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स्वकुं - Wikipedia
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data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button 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">Beginning</div> </a> </li> <li id="toc-प्रकार" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#प्रकार"> <div class="vector-toc-text"> <span class="vector-toc-numb">१</span> <span>प्रकार</span> </div> </a> <ul id="toc-प्रकार-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-साधारण_ज्याखंतः" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#साधारण_ज्याखंतः"> <div class="vector-toc-text"> <span class="vector-toc-numb">२</span> <span>साधारण ज्याखंतः</span> </div> </a> <ul id="toc-साधारण_ज्याखंतः-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Points,_lines_and_circles_associated_with_a_triangle" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Points,_lines_and_circles_associated_with_a_triangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">३</span> <span>Points, lines and circles associated with a triangle</span> </div> </a> <ul id="toc-Points,_lines_and_circles_associated_with_a_triangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computing_the_area_of_a_triangle" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computing_the_area_of_a_triangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">४</span> <span>Computing the area of a triangle</span> </div> </a> <button aria-controls="toc-Computing_the_area_of_a_triangle-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Computing the area of a triangle subsection</span> </button> <ul id="toc-Computing_the_area_of_a_triangle-sublist" class="vector-toc-list"> <li id="toc-Using_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">४.१</span> <span>Using geometry</span> </div> </a> <ul id="toc-Using_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_vectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">४.२</span> <span>Using vectors</span> </div> </a> <ul id="toc-Using_vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_trigonometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_trigonometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">४.३</span> <span>Using trigonometry</span> </div> </a> <ul id="toc-Using_trigonometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">४.४</span> <span>Using coordinates</span> </div> </a> <ul id="toc-Using_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_Heron's_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_Heron's_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">४.५</span> <span>Using Heron's formula</span> </div> </a> <ul id="toc-Using_Heron's_formula-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Non-planar_triangles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-planar_triangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">५</span> <span>Non-planar triangles</span> </div> </a> <ul id="toc-Non-planar_triangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-स्रोत" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#स्रोत"> <div class="vector-toc-text"> <span class="vector-toc-numb">६</span> <span>स्रोत</span> </div> </a> <ul id="toc-स्रोत-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-पिनेयागु_स्वापूतः" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#पिनेयागु_स्वापूतः"> <div class="vector-toc-text"> <span class="vector-toc-numb">७</span> <span>पिनेयागु स्वापूतः</span> </div> </a> <ul id="toc-पिनेयागु_स्वापूतः-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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">स्वकुं</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in १६२ 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-162" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">१६२ languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ab mw-list-item"><a href="https://ab.wikipedia.org/wiki/%D0%90%D1%85%D0%BA%D3%99%D0%B0%D0%BA%D1%8C" title="Ахкәакь – Abkhazian" lang="ab" hreflang="ab" data-title="Ахкәакь" data-language-autonym="Аԥсшәа" data-language-local-name="Abkhazian" class="interlanguage-link-target"><span>Аԥсшәа</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Driehoek" title="Driehoek – Afrikaans" lang="af" hreflang="af" data-title="Driehoek" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Dreieck" title="Dreieck – Alemannic" lang="gsw" hreflang="gsw" data-title="Dreieck" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%B6%E1%88%B5%E1%89%B5_%E1%88%9B%E1%8A%A5%E1%8B%98%E1%8A%95" title="ሶስት ማእዘን – Amharic" lang="am" hreflang="am" data-title="ሶስት ማእዘን" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Trianglo" title="Trianglo – Aragonese" lang="an" hreflang="an" data-title="Trianglo" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/%C3%9Er%C4%ABecge" title="Þrīecge – Old English" lang="ang" hreflang="ang" data-title="Þrīecge" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target"><span>Ænglisc</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB" title="مثلث – Arabic" lang="ar" hreflang="ar" data-title="مثلث" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/%DC%A1%DC%AC%DC%A0%DC%AC%DC%90" title="ܡܬܠܬܐ – Aramaic" lang="arc" hreflang="arc" data-title="ܡܬܠܬܐ" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="Aramaic" class="interlanguage-link-target"><span>ܐܪܡܝܐ</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%AA%D9%84%D8%AA" title="متلت – Moroccan Arabic" lang="ary" hreflang="ary" data-title="متلت" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB" title="مثلث – Egyptian Arabic" lang="arz" hreflang="arz" data-title="مثلث" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A7%B0%E0%A6%BF%E0%A6%AD%E0%A7%81%E0%A6%9C" title="ত্ৰিভুজ – Assamese" lang="as" hreflang="as" data-title="ত্ৰিভুজ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Tri%C3%A1ngulu" title="Triángulu – Asturian" lang="ast" hreflang="ast" data-title="Triángulu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Mujina" title="Mujina – Aymara" lang="ay" hreflang="ay" data-title="Mujina" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%9C%C3%A7bucaq" title="Üçbucaq – Azerbaijani" lang="az" hreflang="az" data-title="Üçbucaq" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%88%DA%86%E2%80%8C%D8%A8%D9%88%D8%AC%D8%A7%D9%82" title="اوچبوجاق – South Azerbaijani" lang="azb" hreflang="azb" data-title="اوچبوجاق" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D3%A8%D1%81%D0%BC%D3%A9%D0%B9%D3%A9%D1%88" title="Өсмөйөш – Bashkir" lang="ba" hreflang="ba" data-title="Өсмөйөш" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Dreieck" title="Dreieck – Bavarian" lang="bar" hreflang="bar" data-title="Dreieck" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Tr%C4%97kompis" title="Trėkompis – Samogitian" lang="sgs" hreflang="sgs" data-title="Trėkompis" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Trianggulo" title="Trianggulo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Trianggulo" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D1%85%D0%B2%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D1%96%D0%BA" title="Трохвугольнік – Belarusian" lang="be" hreflang="be" data-title="Трохвугольнік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%BA%D1%83%D1%82%D0%BD%D1%96%D0%BA" title="Трыкутнік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Трыкутнік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D1%8A%D0%B3%D1%8A%D0%BB%D0%BD%D0%B8%D0%BA" title="Триъгълник – Bulgarian" lang="bg" hreflang="bg" data-title="Триъгълник" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C" title="त्रिभुज – Bhojpuri" lang="bh" hreflang="bh" data-title="त्रिभुज" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%AD%E0%A7%81%E0%A6%9C" title="ত্রিভুজ – Bangla" lang="bn" hreflang="bn" data-title="ত্রিভুজ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A6%E0%BD%9F%E0%BD%B4%E0%BD%A2%E0%BC%8B%E0%BD%82%E0%BD%A6%E0%BD%B4%E0%BD%98%E0%BC%8B%E0%BD%91%E0%BD%96%E0%BD%96%E0%BE%B1%E0%BD%B2%E0%BC%8D" title="སཟུར་གསུམ་དབབྱི། – Tibetan" lang="bo" hreflang="bo" data-title="སཟུར་གསུམ་དབབྱི།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Tric%27horn" title="Tric'horn – Breton" lang="br" hreflang="br" data-title="Tric'horn" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Trougao" title="Trougao – Bosnian" lang="bs" hreflang="bs" data-title="Trougao" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Triangle" title="Triangle – Catalan" lang="ca" hreflang="ca" data-title="Triangle" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/S%C4%83ng-g%C3%A1e%CC%A4k-h%C3%ACng" title="Săng-gáe̤k-hìng – Mindong" lang="cdo" hreflang="cdo" data-title="Săng-gáe̤k-hìng" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8F%A6%E1%8E%A2_%E1%8F%A7%E1%8F%85%E1%8F%8F%E1%8F%AF_%E1%8E%A4%E1%8F%83%E1%8F%B4%E1%8E%A9" title="ᏦᎢ ᏧᏅᏏᏯ ᎤᏃᏴᎩ – Cherokee" lang="chr" hreflang="chr" data-title="ᏦᎢ ᏧᏅᏏᏯ ᎤᏃᏴᎩ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="Cherokee" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95" title="سێگۆشە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="سێگۆشە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Triangulu" title="Triangulu – Corsican" lang="co" hreflang="co" data-title="Triangulu" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Troj%C3%BAheln%C3%ADk" title="Trojúhelník – Czech" lang="cs" hreflang="cs" data-title="Trojúhelník" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-csb mw-list-item"><a href="https://csb.wikipedia.org/wiki/Trz%C3%ABn%C3%B3rt" title="Trzënórt – Kashubian" lang="csb" hreflang="csb" data-title="Trzënórt" data-language-autonym="Kaszëbsczi" data-language-local-name="Kashubian" class="interlanguage-link-target"><span>Kaszëbsczi</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B8%C3%A7%D0%BA%C4%95%D1%82%D0%B5%D1%81%D0%BB%C4%95%D1%85" title="Виçкĕтеслĕх – Chuvash" lang="cv" hreflang="cv" data-title="Виçкĕтеслĕх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Triongl" title="Triongl – Welsh" lang="cy" hreflang="cy" data-title="Triongl" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Trekant" title="Trekant – Danish" lang="da" hreflang="da" data-title="Trekant" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dreieck" title="Dreieck – German" lang="de" hreflang="de" data-title="Dreieck" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Hir%C3%AAk%C4%B1nari" title="Hirêkınari – Zazaki" lang="diq" hreflang="diq" data-title="Hirêkınari" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/T%C5%9Biro%C5%BEk" title="Tśirožk – Lower Sorbian" lang="dsb" hreflang="dsb" data-title="Tśirožk" data-language-autonym="Dolnoserbski" data-language-local-name="Lower Sorbian" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Τρίγωνο – Greek" lang="el" hreflang="el" data-title="Τρίγωνο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Triangle" title="Triangle – English" lang="en" hreflang="en" data-title="Triangle" data-language-autonym="English" data-language-local-name="English" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Triangulo" title="Triangulo – Esperanto" lang="eo" hreflang="eo" data-title="Triangulo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tri%C3%A1ngulo" title="Triángulo – Spanish" lang="es" hreflang="es" data-title="Triángulo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kolmnurk" title="Kolmnurk – Estonian" lang="et" hreflang="et" data-title="Kolmnurk" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Triangelu" title="Triangelu – Basque" lang="eu" hreflang="eu" data-title="Triangelu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB" title="مثلث – Persian" lang="fa" hreflang="fa" data-title="مثلث" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kolmio" title="Kolmio – Finnish" lang="fi" hreflang="fi" data-title="Kolmio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kolmnukk" title="Kolmnukk – Võro" lang="vro" hreflang="vro" data-title="Kolmnukk" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Tututolu" title="Tututolu – Fijian" lang="fj" hreflang="fj" data-title="Tututolu" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Tr%C3%ADkantur" title="Tríkantur – Faroese" lang="fo" hreflang="fo" data-title="Tríkantur" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Triangle" title="Triangle – French" lang="fr" hreflang="fr" data-title="Triangle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Triihuk" title="Triihuk – Northern Frisian" lang="frr" hreflang="frr" data-title="Triihuk" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Triant%C3%A1n_(c%C3%A9imseata)" title="Triantán (céimseata) – Irish" lang="ga" hreflang="ga" data-title="Triantán (céimseata)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形 – Gan" lang="gan" hreflang="gan" data-title="三角形" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Triyang" title="Triyang – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Triyang" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Tri%C3%A1ngulo" title="Triángulo – Galician" lang="gl" hreflang="gl" data-title="Triángulo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%A4%E0%AB%8D%E0%AA%B0%E0%AA%BF%E0%AA%95%E0%AB%8B%E0%AA%A3" title="ત્રિકોણ – Gujarati" lang="gu" hreflang="gu" data-title="ત્રિકોણ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-guc mw-list-item"><a href="https://guc.wikipedia.org/wiki/Ap%C3%BCn%C3%BCinsheke%27einr%C3%BC" title="Apünüinsheke'einrü – Wayuu" lang="guc" hreflang="guc" data-title="Apünüinsheke'einrü" data-language-autonym="Wayuunaiki" data-language-local-name="Wayuu" class="interlanguage-link-target"><span>Wayuunaiki</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Troorane" title="Troorane – Manx" lang="gv" hreflang="gv" data-title="Troorane" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/S%C3%A2m-kok-h%C3%ACn" title="Sâm-kok-hìn – Hakka Chinese" lang="hak" hreflang="hak" data-title="Sâm-kok-hìn" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%9C%D7%A9" title="משולש – Hebrew" lang="he" hreflang="he" data-title="משולש" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C" title="त्रिभुज – Hindi" lang="hi" hreflang="hi" data-title="त्रिभुज" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Trokut" title="Trokut – Croatian" lang="hr" hreflang="hr" data-title="Trokut" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/T%C5%99ir%C3%B3%C5%BEk" title="Třiróžk – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Třiróžk" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Triyang" title="Triyang – Haitian Creole" lang="ht" hreflang="ht" data-title="Triyang" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/H%C3%A1romsz%C3%B6g" title="Háromszög – Hungarian" lang="hu" hreflang="hu" data-title="Háromszög" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6" title="Եռանկյուն – Armenian" lang="hy" hreflang="hy" data-title="Եռանկյուն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Triangulo" title="Triangulo – Interlingua" lang="ia" hreflang="ia" data-title="Triangulo" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Segitiga" title="Segitiga – Indonesian" lang="id" hreflang="id" data-title="Segitiga" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Triangulo" title="Triangulo – Ido" lang="io" hreflang="io" data-title="Triangulo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%9Er%C3%ADhyrningur" title="Þríhyrningur – Icelandic" lang="is" hreflang="is" data-title="Þríhyrningur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Triangolo" title="Triangolo – Italian" lang="it" hreflang="it" data-title="Triangolo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形 – Japanese" lang="ja" hreflang="ja" data-title="三角形" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Chrayanggl" title="Chrayanggl – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Chrayanggl" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pasagi_telu" title="Pasagi telu – Javanese" lang="jv" hreflang="jv" data-title="Pasagi telu" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%90%E1%83%9B%E1%83%99%E1%83%A3%E1%83%97%E1%83%AE%E1%83%94%E1%83%93%E1%83%98" title="სამკუთხედი – Georgian" lang="ka" hreflang="ka" data-title="სამკუთხედი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/%C3%9Ashm%C3%BAyeshlik" title="Úshmúyeshlik – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Úshmúyeshlik" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%A9%D0%B8%D0%BC%D1%8D" title="Щимэ – Kabardian" lang="kbd" hreflang="kbd" data-title="Щимэ" data-language-autonym="Адыгэбзэ" data-language-local-name="Kabardian" class="interlanguage-link-target"><span>Адыгэбзэ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%AE%D1%88%D0%B1%D2%B1%D1%80%D1%8B%D1%88" title="Үшбұрыш – Kazakh" lang="kk" hreflang="kk" data-title="Үшбұрыш" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://km.wikipedia.org/wiki/%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%B8%E1%9E%80%E1%9F%84%E1%9E%8E" title="ត្រីកោណ – Khmer" lang="km" hreflang="km" data-title="ត្រីកោណ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A4%E0%B3%8D%E0%B2%B0%E0%B2%BF%E0%B2%95%E0%B3%8B%E0%B2%A8" title="ತ್ರಿಕೋನ – Kannada" lang="kn" hreflang="kn" data-title="ತ್ರಿಕೋನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EA%B0%81%ED%98%95" title="삼각형 – Korean" lang="ko" hreflang="ko" data-title="삼각형" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/S%C3%AAgo%C5%9Fe" title="Sêgoşe – Kurdish" lang="ku" hreflang="ku" data-title="Sêgoşe" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Trihorn" title="Trihorn – Cornish" lang="kw" hreflang="kw" data-title="Trihorn" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D2%AE%D1%87_%D0%B1%D1%83%D1%80%D1%87%D1%82%D1%83%D0%BA" title="Үч бурчтук – Kyrgyz" lang="ky" hreflang="ky" data-title="Үч бурчтук" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Triangulum" title="Triangulum – Latin" lang="la" hreflang="la" data-title="Triangulum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Triangulo" title="Triangulo – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Triangulo" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Driehook" title="Driehook – Limburgish" lang="li" hreflang="li" data-title="Driehook" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Triangolo" title="Triangolo – Ligurian" lang="lij" hreflang="lij" data-title="Triangolo" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Tri%C3%A0ngol" title="Triàngol – Lombard" lang="lmo" hreflang="lmo" data-title="Triàngol" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Mpanzi-mis%C3%A1to" title="Mpanzi-misáto – Lingala" lang="ln" hreflang="ln" data-title="Mpanzi-misáto" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%AE%E0%BA%B9%E0%BA%9A%E0%BA%AA%E0%BA%B2%E0%BA%A1%E0%BB%81%E0%BA%88" title="ຮູບສາມແຈ – Lao" lang="lo" hreflang="lo" data-title="ຮູບສາມແຈ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trikampis" title="Trikampis – Lithuanian" lang="lt" hreflang="lt" data-title="Trikampis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Trijst%C5%ABris" title="Trijstūris – Latvian" lang="lv" hreflang="lv" data-title="Trijstūris" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Telolafy" title="Telolafy – Malagasy" lang="mg" hreflang="mg" data-title="Telolafy" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></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%BB%D1%83%D0%BA" title="Кумлук – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Кумлук" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Sagitigo" title="Sagitigo – Minangkabau" lang="min" hreflang="min" data-title="Sagitigo" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B0%D0%B3%D0%BE%D0%BB%D0%BD%D0%B8%D0%BA" title="Триаголник – Macedonian" lang="mk" hreflang="mk" data-title="Триаголник" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8B%E0%B4%A3%E0%B4%82" title="ത്രികോണം – Malayalam" lang="ml" hreflang="ml" data-title="ത്രികോണം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://mn.wikipedia.org/wiki/%D0%93%D1%83%D1%80%D0%B2%D0%B0%D0%BB%D0%B6%D0%B8%D0%BD" title="Гурвалжин – Mongolian" lang="mn" hreflang="mn" data-title="Гурвалжин" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3" title="त्रिकोण – Marathi" lang="mr" hreflang="mr" data-title="त्रिकोण" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Segi_tiga" title="Segi