-file-description"><img alt="A square made of PVC pipe on grass" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SquareMeterQuadrat.JPG/220px-SquareMeterQuadrat.JPG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SquareMeterQuadrat.JPG/330px-SquareMeterQuadrat.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/SquareMeterQuadrat.JPG/440px-SquareMeterQuadrat.JPG 2x" data-file-width="2048" data-file-height="1536" /></a><figcaption>A square metre <a href="/wiki/Quadrat" title="Quadrat">quadrat</a> made of PVC pipe</figcaption></figure> Every <a href="/wiki/Unit_of_length" title="Unit of length">unit of length</a> has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in <a href="/wiki/Square_metre" title="Square metre">square metres</a> (m2), square centimetres (cm2), square millimetres (mm2), <a href="/wiki/Square_kilometre" title="Square kilometre">square kilometres</a> (km2), <a href="/wiki/Square_foot" title="Square foot">square feet</a> (ft2), <a href="/wiki/Square_yard" title="Square yard">square yards</a> (yd2), <a href="/wiki/Square_mile" title="Square mile">square miles</a> (mi2), and so forth.<a href="#cite_note-BIPM2006Ch5-13">[13]</a> Algebraically, these units can be thought of as the <a href="/wiki/Square_(algebra)" title="Square (algebra)">squares</a> of the corresponding length units. The SI unit of area is the square metre, which is considered an <a href="/wiki/SI_derived_unit" title="SI derived unit">SI derived unit</a>.<a href="#cite_note-B-3">[3]</a> <div class="mw-heading mw-heading3"><h3 id="Conversions">Conversions</h3>[<a href="/w/index.php?title=Area&action=edit&section=3" title="Edit section: Conversions">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Area_conversion_-_square_mm_in_a_square_cm.png" class="mw-file-description"><img alt="A diagram showing the conversion factor between different areas" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Area_conversion_-_square_mm_in_a_square_cm.png/220px-Area_conversion_-_square_mm_in_a_square_cm.png" decoding="async" width="220" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Area_conversion_-_square_mm_in_a_square_cm.png/330px-Area_conversion_-_square_mm_in_a_square_cm.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Area_conversion_-_square_mm_in_a_square_cm.png/440px-Area_conversion_-_square_mm_in_a_square_cm.png 2x" data-file-width="831" data-file-height="699" /></a><figcaption>Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.</figcaption></figure> Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m2 and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m2. This is equivalent to 6 million square millimetres. Other useful conversions are: <ul><li>1 square kilometre = <a href="/wiki/Million" class="mw-redirect" title="Million">1,000,000</a> square metres</li> <li>1 square metre = <a href="/wiki/10000_(number)" class="mw-redirect" title="10000 (number)">10,000</a> square centimetres = 1,000,000 square millimetres</li> <li>1 square centimetre = <a href="/wiki/100_(number)" class="mw-redirect" title="100 (number)">100</a> square millimetres.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Non-metric_units">Non-metric units</h4>[<a href="/w/index.php?title=Area&action=edit&section=4" title="Edit section: Non-metric units">edit</a>]</div> In non-metric units, the conversion between two square units is the <a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a> of the conversion between the corresponding length units. <dl><dd>1 <a href="/wiki/Foot_(unit)" title="Foot (unit)">foot</a> = 12 <a href="/wiki/Inch" title="Inch">inches</a>,</dd></dl> the relationship between square feet and square inches is <dl><dd>1 square foot = 144 square inches,</dd></dl> where 144 = 122 = 12 × 12. Similarly: <ul><li>1 square yard = <a href="/wiki/9_(number)" class="mw-redirect" title="9 (number)">9</a> square feet</li> <li>1 square mile = 3,097,600 square yards = 27,878,400 square feet</li></ul> In addition, conversion factors include: <ul><li>1 square inch = 6.4516 square centimetres</li> <li>1 square foot = 0.09290304 square metres</li> <li>1 square yard = 0.83612736 square metres</li> <li>1 square mile = 2.589988110336 square kilometres</li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_units_including_historical">Other units including historical</h3>[<a href="/w/index.php?title=Area&action=edit&section=5" title="Edit section: Other units including historical">edit</a>]</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/Category:Units_of_area" title="Category:Units of area">Category:Units of area</a></div> There are several other common units for area. The <a href="/wiki/Are_(unit)" class="mw-redirect" title="Are (unit)">are</a> was the original unit of area in the <a href="/wiki/Metric_system" title="Metric system">metric system</a>, with: <ul><li>1 are = 100 square metres</li></ul> Though the are has fallen out of use, the <a href="/wiki/Hectare" title="Hectare">hectare</a> is still commonly used to measure land:<a href="#cite_note-BIPM2006Ch5-13">[13]</a> <ul><li>1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres</li></ul> Other uncommon metric units of area include the <a href="/wiki/Tetrad_(unit_of_area)" class="mw-redirect" title="Tetrad (unit of area)">tetrad</a>, the <a href="/wiki/Hectad" title="Hectad">hectad</a>, and the <a href="/wiki/Myriad_(area)" title="Myriad (area)">myriad</a>. The <a href="/wiki/Acre" title="Acre">acre</a> is also commonly used to measure land areas, where <ul><li>1 acre = 4,840 square yards = 43,560 square feet.</li></ul> An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of <a href="/wiki/Barn_(unit)" title="Barn (unit)">barns</a>, such that:<a href="#cite_note-BIPM2006Ch5-13">[13]</a> <ul><li>1 barn = 10−28 square meters.</li></ul> The barn is commonly used in describing the cross-sectional area of interaction in <a href="/wiki/Nuclear_physics" title="Nuclear physics">nuclear physics</a>.<a href="#cite_note-BIPM2006Ch5-13">[13]</a> In <a href="/wiki/South_Asia" title="South Asia">South Asia</a> (mainly Indians), although the countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a> Some traditional South Asian units that have fixed value: <ul><li>1 Killa = 1 acre</li> <li>1 Ghumaon = 1 acre</li> <li>1 Kanal = 0.125 acre (1 acre = 8 kanal)</li> <li>1 Decimal = 48.4 square yards</li> <li>1 Chatak = 180 square feet</li></ul> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Area&action=edit&section=6" title="Edit section: History">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Circle_area">Circle area</h3>[<a href="/w/index.php?title=Area&action=edit&section=7" title="Edit section: Circle area">edit</a>]</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/Area_of_a_circle#History" title="Area of a circle">Area of a circle § History</a></div> In the 5th century BCE, <a href="/wiki/Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates of Chios</a> was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his <a href="/wiki/Quadrature_(mathematics)" class="mw-redirect" title="Quadrature (mathematics)">quadrature</a> of the <a href="/wiki/Lune_of_Hippocrates" title="Lune of Hippocrates">lune of Hippocrates</a>,<a href="#cite_note-heath-18">[18]</a> but did not identify the <a href="/wiki/Constant_of_proportionality" class="mw-redirect" title="Constant of proportionality">constant of proportionality</a>. <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a>, also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.<a href="#cite_note-19">[19]</a> Subsequently, Book I of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a> dealt with equality of areas between two-dimensional figures. The mathematician <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> used the tools of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> to show that the area inside a circle is equal to that of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book <a href="/wiki/Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a>. (The circumference is 2πr, and the area of a triangle is half the base times the height, yielding the area πr2 for the disk.) Archimedes approximated the value of π (and hence the area of a unit-radius circle) with <a href="/wiki/Area_of_a_disk#Archimedes'_doubling_method" class="mw-redirect" title="Area of a disk">his doubling method</a>, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular <a href="/wiki/Hexagon" title="Hexagon">hexagon</a>, then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with <a href="/wiki/Circumscribed_polygon" class="mw-redirect" title="Circumscribed polygon">circumscribed polygons</a>). <div class="mw-heading mw-heading3"><h3 id="Triangle_area">Triangle area</h3>[<a href="/w/index.php?title=Area&action=edit&section=8" title="Edit section: Triangle area">edit</a>]</div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Area_of_a_triangle#History" title="Area of a triangle">Area of a triangle § History</a>.[<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Area_of_a_triangle&action=edit#History">edit</a>]</div><div class="excerpt"> <a href="/wiki/Heron_of_Alexandria" class="mw-redirect" title="Heron of Alexandria">Heron of Alexandria</a> found what is known as <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica, written around 60 CE. It has been suggested that <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> knew the formula over two centuries earlier,<a href="#cite_note-20">[20]</a> and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.<a href="#cite_note-21">[21]</a> In 300 BCE Greek mathematician <a href="/wiki/Euclid" title="Euclid">Euclid</a> proved that the area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry.<a href="#cite_note-22">[22]</a> In 499 <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>, a great <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>-<a href="/wiki/Astronomer" title="Astronomer">astronomer</a> from the classical age of <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a> and <a href="/wiki/Indian_astronomy" title="Indian astronomy">Indian astronomy</a>, expressed the area of a triangle as one-half the base times the height in the <a href="/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a>.<a href="#cite_note-23">[23]</a> A formula equivalent to Heron's was discovered by the Chinese independently of the Greeks. It was published in 1247 in Shushu Jiuzhang ("<a href="/wiki/Mathematical_Treatise_in_Nine_Sections" title="Mathematical Treatise in Nine Sections">Mathematical Treatise in Nine Sections</a>"), written by <a href="/wiki/Qin_Jiushao" title="Qin Jiushao">Qin Jiushao</a>.