tiga – Malay" lang="ms" hreflang="ms" data-title="Segi tiga" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Trijangolu" title="Trijangolu – Maltese" lang="mt" hreflang="mt" data-title="Trijangolu" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%90%E1%80%BC%E1%80%AD%E1%80%82%E1%80%B6" title="တြိဂံ – Burmese" lang="my" hreflang="my" data-title="တြိဂံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C" title="त्रिभुज – Nepali" lang="ne" hreflang="ne" data-title="त्रिभुज" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Driehoek_(meetkunde)" title="Driehoek (meetkunde) – Dutch" lang="nl" hreflang="nl" data-title="Driehoek (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Trekant" title="Trekant – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Trekant" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Trekant" title="Trekant – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Trekant" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nrm mw-list-item"><a href="https://nrm.wikipedia.org/wiki/Trian" title="Trian – Norman" lang="nrf" hreflang="nrf" data-title="Trian" data-language-autonym="Nouormand" data-language-local-name="Norman" class="interlanguage-link-target"><span>Nouormand</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Triangle" title="Triangle – Occitan" lang="oc" hreflang="oc" data-title="Triangle" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%A4%E0%AD%8D%E0%AC%B0%E0%AC%BF%E0%AC%AD%E0%AD%81%E0%AC%9C" title="ତ୍ରିଭୁଜ – Odia" lang="or" hreflang="or" data-title="ତ୍ରିଭୁଜ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A4%E0%A8%BF%E0%A8%95%E0%A9%8B%E0%A8%A8" title="ਤਿਕੋਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਤਿਕੋਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Dreieck" title="Dreieck – Palatine German" lang="pfl" hreflang="pfl" data-title="Dreieck" data-language-autonym="Pälzisch" data-language-local-name="Palatine German" class="interlanguage-link-target"><span>Pälzisch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Tr%C3%B3jk%C4%85t" title="Trójkąt – Polish" lang="pl" hreflang="pl" data-title="Trójkąt" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AA%DA%A9%D9%88%D9%86" title="تکون – Western Punjabi" lang="pnb" hreflang="pnb" data-title="تکون" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF%D8%B1%DB%90%DA%85%D9%86%DA%89%DB%8C" title="درېڅنډی – Pashto" lang="ps" hreflang="ps" data-title="درېڅنډی" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tri%C3%A2ngulo" title="Triângulo – Portuguese" lang="pt" hreflang="pt" data-title="Triângulo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Kimsak%27uchu" title="Kimsak'uchu – Quechua" lang="qu" hreflang="qu" data-title="Kimsak'uchu" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Triunghi" title="Triunghi – Romanian" lang="ro" hreflang="ro" data-title="Triunghi" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA" title="Треугольник – Russian" lang="ru" hreflang="ru" data-title="Треугольник" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D1%83%D0%B3%D0%BE%D0%BB%D0%BD%D0%B8%D0%BA" title="Триуголник – Rusyn" lang="rue" hreflang="rue" data-title="Триуголник" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tri%C3%A0nculu" title="Triànculu – Sicilian" lang="scn" hreflang="scn" data-title="Triànculu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Triangle" title="Triangle – Scots" lang="sco" hreflang="sco" data-title="Triangle" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%BD%DA%AA%D9%86%DA%8A%D9%88" title="ٽڪنڊو – Sindhi" lang="sd" hreflang="sd" data-title="ٽڪنڊو" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Golmma%C4%8Diegat" title="Golmmačiegat – Northern Sami" lang="se" hreflang="se" data-title="Golmmačiegat" data-language-autonym="Davvisámegiella" data-language-local-name="Northern Sami" class="interlanguage-link-target"><span>Davvisámegiella</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Trokut" title="Trokut – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Trokut" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%9A%E0%B7%9D%E0%B6%AB" title="ත්රිකෝණ – Sinhala" lang="si" hreflang="si" data-title="ත්රිකෝණ" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Triangle" title="Triangle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Triangle" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Trojuholn%C3%ADk" title="Trojuholník – Slovak" lang="sk" hreflang="sk" data-title="Trojuholník" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Trikotnik" title="Trikotnik – Slovenian" lang="sl" hreflang="sl" data-title="Trikotnik" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Kulm%C3%A2h%C3%A2%C5%A1" title="Kulmâhâš – Inari Sami" lang="smn" hreflang="smn" data-title="Kulmâhâš" data-language-autonym="Anarâškielâ" data-language-local-name="Inari Sami" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Gonyonhatu" title="Gonyonhatu – Shona" lang="sn" hreflang="sn" data-title="Gonyonhatu" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Saddexagal" title="Saddexagal – Somali" lang="so" hreflang="so" data-title="Saddexagal" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Trek%C3%ABnd%C3%ABshi" title="Trekëndëshi – Albanian" lang="sq" hreflang="sq" data-title="Trekëndëshi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D1%83%D0%B3%D0%B0%D0%BE" title="Троугао – Serbian" lang="sr" hreflang="sr" data-title="Троугао" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Juru_tilu" title="Juru tilu – Sundanese" lang="su" hreflang="su" data-title="Juru tilu" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Triangel" title="Triangel – Swedish" lang="sv" hreflang="sv" data-title="Triangel" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Pembetatu" title="Pembetatu – Swahili" lang="sw" hreflang="sw" data-title="Pembetatu" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Trziek" title="Trziek – Silesian" lang="szl" hreflang="szl" data-title="Trziek" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%AE%E0%AF%8D" title="முக்கோணம் – Tamil" lang="ta" hreflang="ta" data-title="முக்கோணம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%AD%E0%B1%81%E0%B0%9C%E0%B0%82" title="త్రిభుజం – Telugu" lang="te" hreflang="te" data-title="త్రిభుజం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A1%D0%B5%D0%BA%D1%83%D0%BD%D2%B7%D0%B0" title="Секунҷа – Tajik" lang="tg" hreflang="tg" data-title="Секунҷа" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%AA%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1" title="รูปสามเหลี่ยม – Thai" lang="th" hreflang="th" data-title="รูปสามเหลี่ยม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tatsulok" title="Tatsulok – Tagalog" lang="tl" hreflang="tl" data-title="Tatsulok" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9C%C3%A7gen" title="Üçgen – Turkish" lang="tr" hreflang="tr" data-title="Üçgen" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D3%A8%D1%87%D0%BF%D0%BE%D1%87%D0%BC%D0%B0%D0%BA" title="Өчпочмак – Tatar" lang="tt" hreflang="tt" data-title="Өчпочмак" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA" title="Трикутник – Ukrainian" lang="uk" hreflang="uk" data-title="Трикутник" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB" title="مثلث – Urdu" lang="ur" hreflang="ur" data-title="مثلث" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Uchburchak" title="Uchburchak – Uzbek" lang="uz" hreflang="uz" data-title="Uchburchak" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Triango%C5%82o" title="Triangoło – Venetian" lang="vec" hreflang="vec" data-title="Triangoło" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Tam_gi%C3%A1c" title="Tam giác – Vietnamese" lang="vi" hreflang="vi" data-title="Tam giác" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Drieoek" title="Drieoek – West Flemish" lang="vls" hreflang="vls" data-title="Drieoek" data-language-autonym="West-Vlams" 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title="कुं">कुं</a> छ्यला कुनातःगु, द्विआयामिक ख्यः ख। थ्व छगु आधारभूत रेखागणितीय <a href="/w/index.php?title=%E0%A4%AA%E0%A5%8B%E0%A4%B2%E0%A4%BF%E0%A4%97%E0%A4%A8&action=edit&redlink=1" class="new" title="पोलिगन (पौ मदु)">पोलिगन</a> खः। थ्व पोलिगनय् स्वंगु <a href="/w/index.php?title=%E0%A4%AD%E0%A4%B0%E0%A5%8D%E0%A4%9F%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%B8&action=edit&redlink=1" class="new" title="भर्टेक्स (पौ मदु)">भर्टेक्स</a> स्वंगु साइड व स्वंगु कोण दै। </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="प्रकार"><span id=".E0.A4.AA.E0.A5.8D.E0.A4.B0.E0.A4.95.E0.A4.BE.E0.A4.B0"></span>प्रकार</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=1" title="खण्ड सम्पादन: प्रकार"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>स्वकुंया ल्हा कथं थुकित ३गु भायय् बाय् छिं </p> <ul><li><b>समबाहू</b> <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>१<span class="cite-bracket">]</span></a></sup></li> <li><b>समद्विबाहू</b> <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>२<span class="cite-bracket">]</span></a></sup></li> <li><b>विषमबाहू</b> <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>३<span class="cite-bracket">]</span></a></sup></li></ul> <table align="center"><tbody><tr align="center"> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Equilateral.svg" class="mw-file-description" title="समबाहू त्रिकोण"><img alt="समबाहू त्रिकोण" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/512px-Triangle.Equilateral.svg.png" decoding="async" width="512" height="415" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/768px-Triangle.Equilateral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/1024px-Triangle.Equilateral.svg.png 2x" data-file-width="512" data-file-height="415" /></a></span></td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Isosceles.svg" class="mw-file-description" title="समद्विबाहू त्रिकोण"><img alt="समद्विबाहू त्रिकोण" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/74px-Triangle.Isosceles.svg.png" decoding="async" width="74" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/111px-Triangle.Isosceles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/148px-Triangle.Isosceles.svg.png 2x" data-file-width="74" data-file-height="114" /></a></span></td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Scalene.svg" class="mw-file-description" title="विषमबाहू त्रिकोण"><img alt="विषमबाहू त्रिकोण" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/245px-Triangle.Scalene.svg.png" decoding="async" width="245" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/368px-Triangle.Scalene.