<a href="#cite_note-24">[24]</a></div></div> <div class="mw-heading mw-heading3"><h3 id="Quadrilateral_area">Quadrilateral area</h3>[<a href="/w/index.php?title=Area&action=edit&section=9" title="Edit section: Quadrilateral area">edit</a>]</div> In the 7th century CE, <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> developed a formula, now known as <a href="/wiki/Brahmagupta%27s_formula" title="Brahmagupta's formula">Brahmagupta's formula</a>, for the area of a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a> (a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> <a href="/wiki/Inscribed_figure" title="Inscribed figure">inscribed</a> in a circle) in terms of its sides. In 1842, the German mathematicians <a href="/wiki/Carl_Anton_Bretschneider" title="Carl Anton Bretschneider">Carl Anton Bretschneider</a> and <a href="/wiki/Karl_Georg_Christian_von_Staudt" title="Karl Georg Christian von Staudt">Karl Georg Christian von Staudt</a> independently found a formula, known as <a href="/wiki/Bretschneider%27s_formula" title="Bretschneider's formula">Bretschneider's formula</a>, for the area of any quadrilateral. <div class="mw-heading mw-heading3"><h3 id="General_polygon_area">General polygon area</h3>[<a href="/w/index.php?title=Area&action=edit&section=10" title="Edit section: General polygon area">edit</a>]</div> The development of <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinates</a> by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> in the 17th century allowed the development of the <a href="/wiki/Shoelace_formula" title="Shoelace formula">surveyor's formula</a> for the area of any polygon with known <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a> locations by <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a> in the 19th century. <div class="mw-heading mw-heading3"><h3 id="Areas_determined_using_calculus">Areas determined using calculus</h3>[<a href="/w/index.php?title=Area&action=edit&section=11" title="Edit section: Areas determined using calculus">edit</a>]</div> The development of <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a> in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an <a href="/wiki/Ellipse#Area" title="Ellipse">ellipse</a> and the <a href="/wiki/Surface_area" title="Surface area">surface areas</a> of various curved three-dimensional objects. <div class="mw-heading mw-heading2"><h2 id="Area_formulas">Area formulas</h2>[<a href="/w/index.php?title=Area&action=edit&section=12" title="Edit section: Area formulas">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Polygon_formulas">Polygon formulas</h3>[<a href="/w/index.php?title=Area&action=edit&section=13" title="Edit section: Polygon formulas">edit</a>]</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/Polygon#Area" title="Polygon">Polygon § Area</a></div> For a non-self-intersecting (<a href="/wiki/Simple_polygon" title="Simple polygon">simple</a>) polygon, the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinates</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},y_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i},y_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6dbb919b91ccacf17ed47898048428a1baf9703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:2.843ex;" alt="{\displaystyle (x_{i},y_{i})}"> (i=0, 1, ..., n-1) of whose n <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> are known, the area is given by the <a href="/wiki/Shoelace_formula" title="Shoelace formula">surveyor's formula</a>:<a href="#cite_note-25">[25]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {1}{2}}{\Biggl \vert }\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i}){\Biggr \vert }}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}{\Biggl \vert }\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i}){\Biggr \vert }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a7a8f3daee3a89bf30d86734f007b563007a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.863ex; height:7.343ex;" alt="{\displaystyle A={\frac {1}{2}}{\Biggl \vert }\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i}){\Biggr \vert }}"></dd></dl> where when i=n-1, then i+1 is expressed as <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulus</a> n and so refers to 0. <div class="mw-heading mw-heading4"><h4 id="Rectangles">Rectangles</h4>[<a href="/w/index.php?title=Area&action=edit&section=14" title="Edit section: Rectangles">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:RectangleLengthWidth.svg" class="mw-file-description"><img alt="A rectangle with length and width labelled" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/RectangleLengthWidth.svg/170px-RectangleLengthWidth.svg.png" decoding="async" width="170" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/RectangleLengthWidth.svg/255px-RectangleLengthWidth.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/RectangleLengthWidth.svg/340px-RectangleLengthWidth.svg.png 2x" data-file-width="220" data-file-height="170" /></a><figcaption>The area of this rectangle is lw.</figcaption></figure> The most basic area formula is the formula for the area of a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>. Given a rectangle with length l and width w, the formula for the area is:<a href="#cite_note-AF-2">[2]</a> <dl><dd>A = lw  (rectangle).</dd></dl> That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula:<a href="#cite_note-MathWorld-1">[1]</a><a href="#cite_note-AF-2">[2]</a> <dl><dd>A = s2  (square).</dd></dl> The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a <a href="/wiki/Definition" title="Definition">definition</a> or <a href="/wiki/Axiom" title="Axiom">axiom</a>. On the other hand, if <a href="/wiki/Geometry" title="Geometry">geometry</a> is developed before <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, this formula can be used to define <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a>. <div class="mw-heading mw-heading4"><h4 id="Dissection,_parallelograms,_and_triangles">Dissection, parallelograms, and triangles</h4>[<a href="/w/index.php?title=Area&action=edit&section=15" title="Edit section: Dissection, parallelograms, and triangles">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Triangle_area" class="mw-redirect" title="Triangle area">Triangle area</a> and <a href="/wiki/Parallelogram#Area_formula" title="Parallelogram">Parallelogram § Area formula</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ParallelogramArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/170px-ParallelogramArea.svg.png" decoding="async" width="170" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/255px-ParallelogramArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/340px-ParallelogramArea.svg.png 2x" data-file-width="204" data-file-height="268" /></a><figcaption>A parallelogram can be cut up and re-arranged to form a rectangle.</figcaption></figure> Most other simple formulas for area follow from the method of <a href="/wiki/Dissection_(geometry)" class="mw-redirect" title="Dissection (geometry)">dissection</a>. This involves cutting a shape into pieces, whose areas must <a href="/wiki/Addition" title="Addition">sum</a> to the area of the original shape. For an example, any <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> can be subdivided into a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a> and a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a>, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:<a href="#cite_note-AF-2">[2]</a> <dl><dd>A = bh  (parallelogram).</dd></dl> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:TriangleArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/TriangleArea.svg/170px-TriangleArea.svg.png" decoding="async" width="170" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/TriangleArea.svg/255px-TriangleArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/TriangleArea.svg/340px-TriangleArea.svg.png 2x" data-file-width="220" data-file-height="160" /></a><figcaption>A parallelogram split into two equal triangles</figcaption></figure> However, the same parallelogram can also be cut along a <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> into two <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> triangles, as shown in the figure to the right. It follows that the area of each <a href="/wiki/Triangle" title="Triangle">triangle</a> is half the area of the parallelogram:<a href="#cite_note-AF-2">[2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {1}{2}}bh}"> <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>b</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}bh}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78bb1af1b9493a6b26955f711ae9b95a68f671be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.177ex; height:5.176ex;" alt="{\displaystyle A={\frac {1}{2}}bh}">  (triangle).</dd></dl> Similar arguments can be used to find area formulas for the <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a><a href="#cite_note-26">[26]</a> as well as more complicated <a href="/wiki/Polygon" title="Polygon">polygons</a>.<a href="#cite_note-27">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Area_of_curved_shapes">Area of curved shapes</h3>[<a href="/w/index.php?title=Area&action=edit&section=16" title="Edit section: Area of curved shapes">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Circles">Circles</h4>[<a href="/w/index.php?title=Area&action=edit&section=17" title="Edit section: Circles">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CircleArea.svg" class="mw-file-description"><img alt="A circle divided into many sectors can be re-arranged roughly to form a parallelogram" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/220px-CircleArea.svg.png" decoding="async" width="220" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/330px-CircleArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/CircleArea.svg/440px-CircleArea.svg.png 2x" data-file-width="240" data-file-height="260" /></a><figcaption>A circle can be divided into <a href="/wiki/Circular_sector" title="Circular sector">sectors</a> which rearrange to form an approximate <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>.</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> The formula for the area of a <a href="/wiki/Circle" title="Circle">circle</a> (more properly called the area enclosed by a circle or the area of a <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a>) is based on a similar method. Given a circle of radius r, it is possible to partition the circle into <a href="/wiki/Circular_sector" title="Circular sector">sectors</a>, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r, and the width is half the <a href="/wiki/Circumference" title="Circumference">circumference</a> of the circle, or πr. Thus, the total area of the circle is πr2:<a href="#cite_note-AF-2">[2]</a> <dl><dd>A = πr2  (circle).</dd></dl> Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of the areas of the approximate parallelograms is exactly πr2, which is the area of the circle.<a href="#cite_note-Surveyor-28">[28]</a> This argument is actually a simple application of the ideas of <a href="/wiki/Calculus" title="Calculus">calculus</a>. In ancient times, the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a>. Using modern methods, the area of a circle can be computed using a <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;=\;2\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\;=\;2\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b8d5ed507b330b0f07bf4310d64c8b687ea2ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.668ex; height:6.009ex;" alt="{\displaystyle A\;=\;2\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}.}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Ellipses">Ellipses</h4>[<a href="/w/index.php?title=Area&action=edit&section=18" title="Edit section: Ellipses">edit</a>]</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/Ellipse#Area" title="Ellipse">Ellipse § Area</a></div> The formula for the area enclosed by an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> is related to the formula of a circle; for an ellipse with <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major</a> and <a href="/wiki/Semi-minor_axis" class="mw-redirect" title="Semi-minor axis">semi-minor</a> axes x and y the formula is:<a href="#cite_note-AF-2">[2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\pi xy.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <mi>x</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi xy.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a81da558d3d525f9ff7688980d2788087713390f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.306ex; height:2.509ex;" alt="{\displaystyle A=\pi xy.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Non-planar_surface_area">Non-planar surface area</h3>[<a href="/w/index.php?title=Area&action=edit&section=19" title="Edit section: Non-planar surface area">edit</a>]</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/Surface_area" title="Surface area">Surface area</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Archimedes_sphere_and_cylinder.svg" class="mw-file-description"><img alt="A blue sphere inside a cylinder of the same height and radius" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/220px-Archimedes_sphere_and_cylinder.svg.png" decoding="async" width="220" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/330px-Archimedes_sphere_and_cylinder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Archimedes_sphere_and_cylinder.svg/440px-Archimedes_sphere_and_cylinder.svg.png 2x" data-file-width="426" data-file-height="461" /></a><figcaption><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> showed that the surface area of a <a href="/wiki/Sphere" title="Sphere">sphere</a> is exactly four times the area of a flat <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a> of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a> of the same height and radius.</figcaption></figure> Most basic formulas for <a href="/wiki/Surface_area" title="Surface area">surface area</a> can be obtained by cutting surfaces and flattening them out (see: <a href="/wiki/Developable_surface" title="Developable surface">developable surfaces</a>). For example, if the side surface of a <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a> (or any <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prism</a>) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cone</a>, the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a <a href="/wiki/Sphere" title="Sphere">sphere</a> is more difficult to derive: because a sphere has nonzero <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> in his work <a href="/wiki/On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a>. The formula is:<a href="#cite_note-MathWorldSurfaceArea-6">[6]</a> <dl><dd>A = 4πr2  (sphere),</dd></dl> where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to <a href="/wiki/Calculus" title="Calculus">calculus</a>. <div class="mw-heading mw-heading3"><h3 id="General_formulas">General formulas</h3>[<a href="/w/index.php?title=Area&action=edit&section=20" title="Edit section: General formulas">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Areas_of_2-dimensional_figures">Areas of 2-dimensional figures</h4>[<a href="/w/index.php?title=Area&action=edit&section=21" title="Edit section: Areas of 2-dimensional figures">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Triangle_GeometryArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Triangle_GeometryArea.svg/220px-Triangle_GeometryArea.svg.png" decoding="async" width="220" height="55" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Triangle_GeometryArea.svg/330px-Triangle_GeometryArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Triangle_GeometryArea.svg/440px-Triangle_GeometryArea.svg.png 2x" data-file-width="473" data-file-height="119" /></a><figcaption>Triangle area <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\tfrac {b\cdot h}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>h</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\tfrac {b\cdot h}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4d00ca1155c1418f0f6dcef9bdc0c9e4bb6236" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.787ex; height:3.676ex;" alt="{\displaystyle A={\tfrac {b\cdot h}{2}}}"></figcaption></figure> <ul><li>A <a href="/wiki/Triangle" title="Triangle">triangle</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}Bh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>B</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}Bh}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f09dd787bfe06fe8b0eb87d06b87f8d2c05b91ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.761ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}Bh}"> (where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known. If the lengths of the three sides are known then <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> can be used: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7e3feb227eeda3c19b052c797ee19a01a6c467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.869ex; height:4.843ex;" alt="{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}"> where a, b, c are the sides of the triangle, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\tfrac {1}{2}}(a+b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\tfrac {1}{2}}(a+b+c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed2c4193212526a50585182f301e85e2f1cdfde8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.571ex; height:3.509ex;" alt="{\displaystyle s={\tfrac {1}{2}}(a+b+c)}"> is half of its perimeter.<a href="#cite_note-AF-2">[2]</a> If an angle and its two included sides are given, the area is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}ab\sin(C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}ab\sin(C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3719986bc49ddbdd159b5920e995f1c79f0b6c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.704ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}ab\sin(C)}"> where C is the given angle and a and b are its included sides.<a href="#cite_note-AF-2">[2]</a> If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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 {\tfrac {1}{2}}(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc87b3de461b79f0fc4f5fde09a0cb00ced460e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:45.134ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3})}">. This formula is also known as the <a href="/wiki/Shoelace_formula" title="Shoelace formula">shoelace formula</a> and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points (x1,y1), (x2,y2), and (x3,y3). The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use <a href="/wiki/Calculus" title="Calculus">calculus</a> to find the area.</li> <li>A <a href="/wiki/Simple_polygon" title="Simple polygon">simple polygon</a> constructed on a grid of equal-distanced points (i.e., points with <a href="/wiki/Integer" title="Integer">integer</a> coordinates) such that all the polygon's vertices are grid points: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i+{\frac {b}{2}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i+{\frac {b}{2}}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f6bf0fc9eaa8a4693b7dacc2a9e1a420dcedc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.644ex; height:5.343ex;" alt="{\displaystyle i+{\frac {b}{2}}-1}">, where i is the number of grid points inside the polygon and b is the number of boundary points. This result is known as <a href="/wiki/Pick%27s_theorem" title="Pick's theorem">Pick's theorem</a>.<a href="#cite_note-Pick-29">[29]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Area_in_calculus">Area in calculus</h4>[<a href="/w/index.php?title=Area&action=edit&section=22" title="Edit section: Area in calculus">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Integral_as_region_under_curve.svg" class="mw-file-description"><img alt="A diagram showing the area between a given curve and the x-axis" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/220px-Integral_as_region_under_curve.svg.png" decoding="async" width="220" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/330px-Integral_as_region_under_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/440px-Integral_as_region_under_curve.svg.png 2x" data-file-width="750" data-file-height="700" /></a><figcaption>Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Areabetweentwographs.svg" class="mw-file-description"><img alt="A diagram showing the area between two functions" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Areabetweentwographs.svg/220px-Areabetweentwographs.svg.png" decoding="async" width="220" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Areabetweentwographs.svg/330px-Areabetweentwographs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Areabetweentwographs.svg/440px-Areabetweentwographs.svg.png 2x" data-file-width="1000" data-file-height="900" /></a><figcaption>The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions</figcaption></figure> <ul><li>The area between a positive-valued curve and the horizontal axis, measured between two values a and b (b is defined as the larger of the two values) on the horizontal axis, is given by the integral from a to b of the function that represents the curve:<a href="#cite_note-MathWorld-1">[1]</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\int _{a}^{b}f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\int _{a}^{b}f(x)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c7fde7f01e5a0537613d14280b7811329b3c383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.628ex; height:6.343ex;" alt="{\displaystyle A=\int _{a}^{b}f(x)\,dx.}"></dd></dl> <ul><li>The area between the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graphs</a> of two functions is <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equal</a> to the <a href="/wiki/Integral" title="Integral">integral</a> of one <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, f(x), <a href="/wiki/Subtraction" title="Subtraction">minus</a> the integral of the other function, g(x):</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\int _{a}^{b}(f(x)-g(x))\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\int _{a}^{b}(f(x)-g(x))\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ba0f682fd365a15c175a3e2752970178d09edb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.145ex; height:6.343ex;" alt="{\displaystyle A=\int _{a}^{b}(f(x)-g(x))\,dx,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is the curve with the greater y-value.</dd></dl> <ul><li>An area bounded by a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/979c97f7e8c52e8ee26aa2e5ccd517cc76e31508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.096ex; height:2.843ex;" alt="{\displaystyle r=r(\theta )}"> expressed in <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a> is:<a href="#cite_note-MathWorld-1">[1]</a></li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={1 \over 2}\int r^{2}\,d\theta .}"> <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> <mo>∫</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={1 \over 2}\int r^{2}\,d\theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ceec9961f3bfb4a2e5edfe77b4465adff35c37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.251ex; height:5.676ex;" alt="{\displaystyle A={1 \over 2}\int r^{2}\,d\theta .}"></dd></dl> <ul><li>The area enclosed by a <a href="/wiki/Parametric_curve" class="mw-redirect" title="Parametric curve">parametric curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}(t)=(x(t),y(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}(t)=(x(t),y(t))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60fca3145cbdd2d5c7ecfefa8e07a8041e39c449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.703ex; height:2.843ex;" alt="{\displaystyle {\vec {u}}(t)=(x(t),y(t))}"> with endpoints <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}(t_{0})={\vec {u}}(t_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}(t_{0})={\vec {u}}(t_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8b51a1236ada19d3e70d976b6d617158edfadb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.164ex; height:2.843ex;" alt="{\displaystyle {\vec {u}}(t_{0})={\vec {u}}(t_{1})}"> is given by the <a href="/wiki/Line_integral" title="Line integral">line integrals</a>:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{t_{0}}^{t_{1}}x{\dot {y}}\,dt=-\oint _{t_{0}}^{t_{1}}y{\dot {x}}\,dt={1 \over 2}\oint _{t_{0}}^{t_{1}}(x{\dot {y}}-y{\dot {x}})\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{t_{0}}^{t_{1}}x{\dot {y}}\,dt=-\oint _{t_{0}}^{t_{1}}y{\dot {x}}\,dt={1 \over 2}\oint _{t_{0}}^{t_{1}}(x{\dot {y}}-y{\dot {x}})\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940fad5094a1ce067667a15559feac98c0f02d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.129ex; height:6.509ex;" alt="{\displaystyle \oint _{t_{0}}^{t_{1}}x{\dot {y}}\,dt=-\oint _{t_{0}}^{t_{1}}y{\dot {x}}\,dt={1 \over 2}\oint _{t_{0}}^{t_{1}}(x{\dot {y}}-y{\dot {x}})\,dt}"></dd></dl></dd> <dd>or the z-component of <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over 2}\oint _{t_{0}}^{t_{1}}{\vec {u}}\times {\dot {\vec {u}}}\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over 2}\oint _{t_{0}}^{t_{1}}{\vec {u}}\times {\dot {\vec {u}}}\,dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6bf91b1745a4a27fe07a8b76f66324fb212233" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.484ex; height:6.509ex;" alt="{\displaystyle {1 \over 2}\oint _{t_{0}}^{t_{1}}{\vec {u}}\times {\dot {\vec {u}}}\,dt.}"></dd></dl></dd> <dd>(For details, see <a href="/wiki/Green%27s_theorem#Area_calculation" title="Green's theorem">Green's theorem § Area calculation</a>.) This is the principle of the <a href="/wiki/Planimeter" title="Planimeter">planimeter</a> mechanical device.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Bounded_area_between_two_quadratic_functions">Bounded area between two quadratic functions</h4>[<a href="/w/index.php?title=Area&action=edit&section=23" title="Edit section: Bounded area between two quadratic functions">edit</a>]</div> To find the bounded area between two <a href="/wiki/Quadratic_function" title="Quadratic function">quadratic functions</a>, we first subtract one from the other, writing the difference as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)-g(x)=ax^{2}+bx+c=a(x-\alpha )(x-\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)-g(x)=ax^{2}+bx+c=a(x-\alpha )(x-\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db282d248f9398e0a6fbec8fe342148ba41a4a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.347ex; height:3.176ex;" alt="{\displaystyle f(x)-g(x)=ax^{2}+bx+c=a(x-\alpha )(x-\beta )}"> where f(x) is the quadratic upper bound and g(x) is the quadratic lower bound. By the area integral formulas above and <a href="/wiki/Vieta%27s_formulas" title="Vieta's formulas">Vieta's formula</a>, we can obtain that<a href="#cite_note-30">[30]</a><a href="#cite_note-31">[31]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {(b^{2}-4ac)^{3/2}}{6a^{2}}}={\frac {a}{6}}(\beta -\alpha )^{3},\qquad a\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>6</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>6</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <mi>a</mi> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {(b^{2}-4ac)^{3/2}}{6a^{2}}}={\frac {a}{6}}(\beta -\alpha )^{3},\qquad a\neq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3244866c46c29a0bb151501fc90f375a1ec3a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:43.981ex; height:6.343ex;" alt="{\displaystyle A={\frac {(b^{2}-4ac)^{3/2}}{6a^{2}}}={\frac {a}{6}}(\beta -\alpha )^{3},\qquad a\neq 0.}"> The above remains valid if one of the bounding functions is linear instead of quadratic. <div class="mw-heading mw-heading4"><h4 id="Surface_area_of_3-dimensional_figures">Surface area of 3-dimensional figures</h4>[<a href="/w/index.php?title=Area&action=edit&section=24" title="Edit section: Surface area of 3-dimensional figures">edit</a>]</div> <ul><li><a href="/wiki/Cone" title="Cone">Cone</a>:<a href="#cite_note-MathWorldCone-32">[32]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41aca8f13a1e2996128c0ae640d7261682edf793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.093ex; height:4.843ex;" alt="{\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)}">, where r is the radius of the circular base, and h is the height. That can also be rewritten as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi r^{2}+\pi rl}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>π</mi> <mi>r</mi> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{2}+\pi rl}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9939e45942874b1579592e41aacbe316226bad99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.349ex; height:2.843ex;" alt="{\displaystyle \pi r^{2}+\pi rl}"><a href="#cite_note-MathWorldCone-32">[32]</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi r(r+l)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r(r+l)\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eda1165be2830dc4647cd6d3ce53b7aa6d345ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.159ex; height:2.843ex;" alt="{\displaystyle \pi r(r+l)\,\!}"> where r is the radius and l is the slant height of the cone. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd37db3982ad4e1157dcf8ddbfb280e7bae3b192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.676ex;" alt="{\displaystyle \pi r^{2}}"> is the base area while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi rl}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>r</mi> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi rl}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150bc25be0060c8e640e924e29242b9a1a8af64c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.074ex; height:2.176ex;" alt="{\displaystyle \pi rl}"> is the lateral surface area of the cone.<a href="#cite_note-MathWorldCone-32">[32]</a></li> <li><a href="/wiki/Cube" title="Cube">Cube</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6s^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c8686e0f2659ed3e427883b9863e411b8832d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.307ex; height:2.676ex;" alt="{\displaystyle 6s^{2}}">, where s is the length of an edge.<a href="#cite_note-MathWorldSurfaceArea-6">[6]</a></li> <li><a href="/wiki/Cylinder" title="Cylinder">Cylinder</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi r(r+h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi r(r+h)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540fd341370c845484ed38751872e27df0219682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.58ex; height:2.843ex;" alt="{\displaystyle 2\pi r(r+h)}">, where r is the radius of a base and h is the height. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e811131a9c6c5f45e6657e0fc506e7e2a37f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.176ex;" alt="{\displaystyle 2\pi r}"> can also be rewritten as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc818436da025a58d10638dc12c92ad1c4e3ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.176ex;" alt="{\displaystyle \pi d}">, where d is the diameter.</li> <li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">Prism</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2B+Ph}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>B</mi> <mo>+</mo> <mi>P</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2B+Ph}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881aa43d6a2ffa10f58717ca57ba2c8125f0d1c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.851ex; height:2.343ex;" alt="{\displaystyle 2B+Ph}">, where B is the area of a base, P is the perimeter of a base, and h is the height of the prism.</li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B+{\frac {PL}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>L</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B+{\frac {PL}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d5cac9cbaea2d315ce42caf71d241056c043b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.769ex; height:5.176ex;" alt="{\displaystyle B+{\frac {PL}{2}}}">, where B is the area of the base, P is the perimeter of the base, and L is the length of the slant.</li> <li><a href="/wiki/Rectangular_prism" class="mw-redirect" title="Rectangular prism">Rectangular prism</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(\ell w+\ell h+wh)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mi>w</mi> <mo>+</mo> <mi>ℓ</mi> <mi>h</mi> <mo>+</mo> <mi>w</mi> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2(\ell w+\ell h+wh)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7db04a449659be887d57e8c8d175a0bc6f720e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.598ex; height:2.843ex;" alt="{\displaystyle 2(\ell w+\ell h+wh)}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> is the length, w is the width, and h is the height.</li></ul> <div class="mw-heading mw-heading4"><h4 id="General_formula_for_surface_area">General formula for surface area</h4>[<a href="/w/index.php?title=Area&action=edit&section=25" title="Edit section: General formula for surface area">edit</a>]</div> The general formula for the surface area of the graph of a continuously differentiable function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(x,y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(x,y),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1269bfd7a8dbf5a109363ce2a7992efdf8e406a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.44ex; height:2.843ex;" alt="{\displaystyle z=f(x,y),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in D\subset \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>D</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in D\subset \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/889ab97d6b3f931120e277a925164d57cda96072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.924ex; height:3.176ex;" alt="{\displaystyle (x,y)\in D\subset \mathbb {R} ^{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> is a region in the xy-plane with the smooth boundary: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\iint _{D}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}+1}}\,dx\,dy.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\iint _{D}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}+1}}\,dx\,dy.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1ccc930568eb5c289a5a2e6a29c774b6138e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.713ex; height:7.676ex;" alt="{\displaystyle A=\iint _{D}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}+1}}\,dx\,dy.}"></dd></dl> An even more general formula for the area of the graph of a <a href="/wiki/Parametric_surface" title="Parametric surface">parametric surface</a> in the vector form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =\mathbf {r} (u,v),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {r} (u,v),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aeec617173611dac8fc72db0dff05aa533de232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.25ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} =\mathbf {r} (u,v),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> is a continuously differentiable vector function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)\in D\subset \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>D</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u,v)\in D\subset \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840754272f6c3d17618fe2a2ca80697700dc9adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.896ex; height:3.176ex;" alt="{\displaystyle (u,v)\in D\subset \mathbb {R} ^{2}}"> is:<a href="#cite_note-doCarmo-8">[8]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\iint _{D}\left|{\frac {\partial \mathbf {r} }{\partial u}}\times {\frac {\partial \mathbf {r} }{\partial v}}\right|\,du\,dv.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\iint _{D}\left|{\frac {\partial \mathbf {r} }{\partial u}}\times {\frac {\partial \mathbf {r} }{\partial v}}\right|\,du\,dv.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0b4024b8e933d55893aad41806b2e8bedf6ae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.937ex; height:5.843ex;" alt="{\displaystyle A=\iint _{D}\left|{\frac {\partial \mathbf {r} }{\partial u}}\times {\frac {\partial \mathbf {r} }{\partial v}}\right|\,du\,dv.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="List_of_formulas">List of formulas</h3>[<a href="/w/index.php?title=Area&action=edit&section=26" title="Edit section: List of formulas">edit</a>]</div> <table class="wikitable"> <caption>Additional common formulas for area: </caption> <tbody><tr> <th>Shape </th> <th>Formula </th> <th>Variables </th></tr> <tr> <td><a href="/wiki/Square" title="Square">Square</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=s^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1607109864e43a307cc541ff06b73a7bfe423500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:2.676ex;" alt="{\displaystyle A=s^{2}}"> </td> <td><a href="/wiki/File:Simple_square_with_sides_marked.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Simple_square_with_sides_marked.svg/120px-Simple_square_with_sides_marked.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Simple_square_with_sides_marked.svg/180px-Simple_square_with_sides_marked.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Simple_square_with_sides_marked.svg/240px-Simple_square_with_sides_marked.svg.png 2x" data-file-width="120" data-file-height="120" /></a> </td></tr> <tr> <td><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d1d98eb0f7365f44de12cd8d7198161ebb1a48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.069ex; height:2.176ex;" alt="{\displaystyle A=ab}"> </td> <td><a href="/wiki/File:Rechteck-ab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Rechteck-ab.svg/120px-Rechteck-ab.svg.png" decoding="async" width="120" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Rechteck-ab.svg/180px-Rechteck-ab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Rechteck-ab.svg/240px-Rechteck-ab.svg.png 2x" data-file-width="126" data-file-height="72" /></a> </td></tr> <tr> <td><a href="/wiki/Triangle" title="Triangle">Triangle</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {1}{2}}bh\,\!}"> <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>b</mi> <mi>h</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}bh\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d797f5563af8f5d34ea0bb8713dcee923f8836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:9.564ex; height:5.176ex;" alt="{\displaystyle A={\frac {1}{2}}bh\,\!}"> </td> <td><a href="/wiki/File:Dreieck-allg-bh.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Dreieck-allg-bh.svg/100px-Dreieck-allg-bh.svg.png" decoding="async" width="100" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Dreieck-allg-bh.svg/150px-Dreieck-allg-bh.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Dreieck-allg-bh.svg/200px-Dreieck-allg-bh.svg.png 2x" data-file-width="144" data-file-height="126" /></a> </td></tr> <tr> <td><a href="/wiki/Triangle" title="Triangle">Triangle</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {1}{2}}ab\sin(\gamma )\,\!}"> <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>a</mi> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}ab\sin(\gamma )\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91134999becbb6016f19e13d63c7eba807007b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:15.769ex; height:5.176ex;" alt="{\displaystyle A={\frac {1}{2}}ab\sin(\gamma )\,\!}"> </td> <td><a href="/wiki/File:Dreieck-allg-w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Dreieck-allg-w.svg/100px-Dreieck-allg-w.svg.png" decoding="async" width="100" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Dreieck-allg-w.svg/150px-Dreieck-allg-w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Dreieck-allg-w.svg/200px-Dreieck-allg-w.svg.png 2x" data-file-width="144" data-file-height="126" /></a> </td></tr> <tr> <td><a href="/wiki/Triangle" title="Triangle">Triangle</a> (<a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a>) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce11654ff137a2f313184e36f20768b0890d3759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:29.098ex; height:4.843ex;" alt="{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}\,\!}"> </td> <td><a href="/wiki/File:Dreieck-allg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Dreieck-allg.svg/100px-Dreieck-allg.svg.png" decoding="async" width="100" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Dreieck-allg.svg/150px-Dreieck-allg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Dreieck-allg.svg/200px-Dreieck-allg.svg.png 2x" data-file-width="144" data-file-height="126" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\tfrac {1}{2}}(a+b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\tfrac {1}{2}}(a+b+c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed2c4193212526a50585182f301e85e2f1cdfde8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.571ex; height:3.509ex;" alt="{\displaystyle s={\tfrac {1}{2}}(a+b+c)}"> </td></tr> <tr> <td><a href="/wiki/Isosceles_triangle" title="Isosceles triangle">Isosceles triangle</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {c}{4}}{\sqrt {4a^{2}-c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {c}{4}}{\sqrt {4a^{2}-c^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cdc3d444be871d87a9088601f933a9fcdf39733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.512ex; height:4.676ex;" alt="{\displaystyle A={\frac {c}{4}}{\sqrt {4a^{2}-c^{2}}}}"> </td> <td><a href="/wiki/File:Dreieck-gsch.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Dreieck-gsch.svg/80px-Dreieck-gsch.svg.png" decoding="async" width="80" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Dreieck-gsch.svg/120px-Dreieck-gsch.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Dreieck-gsch.svg/160px-Dreieck-gsch.svg.png 2x" data-file-width="108" data-file-height="126" /></a> </td></tr> <tr> <td>Regular <a href="/wiki/Triangle" title="Triangle">triangle</a> (<a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>4</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8109993210fe0425fe4949b7c8c8ce1faa1d48a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:11.447ex; height:5.843ex;" alt="{\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}\,\!}"> </td> <td><a href="/wiki/File:Dreieck-gseit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Dreieck-gseit.svg/100px-Dreieck-gseit.svg.png" decoding="async" width="100" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Dreieck-gseit.svg/150px-Dreieck-gseit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Dreieck-gseit.svg/200px-Dreieck-gseit.svg.png 2x" data-file-width="144" data-file-height="126" /></a> </td></tr> <tr> <td><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a>/<a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {1}{2}}de}"> <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>d</mi> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}de}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d11afa99e38d284615d1e8110b0a5cb9701976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.14ex; height:5.176ex;" alt="{\displaystyle A={\frac {1}{2}}de}"> </td> <td><a href="/wiki/File:Raute-de.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Raute-de.svg/160px-Raute-de.svg.png" decoding="async" width="160" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Raute-de.svg/240px-Raute-de.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Raute-de.svg/320px-Raute-de.svg.png 2x" data-file-width="216" data-file-height="126" /></a> </td></tr> <tr> <td><a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=ah_{a}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>a</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=ah_{a}\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c33ad35df2d9f53110a71cba103a87797117227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:8.899ex; height:2.509ex;" alt="{\displaystyle A=ah_{a}\,\!}"> </td> <td><a href="/wiki/File:Parallelog-aha.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Parallelog-aha.svg/160px-Parallelog-aha.svg.png" decoding="async" width="160" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Parallelog-aha.svg/240px-Parallelog-aha.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Parallelog-aha.svg/320px-Parallelog-aha.svg.png 2x" data-file-width="162" data-file-height="72" /></a> </td></tr> <tr> <td><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {(a+c)h}{2}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mi>h</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {(a+c)h}{2}}\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8db9433565e4c823948974baab1bb9a07d50a4ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:14.29ex; height:5.676ex;" alt="{\displaystyle A={\frac {(a+c)h}{2}}\,\!}"> </td> <td><a href="/wiki/File:Trapez-abcdh.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Trapez-abcdh.svg/150px-Trapez-abcdh.svg.png" decoding="async" width="150" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Trapez-abcdh.svg/225px-Trapez-abcdh.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Trapez-abcdh.svg/300px-Trapez-abcdh.svg.png 2x" data-file-width="189" data-file-height="108" /></a> </td></tr> <tr> <td>Regular <a href="/wiki/Hexagon" title="Hexagon">hexagon</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {3}{2}}{\sqrt {3}}a^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {3}{2}}{\sqrt {3}}a^{2}\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b1fa9f06cf4c87e28a98a62e42550951c222837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:12.61ex; height:5.176ex;" alt="{\displaystyle A={\frac {3}{2}}{\sqrt {3}}a^{2}\,\!}"> </td> <td><a href="/wiki/File:Hexagon-a.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Hexagon-a.svg/100px-Hexagon-a.svg.png" decoding="async" width="100" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Hexagon-a.svg/150px-Hexagon-a.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Hexagon-a.svg/200px-Hexagon-a.svg.png 2x" data-file-width="126" data-file-height="144" /></a> </td></tr> <tr> <td>Regular <a href="/wiki/Octagon" title="Octagon">octagon</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=2(1+{\sqrt {2}})a^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2(1+{\sqrt {2}})a^{2}\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1d46f7e735047e5309c52b6f2d6fcb183a5cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:17.586ex; height:3.176ex;" alt="{\displaystyle A=2(1+{\sqrt {2}})a^{2}\,\!}"> </td> <td><a href="/wiki/File:Oktagon-a.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Oktagon-a.svg/120px-Oktagon-a.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Oktagon-a.svg/180px-Oktagon-a.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Oktagon-a.svg/240px-Oktagon-a.svg.png 2x" data-file-width="180" data-file-height="180" /></a> </td></tr> <tr> <td><a href="/wiki/Regular_polygon" title="Regular polygon">Regular polygon</a> (<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><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}"> sides) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=n{\frac {ar}{2}}={\frac {pr}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mi>r</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=n{\frac {ar}{2}}={\frac {pr}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/354b89a0f2e027b715f7457dbc20fa2c5066637c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.504ex; height:4.843ex;" alt="{\displaystyle A=n{\frac {ar}{2}}={\frac {pr}{2}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad ={\tfrac {1}{4}}na^{2}\cot({\tfrac {\pi }{n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>n</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad ={\tfrac {1}{4}}na^{2}\cot({\tfrac {\pi }{n}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a890b71c812186d1c7a7b00691f47d8894569e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.231ex; height:3.509ex;" alt="{\displaystyle \quad ={\tfrac {1}{4}}na^{2}\cot({\tfrac {\pi }{n}})}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad =nr^{2}\tan({\tfrac {\pi }{n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo>=</mo> <mi>n</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad =nr^{2}\tan({\tfrac {\pi }{n}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa0c6ff722626e412a1d61972be24dd8e1d3f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.652ex; height:3.343ex;" alt="{\displaystyle \quad =nr^{2}\tan({\tfrac {\pi }{n}})}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad ={\tfrac {1}{4n}}p^{2}\cot({\tfrac {\pi }{n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad ={\tfrac {1}{4n}}p^{2}\cot({\tfrac {\pi }{n}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1792087716dc1ab272bda5419cb0085de55a1b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.762ex; height:3.676ex;" alt="{\displaystyle \quad ={\tfrac {1}{4n}}p^{2}\cot({\tfrac {\pi }{n}})}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad ={\tfrac {1}{2}}nR^{2}\sin({\tfrac {2\pi }{n}})\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>n</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad ={\tfrac {1}{2}}nR^{2}\sin({\tfrac {2\pi }{n}})\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3dde5ca35e56525e8de9aebce33d26affbd5d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:18.686ex; height:3.509ex;" alt="{\displaystyle \quad ={\tfrac {1}{2}}nR^{2}\sin({\tfrac {2\pi }{n}})\,\!}"> </td> <td><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/File:Oktagon-a-r-R.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Oktagon-a-r-R.svg/150px-Oktagon-a-r-R.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Oktagon-a-r-R.svg/225px-Oktagon-a-r-R.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Oktagon-a-r-R.svg/300px-Oktagon-a-r-R.svg.png 2x" data-file-width="189" data-file-height="189" /></a><figcaption></figcaption></figure> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=na\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>n</mi> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=na\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8237110ed23490fd2d7cf587201703fc3d0a091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.562ex; height:2.009ex;" alt="{\displaystyle p=na\ }"> (<a href="/wiki/Perimeter" title="Perimeter">perimeter</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\tfrac {a}{2}}\cot({\tfrac {\pi }{n}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\tfrac {a}{2}}\cot({\tfrac {\pi }{n}}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86622d72f2ee710c3c4a45919c89fdb4fde835af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.618ex; height:3.176ex;" alt="{\displaystyle r={\tfrac {a}{2}}\cot({\tfrac {\pi }{n}}),}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{2}}=r\tan({\tfrac {\pi }{n}})=R\sin({\tfrac {\pi }{n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>r</mi> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{2}}=r\tan({\tfrac {\pi }{n}})=R\sin({\tfrac {\pi }{n}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027949269e1cadcd61be7bd9e9e4d9c2e91a1689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.968ex; height:3.176ex;" alt="{\displaystyle {\tfrac {a}{2}}=r\tan({\tfrac {\pi }{n}})=R\sin({\tfrac {\pi }{n}})}"> <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> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e953871da035a97eb7aded88277ad1a47205893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.341ex; height:1.676ex;" alt="{\displaystyle r:}"> <a href="/wiki/Incircle" class="mw-redirect" title="Incircle">incircle</a> radius <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> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470e63632cae3e604afbf95bd86a9d950ed7e4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.056ex; height:2.176ex;" alt="{\displaystyle R:}"> <a href="/wiki/Circumcircle" title="Circumcircle">circumcircle</a> radius </td></tr> <tr> <td><a href="/wiki/Circle" title="Circle">Circle</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\pi r^{2}={\frac {\pi d^{2}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}={\frac {\pi d^{2}}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5e6bb24b18aa77e1e419ff1e09240456892a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.815ex; height:5.676ex;" alt="{\displaystyle A=\pi r^{2}={\frac {\pi d^{2}}{4}}}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=2r:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mi>r</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=2r:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5ab3607f5070f24d3f689368508e88e45465e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.817ex; height:2.176ex;" alt="{\displaystyle d=2r:}"> <a href="/wiki/Diameter" title="Diameter">diameter</a>) </td> <td><a href="/wiki/File:Kreis-r-tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Kreis-r-tab.svg/100px-Kreis-r-tab.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Kreis-r-tab.svg/150px-Kreis-r-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Kreis-r-tab.svg/200px-Kreis-r-tab.svg.png 2x" data-file-width="94" data-file-height="94" /></a> </td></tr> <tr> <td><a href="/wiki/Circular_sector" title="Circular sector">Circular sector</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {\theta }{2}}r^{2}={\frac {L\cdot r}{2}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>⋅</mo> <mi>r</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {\theta }{2}}r^{2}={\frac {L\cdot r}{2}}\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9100301d343d3fbff54c4eb6579378b2a0d35e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:17.575ex; height:5.343ex;" alt="{\displaystyle A={\frac {\theta }{2}}r^{2}={\frac {L\cdot r}{2}}\,\!}"> </td> <td><a href="/wiki/File:Circle_arc.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Circle_arc.svg/120px-Circle_arc.svg.png" decoding="async" width="120" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Circle_arc.svg/180px-Circle_arc.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Circle_arc.svg/240px-Circle_arc.svg.png 2x" data-file-width="779" data-file-height="703" /></a> </td></tr> <tr> <td><a href="/wiki/Ellipse" title="Ellipse">Ellipse</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\pi ab\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi ab\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054e5e946a7a106ca68c7ca0f3cae845621b678f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:8.788ex; height:2.176ex;" alt="{\displaystyle A=\pi ab\,\!}"> </td> <td><a href="/wiki/File:Ellipse-ab-tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Ellipse-ab-tab.svg/120px-Ellipse-ab-tab.svg.png" decoding="async" width="120" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Ellipse-ab-tab.svg/180px-Ellipse-ab-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Ellipse-ab-tab.svg/240px-Ellipse-ab-tab.svg.png 2x" data-file-width="129" data-file-height="69" /></a> </td></tr> <tr> <td><a href="/wiki/Integral" title="Integral">Integral</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\int _{a}^{b}f(x)\mathrm {d} x,\ f(x)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\int _{a}^{b}f(x)\mathrm {d} x,\ f(x)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d989a91365109ea3815cf4d773358cb626f35c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.963ex; height:6.343ex;" alt="{\displaystyle A=\int _{a}^{b}f(x)\mathrm {d} x,\ f(x)\geq 0}"> </td> <td><a href="/wiki/File:Vase-f-fx-tab.svg" class="mw-file-description" title="hochkant=0.2"><img alt="hochkant=0.2" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Vase-f-fx-tab.svg/218px-Vase-f-fx-tab.svg.png" decoding="async" width="218" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Vase-f-fx-tab.svg/327px-Vase-f-fx-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Vase-f-fx-tab.svg/436px-Vase-f-fx-tab.svg.png 2x" data-file-width="218" data-file-height="82" /></a> </td></tr> <tr> <td> </td> <td><a href="/wiki/Surface_area" title="Surface area">Surface area</a> </td> <td> </td></tr> <tr> <td><a href="/wiki/Sphere_(geometry)" class="mw-redirect" title="Sphere (geometry)">Sphere</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=4\pi r^{2}=\pi d^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>π</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi r^{2}=\pi d^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/825e9926272f5a3c5c4a283b2fa4de7f848f65eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.142ex; height:2.676ex;" alt="{\displaystyle A=4\pi r^{2}=\pi d^{2}}"> </td> <td><a href="/wiki/File:Kugel-1-tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Kugel-1-tab.svg/100px-Kugel-1-tab.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Kugel-1-tab.svg/150px-Kugel-1-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Kugel-1-tab.svg/200px-Kugel-1-tab.svg.png 2x" data-file-width="155" data-file-height="155" /></a> </td></tr> <tr> <td><a href="/wiki/Cuboid" title="Cuboid">Cuboid</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=2(ab+ac+bc)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2(ab+ac+bc)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ffa91220b550ac15eb01ab747e37965bf33f31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.963ex; height:2.843ex;" alt="{\displaystyle A=2(ab+ac+bc)}"> </td> <td><a href="/wiki/File:Quader-1-tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Quader-1-tab.svg/150px-Quader-1-tab.svg.png" decoding="async" width="150" height="85" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Quader-1-tab.svg/225px-Quader-1-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Quader-1-tab.svg/300px-Quader-1-tab.svg.png 2x" data-file-width="187" data-file-height="106" /></a> </td></tr> <tr> <td><a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a> (incl. bottom and top) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=2\pi r(r+h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi r(r+h)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c58f5516eb7757a4ef57213291c912b0230f013d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.422ex; height:2.843ex;" alt="{\displaystyle A=2\pi r(r+h)}"> </td> <td><a href="/wiki/File:Zylinder-1-tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Zylinder-1-tab.svg/120px-Zylinder-1-tab.svg.png" decoding="async" width="120" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Zylinder-1-tab.svg/180px-Zylinder-1-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Zylinder-1-tab.svg/240px-Zylinder-1-tab.svg.png 2x" data-file-width="155" data-file-height="162" /></a> </td></tr> <tr> <td><a href="/wiki/Cone" title="Cone">Cone</a> (incl. bottom) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\pi r(r+{\sqrt {r^{2}+h^{2}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r(r+{\sqrt {r^{2}+h^{2}}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b54715931f51016de066caef9cb9711379bba25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.581ex; height:3.509ex;" alt="{\displaystyle A=\pi r(r+{\sqrt {r^{2}+h^{2}}})}"> </td> <td><a href="/wiki/File:Kegel-1-tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Kegel-1-tab.svg/120px-Kegel-1-tab.svg.png" decoding="async" width="120" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Kegel-1-tab.svg/180px-Kegel-1-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Kegel-1-tab.svg/240px-Kegel-1-tab.svg.png 2x" data-file-width="153" data-file-height="138" /></a> </td></tr> <tr> <td><a href="/wiki/Torus" title="Torus">Torus</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=4\pi ^{2}\cdot R\cdot r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi ^{2}\cdot R\cdot r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f884b29b1cdb02a79746fe57e1107ff81a94ff37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.563ex; height:2.676ex;" alt="{\displaystyle A=4\pi ^{2}\cdot R\cdot r}"> </td> <td><a href="/wiki/File:Torus-1-tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Torus-1-tab.svg/200px-Torus-1-tab.svg.png" decoding="async" width="200" height="59" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Torus-1-tab.svg/300px-Torus-1-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Torus-1-tab.svg/400px-Torus-1-tab.svg.png 2x" data-file-width="285" data-file-height="84" /></a> </td></tr> <tr> <td><a href="/wiki/Surface_of_revolution" title="Surface of revolution">Surface of revolution</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=2\pi \int _{a}^{b}\!f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2\pi \int _{a}^{b}\!f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07cf6da325a77c650339de80d9b00d984ca3751d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.983ex; height:6.343ex;" alt="{\displaystyle A=2\pi \int _{a}^{b}\!f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x}"> (rotation around the x-axis) </td> <td><a href="/wiki/File:Vase-1-tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Vase-1-tab.svg/220px-Vase-1-tab.svg.png" decoding="async" width="220" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Vase-1-tab.svg/330px-Vase-1-tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Vase-1-tab.svg/440px-Vase-1-tab.svg.png 2x" data-file-width="396" data-file-height="210" /></a> </td></tr> </tbody></table> The above calculations show how to find the areas of many common <a href="/wiki/Shapes" class="mw-redirect" title="Shapes">shapes</a>. The areas of irregular (and thus arbitrary) polygons can be calculated using the "<a href="/wiki/Surveyor%27s_formula" class="mw-redirect" title="Surveyor's formula">Surveyor's formula</a>" (shoelace formula).<a href="#cite_note-Surveyor-28">[28]</a> <div class="mw-heading mw-heading3"><h3 id="Relation_of_area_to_perimeter">Relation of area to perimeter</h3>[<a href="/w/index.php?title=Area&action=edit&section=27" title="Edit section: Relation of area to perimeter">edit</a>]</div> The <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a> states that, for a closed curve of length L (so the region it encloses has <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> L) and for area A of the region that it encloses, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi A\leq L^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <mi>A</mi> <mo>≤</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi A\leq L^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8bd2c0d6a1230d3672a79f0974b1488aa89df1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.62ex; height:3.009ex;" alt="{\displaystyle 4\pi A\leq L^{2},}"></dd></dl> and equality holds if and only if the curve is a <a href="/wiki/Circle" title="Circle">circle</a>. Thus a circle has the largest area of any closed figure with a given perimeter. At the other extreme, a figure with given perimeter L could have an arbitrarily small area, as illustrated by a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a> that is "tipped over" arbitrarily far so that two of its <a href="/wiki/Angle" title="Angle">angles</a> are arbitrarily close to 0° and the other two are arbitrarily close to 180°. For a circle, the ratio of the area to the <a href="/wiki/Circumference" title="Circumference">circumference</a> (the term for the perimeter of a circle) equals half the <a href="/wiki/Radius" title="Radius">radius</a> r. This can be seen from the area formula πr2 and the circumference formula 2πr. The area of a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a> is half its perimeter times the <a href="/wiki/Apothem" title="Apothem">apothem</a> (where the apothem is the distance from the center to the nearest point on any side). <div class="mw-heading mw-heading3"><h3 id="Fractals">Fractals</h3>[<a href="/w/index.php?title=Area&action=edit&section=28" title="Edit section: Fractals">edit</a>]</div> Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of a <a href="/wiki/Fractal" title="Fractal">fractal</a> drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called the <a href="/wiki/Fractal_dimension" title="Fractal dimension">fractal dimension</a> of the fractal. <a href="#cite_note-Mandelbrot1983-33">[33]</a> <div class="mw-heading mw-heading2"><h2 id="Area_bisectors">Area bisectors</h2>[<a href="/w/index.php?title=Area&action=edit&section=29" title="Edit section: Area bisectors">edit</a>]</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/Bisection#Area_bisectors_and_perimeter_bisectors" title="Bisection">Bisection § Area bisectors and perimeter bisectors</a></div> There are an infinitude of lines that bisect the area of a triangle. Three of them are the <a href="/wiki/Median_(triangle)" class="mw-redirect" title="Median (triangle)">medians</a> of the triangle (which connect the sides' midpoints with the opposite vertices), and these are <a href="/wiki/Concurrent_lines" title="Concurrent lines">concurrent</a> at the triangle's <a href="/wiki/Centroid" title="Centroid">centroid</a>; indeed, they are the only area bisectors that go through the centroid. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its <a href="/wiki/Incircle" class="mw-redirect" title="Incircle">incircle</a>). There are either one, two, or three of these for any given triangle. Any line through the midpoint of a parallelogram bisects the area. All area bisectors of a circle or other ellipse go through the center, and any <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chords</a> through the center bisect the area. In the case of a circle they are the diameters of the circle. <div class="mw-heading mw-heading2"><h2 id="Optimization">Optimization</h2>[<a href="/w/index.php?title=Area&action=edit&section=30" title="Edit section: Optimization">edit</a>]</div> Given a wire contour, the surface of least area spanning ("filling") it is a <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surface</a>. Familiar examples include <a href="/wiki/Soap_bubble" title="Soap bubble">soap bubbles</a>. The question of the <a href="/wiki/Filling_area_conjecture" title="Filling area conjecture">filling area</a> of the <a href="/wiki/Riemannian_circle" class="mw-redirect" title="Riemannian circle">Riemannian circle</a> remains open.<a href="#cite_note-34">[34]</a> The circle has the largest area of any two-dimensional object having the same perimeter. A <a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">cyclic polygon</a> (one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths. A version of the <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a> for triangles states that the triangle of greatest area among all those with a given perimeter is <a href="/wiki/Equilateral" class="mw-redirect" title="Equilateral">equilateral</a>.<a href="#cite_note-Chakerian-35">[35]</a> The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.<a href="#cite_note-36">[36]</a> The ratio of the area of the incircle to the area of an equilateral triangle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e87580f1fcf9c9b18a8e4a5d9261491e79b8bf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:5.097ex; height:5.676ex;" alt="{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}">, is larger than that of any non-equilateral triangle.<a href="#cite_note-37">[37]</a> The ratio of the area to the square of the perimeter of an equilateral triangle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{12{\sqrt {3}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{12{\sqrt {3}}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/844409c74d8d57e537c55a4c22aeea1a25ecd65a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:6.906ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{12{\sqrt {3}}}},}"> is larger than that for any other triangle.<a href="#cite_note-Chakerian-35">[35]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Area&action=edit&section=31" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Brahmagupta_quadrilateral" class="mw-redirect" title="Brahmagupta quadrilateral">Brahmagupta quadrilateral</a>, a cyclic quadrilateral with integer sides, integer diagonals, and integer area.</li> <li><a href="/wiki/Equiareal_map" title="Equiareal map">Equiareal map</a></li> <li><a href="/wiki/Heronian_triangle" title="Heronian triangle">Heronian triangle</a>, a triangle with integer sides and integer area.</li> <li><a href="/wiki/List_of_triangle_inequalities#Area" title="List of triangle inequalities">List of triangle inequalities</a></li> <li><a href="/wiki/One-seventh_area_triangle" title="One-seventh area triangle">One-seventh area triangle</a>, an inner triangle with one-seventh the area of the reference triangle.</li></ul> <dl><dd><ul><li><a href="/wiki/Routh%27s_theorem" title="Routh's theorem">Routh's theorem</a>, a generalization of the one-seventh area triangle.</li></ul></dd></dl> <ul><li><a href="/wiki/Orders_of_magnitude_(area)" title="Orders of magnitude (area)">Orders of magnitude</a>—A list of areas by size.</li> <li><a href="/wiki/Pentagon#Derivation_of_the_area_formula" title="Pentagon">Derivation of the formula of a pentagon</a></li> <li><a href="/wiki/Planimeter" title="Planimeter">Planimeter</a>, an instrument for measuring small areas, e.g. on maps.</li> <li><a href="/wiki/Quadrilateral#Area_of_a_convex_quadrilateral" title="Quadrilateral">Area of a convex quadrilateral</a></li> <li><a href="/wiki/Robbins_pentagon" title="Robbins pentagon">Robbins pentagon</a>, a cyclic pentagon whose side lengths and area are all rational numbers.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Area&action=edit&section=32" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-MathWorld-1">^ <a href="#cite_ref-MathWorld_1-0">a</a> <a href="#cite_ref-MathWorld_1-1">b</a> <a href="#cite_ref-MathWorld_1-2">c</a> <a href="#cite_ref-MathWorld_1-3">d</a> <a href="#cite_ref-MathWorld_1-4">e</a> <a href="#cite_ref-MathWorld_1-5">f</a> <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 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Retrieved 3 July 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Area&rft.pub=Wolfram+MathWorld&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FArea.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea" class="Z3988"> </li> <li id="cite_note-AF-2">^ <a href="#cite_ref-AF_2-0">a</a> <a href="#cite_ref-AF_2-1">b</a> <a href="#cite_ref-AF_2-2">c</a> <a href="#cite_ref-AF_2-3">d</a> <a href="#cite_ref-AF_2-4">e</a> <a href="#cite_ref-AF_2-5">f</a> <a href="#cite_ref-AF_2-6">g</a> <a href="#cite_ref-AF_2-7">h</a> <a href="#cite_ref-AF_2-8">i</a> <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.math.com/tables/geometry/areas.htm">"Area Formulas"</a>. 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Courier Dover Publications. pp. 121–132. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43231-1" title="Special:BookSources/978-0-486-43231-1"><bdi>978-0-486-43231-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160501215852/https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA121">Archived</a> from the original on 2016-05-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Manual+of+Greek+Mathematics&rft.pages=121-132&rft.pub=Courier+Dover+Publications&rft.date=2003&rft.isbn=978-0-486-43231-1&rft.aulast=Heath&rft.aufirst=Thomas+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_HZNr_mGFzQC%26pg%3DPA121&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2003" class="citation book cs1">Stewart, James (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/singlevariableca00stew/page/3">Single variable calculus early transcendentals</a> (5th. ed.). 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University of Chicago Press. p. 26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Aryabhatiya+of+Aryabhata%3A+An+Ancient+Indian+Work+on+Mathematics+and+Astronomy&rft.pages=26&rft.pub=University+of+Chicago+Press&rft.date=1930&rft.aulast=Clark&rft.aufirst=Walter+Eugene&rft_id=https%3A%2F%2Flibarch.nmu.org.ua%2Fbitstream%2Fhandle%2FGenofondUA%2F26818%2Ff14e7857ef0bdd30ca0fd4ec057fe3c3.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXuYu2013" class="citation journal cs1">Xu, Wenwen; Yu, Ning (May 2013). <a rel="nofollow" class="external text" href="http://www.ams.org/journals/notices/201305/rnoti-p596.pdf">"Bridge Named After the Mathematician Who Discovered the Chinese Remainder Theorem"</a> (PDF). 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PT Grafindo Media Pratama. pp. 51–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-979-758-477-1" title="Special:BookSources/978-979-758-477-1"><bdi>978-979-758-477-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170320100900/https://books.google.com/books?id=NFkVfrZBqpUC&pg=PA51">Archived</a> from the original on 2017-03-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matematika&rft.pages=51-&rft.pub=PT+Grafindo+Media+Pratama&rft.isbn=978-979-758-477-1&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNFkVfrZBqpUC%26pg%3DPA51&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=uwqvITs8OaUC&pg=PA157">Get Success UN +SPMB Matematika</a>. 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(October 2008). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/228698127">"Triangles, ellipses, and cubic polynomials"</a>. <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 115 (8): 679–689: Theorem 4.1. <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%2F00029890.2008.11920581">10.1080/00029890.2008.11920581</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27642581">27642581</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:15049234">15049234</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161104141707/https://www.researchgate.net/publication/228698127_Triangles_ellipses_and_cubic_polynomials">Archived</a> from the original on 2016-11-04.</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=Triangles%2C+ellipses%2C+and+cubic+polynomials&rft.volume=115&rft.issue=8&rft.pages=679-689%3A+Theorem+4.1&rft.date=2008-10&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15049234%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27642581%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F00029890.2008.11920581&rft.au=Minda%2C+D.&rft.au=Phelps%2C+S.&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F228698127&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArea" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a 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style="border:none;backgound:none; font-weight:bold;">Linear/translational quantities</td> <td rowspan="12" style="border:none;backgound:none;"></td> <td colspan="4" style="border:none;backgound:none; font-weight:bold;">Angular/rotational quantities</td> </tr> <tr> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;">L</th> <th style="font-weight:normal;">L2</th> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;">θ</th> <th style="font-weight:normal;">θ2</th> </tr> <tr> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: t <a href="/wiki/Second" title="Second">s</a></td> <td><a href="/wiki/Absement" title="Absement">absement</a>: A <a href="/wiki/Meter_second" class="mw-redirect" title="Meter second">m s</a></td> <td></td> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: t <a href="/wiki/Second" title="Second">s</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Distance" title="Distance">distance</a>: d, <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a>: r, s, x, <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> <a href="/wiki/Metre" title="Metre">m</a></td> <td><a class="mw-selflink selflink">area</a>: A <a href="/wiki/Square_metre" title="Square metre">m2</a></td> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Angle" title="Angle">angle</a>: θ, <a href="/wiki/Angular_displacement" title="Angular displacement">angular displacement</a>: θ <a href="/wiki/Radian" title="Radian">rad</a></td> <td><a href="/wiki/Solid_angle" title="Solid angle">solid angle</a>: Ω <a href="/wiki/Steradian" title="Steradian">rad2, sr</a></td> </tr> <tr> <th style="font-weight:normal;">T−1</th> <td><a href="/wiki/Frequency" title="Frequency">frequency</a>: f <a href="/wiki/Inverse_second" title="Inverse second">s−1</a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Speed" title="Speed">speed</a>: v, <a href="/wiki/Velocity" title="Velocity">velocity</a>: v <a href="/wiki/Metre_per_second" title="Metre per second">m s−1</a></td> <td><a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a>: ν, <a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">specific angular momentum</a>: h m2 s−1</td> <th style="font-weight:normal;">T−1</th> <td><a href="/wiki/Frequency" title="Frequency">frequency</a>: f, <a href="/wiki/Rotational_speed" class="mw-redirect" title="Rotational speed">rotational speed</a>: n, <a href="/wiki/Rotational_velocity" class="mw-redirect" title="Rotational velocity">rotational velocity</a>: n <a href="/wiki/Inverse_second" title="Inverse second">s−1</a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Angular_speed" class="mw-redirect" title="Angular speed">angular speed</a>: ω, <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>: ω <a href="/wiki/Radian_per_second" title="Radian per second">rad s−1</a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T−2</th> <td></td> <td><a href="/wiki/Acceleration" title="Acceleration">acceleration</a>: a <a href="/wiki/Metre_per_second_squared" title="Metre per second squared">m s−2</a></td> <td></td> <th style="font-weight:normal;">T−2</th> <td><a href="/wiki/Rotational_acceleration" class="mw-redirect" title="Rotational acceleration">rotational acceleration</a> <a href="/wiki/Inverse_square_second" class="mw-redirect" title="Inverse square second">s−2</a></td> <td><a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a>: α <a href="/wiki/Radian_per_second_squared" class="mw-redirect" title="Radian per second squared">rad s−2</a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T−3</th> <td></td> <td><a href="/wiki/Jerk_(physics)" title="Jerk (physics)">jerk</a>: j m s−3</td> <td></td> <th style="font-weight:normal;">T−3</th> <td></td> <td><a href="/wiki/Jerk_(physics)#Jerk_in_rotation" title="Jerk (physics)">angular jerk</a>: ζ rad s−3</td> <td></td> </tr> <tr style="border-top: 3px double #a2a9b1;"> <th style="font-weight:normal;">M</th> <td><a href="/wiki/Mass" title="Mass">mass</a>: m <a href="/wiki/Kilogram" title="Kilogram">kg</a></td> <td>weighted position: M ⟨x⟩ = ∑ m x </td> <td></td> <th style="font-weight:normal;">ML2</th> <td><a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>: I <a href="/wiki/Kilogram_square_metre" class="mw-redirect" title="Kilogram square metre">kg m2</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT−1</th> <td><a href="/wiki/Mass_flow_rate" title="Mass flow rate">Mass flow rate</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad59b9876301e8fb75b9ddbf08de594b87251d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.176ex;" alt="{\displaystyle {\dot {m}}}"> <a href="/wiki/Kilogram_per_second" class="mw-redirect" title="Kilogram per second">kg s−1</a></td> <td><a href="/wiki/Momentum" title="Momentum">momentum</a>: p, <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a>: J <a href="/wiki/Kilogram_metre_per_second" class="mw-redirect" title="Kilogram metre per second">kg m s−1</a>, <a href="/wiki/Newton_second" class="mw-redirect" title="Newton second">N s</a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: 𝒮, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: ℵ <a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg m2 s−1</a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> <th style="font-weight:normal;">ML2T−1</th> <td></td> <td><a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>: L, <a href="/wiki/List_of_equations_in_classical_mechanics#Derived_dynamic_quantities" title="List of equations in classical mechanics">angular impulse</a>: ΔL <a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg m2 s−1</a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: 𝒮, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: ℵ <a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg m2 s−1</a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> </tr> <tr> <th style="font-weight:normal;">MT−2</th> <td></td> <td><a href="/wiki/Force" title="Force">force</a>: F, <a href="/wiki/Weight" title="Weight">weight</a>: Fg kg m s−2, <a href="/wiki/Newton_(unit)" title="Newton (unit)">N</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: E, <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>: W, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: L kg m2 s−2, <a href="/wiki/Joule" title="Joule">J</a></td> <th style="font-weight:normal;">ML2T−2</th> <td></td> <td><a href="/wiki/Torque" title="Torque">torque</a>: τ, <a href="/wiki/Torque#Terminology" title="Torque">moment</a>: M kg m2 s−2, <a href="/wiki/Newton-metre" title="Newton-metre">N m</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: E, <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>: W, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: L kg m2 s−2, <a href="/wiki/Joule" title="Joule">J</a></td> </tr> <tr> <th style="font-weight:normal;">MT−3</th> <td></td> <td><a href="/wiki/Yank_(physics)" class="mw-redirect" title="Yank (physics)">yank</a>: Y kg m s−3, N s−1</td> <td><a href="/wiki/Power_(physics)" title="Power (physics)">power</a>: P kg m2 s−3, <a href="/wiki/Watt" title="Watt">W</a></td> <th style="font-weight:normal;">ML2T−3</th> <td></td> <td><a href="/wiki/Rotatum" class="mw-redirect" title="Rotatum">rotatum</a>: P kg m2 s−3, N m s−1</td> <td><a href="/wiki/Power_(physics)" title="Power (physics)">power</a>: P kg m2 s−3, <a href="/wiki/Watt" title="Watt">W</a></td> </tr> </tbody></table></div></td></tr></tbody></table></div> <div 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