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/490px-Triangle.Scalene.svg.png 2x" data-file-width="245" data-file-height="110" /></a></span></td> </tr> <tr align="center"> <td>समबाहू</td><td>समद्विबाहू</td><td>विषमबाहू</td> </tr> </tbody></table> <p>कोणयागु कथं त्रिभूजयात स्वंगु भागय् बाय् छिं- </p> <ul><li><b><a href="/w/index.php?title=%E0%A4%B0%E0%A4%BE%E0%A4%87%E0%A4%9F_%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3&action=edit&redlink=1" class="new" title="राइट त्रिकोण (पौ मदु)">राइट त्रिकोण</a></b></li> <li><b><a href="/w/index.php?title=%E0%A4%85%E0%A4%AC%E0%A5%8D%E0%A4%9F%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%9C_%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3&action=edit&redlink=1" class="new" title="अब्ट्युज त्रिकोण (पौ मदु)">अब्ट्युज त्रिकोण</a></b> .</li> <li><b><a href="/w/index.php?title=%E0%A4%8F%E0%A4%95%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%9F_%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3&action=edit&redlink=1" class="new" title="एक्युट त्रिकोण (पौ मदु)">एक्युट त्रिकोण</a></b> .</li></ul> <table align="center"> <tbody><tr align="center"> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Right.svg" class="mw-file-description" title="राइट त्रिकोण"><img alt="राइट त्रिकोण" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/150px-Triangle.Right.svg.png" decoding="async" width="150" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/225px-Triangle.Right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/300px-Triangle.Right.svg.png 2x" data-file-width="150" data-file-height="113" /></a></span></td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Obtuse.svg" class="mw-file-description" title="अब्ट्युज त्रिकोण"><img alt="अब्ट्युज त्रिकोण" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/113px-Triangle.Obtuse.svg.png" decoding="async" width="113" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/170px-Triangle.Obtuse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/226px-Triangle.Obtuse.svg.png 2x" data-file-width="113" data-file-height="113" /></a></span></td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Acute.svg" class="mw-file-description" title="एक्युट त्रिकोण"><img alt="एक्युट त्रिकोण" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/794px-Triangle.Acute.svg.png" decoding="async" width="794" height="491" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/1191px-Triangle.Acute.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/1588px-Triangle.Acute.svg.png 2x" data-file-width="794" data-file-height="491" /></a></span></td> </tr> <tr align="center"> <td>राइट त्रिकोण</td><td>अब्ट्युज त्रिकोण</td><td>एक्युट त्रिकोण</td> </tr> </tbody></table> <div class="mw-heading mw-heading2"><h2 id="साधारण_ज्याखंतः"><span id=".E0.A4.B8.E0.A4.BE.E0.A4.A7.E0.A4.BE.E0.A4.B0.E0.A4.A3_.E0.A4.9C.E0.A5.8D.E0.A4.AF.E0.A4.BE.E0.A4.96.E0.A4.82.E0.A4.A4.E0.A4.83"></span>साधारण ज्याखंतः</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=2" title="खण्ड सम्पादन: साधारण ज्याखंतः"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>त्रिकोणयागु छुं खंतेत <a href="/w/index.php?title=%E0%A4%AF%E0%A5%81%E0%A4%95%E0%A5%8D%E0%A4%B2%E0%A4%BF%E0%A4%A1&action=edit&redlink=1" class="new" title="युक्लिड (पौ मदु)">युक्लिड</a>नं वेकयागु <i><a href="/w/index.php?title=%E0%A4%AF%E0%A5%81%E0%A4%95%E0%A5%8D%E0%A4%B2%E0%A4%BF%E0%A4%A1%E0%A4%AF%E0%A4%BE%E0%A4%97%E0%A5%81_%E0%A4%8F%E0%A4%B2%E0%A5%87%E0%A4%AE%E0%A5%87%E0%A4%A8%E0%A5%8D%E0%A4%9F%E0%A4%A4%E0%A4%83&action=edit&redlink=1" class="new" title="युक्लिडयागु एलेमेन्टतः (पौ मदु)">युक्लिडयागु एलेमेन्टतः</a></i> सफूयु भाग १-४य् थ्यं-मथ्यं <a href="/w/index.php?title=%E0%A5%A9%E0%A5%A6%E0%A5%A6_%E0%A4%87_%E0%A4%AA%E0%A5%82&action=edit&redlink=1" class="new" title="३०० इ पू (पौ मदु)">३०० इ पू</a>य् च्वयादिगु दु। </p><p>A triangle is a <a href="/w/index.php?title=Polygon&action=edit&redlink=1" class="new" title="Polygon (पौ मदु)">polygon</a> and a 2-<a href="/w/index.php?title=Simplex&action=edit&redlink=1" class="new" title="Simplex (पौ मदु)">simplex</a> (see <a href="/w/index.php?title=Polytope&action=edit&redlink=1" class="new" title="Polytope (पौ मदु)">polytope</a>). All triangles are two-<a href="/w/index.php?title=Dimension&action=edit&redlink=1" class="new" title="Dimension (पौ मदु)">dimensional</a>. </p><p>Two triangles are said to be <i><a href="/w/index.php?title=Similarity_(mathematics)&action=edit&redlink=1" class="new" title="Similarity (mathematics) (पौ मदु)">similar</a></i> if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are <a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&redlink=1" class="new" title="Proportionality (mathematics) (पौ मदु)">proportional</a>. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. </p><p>Using right triangles and the concept of similarity, the <a href="/w/index.php?title=Trigonometric_function&action=edit&redlink=1" class="new" title="Trigonometric function (पौ मदु)">trigonometric functions</a> sine and cosine can be defined. These are functions of an <a href="/wiki/Angle" class="mw-redirect" title="Angle">angle</a> which are investigated in <a href="/w/index.php?title=Trigonometry&action=edit&redlink=1" class="new" title="Trigonometry (पौ मदु)">trigonometry</a>. </p><p>In the remainder we will consider a triangle with vertices A, B and C, angles α, β and γ and sides <i>a</i>, <i>b</i> and <i>c</i>. The side <i>a</i> is opposite to the vertex <i>A</i> and angle α and analogously for the other sides. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Labels.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Triangle.Labels.svg/200px-Triangle.Labels.svg.png" decoding="async" width="200" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Triangle.Labels.svg/300px-Triangle.Labels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Triangle.Labels.svg/400px-Triangle.Labels.svg.png 2x" data-file-width="142" data-file-height="70" /></a><figcaption>A triangle with vertices, sides and angles labelled</figcaption></figure> <p>In Euclidean geometry, the sum of the internal angles α + β + γ is equal to two right angles (180° or π radians). This allows determination of the third angle of any triangle as soon as two angles are known. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Pythagorean.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/220px-Pythagorean.svg.png" decoding="async" width="220" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/330px-Pythagorean.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/440px-Pythagorean.svg.png 2x" data-file-width="512" data-file-height="466" /></a><figcaption>पाइथागोरस थियोरम</figcaption></figure> <p>A central theorem is the <a href="/w/index.php?title=Pythagorean_theorem&action=edit&redlink=1" class="new" title="Pythagorean theorem (पौ मदु)">Pythagorean theorem</a> stating that in any right triangle, the area of the square on the <a href="/w/index.php?title=Hypotenuse&action=edit&redlink=1" class="new" title="Hypotenuse (पौ मदु)">hypotenuse</a> is equal to the sum of the areas of the squares on the other two sides. If side C is the hypotenuse, we can write this as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=a^{2}+b^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=a^{2}+b^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034ff4dc0a2a475255be544edb097b66c67935b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.723ex; height:2.843ex;" alt="{\displaystyle c^{2}=a^{2}+b^{2}\,}"></span></dd></dl> <p>This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third—something unique to right triangles. The Pythagorean theorem can be generalized to the <a href="/w/index.php?title=Law_of_cosines&action=edit&redlink=1" class="new" title="Law of cosines (पौ मदु)">law of cosines</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841e578e61b316a46d393d8e05c24aa2fbbae891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.101ex; height:3.176ex;" alt="{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma \,}"></span></dd></dl> <p>which is valid for all triangles, even if γ is not a right angle. The law of cosines can be used to compute the side lengths and angles of a triangle as soon as all three sides or two sides and an enclosed angle are known. </p><p><a href="/w/index.php?title=%E0%A4%B8%E0%A4%BE%E0%A4%87%E0%A4%A8%E0%A4%AF%E0%A5%81_%E0%A4%B2%E0%A4%83&action=edit&redlink=1" class="new" title="साइनयु लः (पौ मदु)">साइनयु लः</a> कथं : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}={\frac {1}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}={\frac {1}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92295757668f14cac6b4e1e747fb0eeeba5d2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.666ex; height:5.509ex;" alt="{\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}={\frac {1}{d}}}"></span></dd></dl> <p>where <i>d</i> is the diameter of the <a href="/w/index.php?title=Circumcircle&action=edit&redlink=1" class="new" title="Circumcircle (पौ मदु)">circumcircle</a> (the circle which passes through all three points of the triangle). The law of sines can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions. </p><p>There are two <a href="/w/index.php?title=Special_right_triangles&action=edit&redlink=1" class="new" title="Special right triangles (पौ मदु)">special right triangles</a> that appear commonly in geometry. The so-called "45-45-90 triangle" has angles with those angle measures and the ratio of its sides is : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1:1:{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:1:{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b01392f18a00aea86e8f831c73191930baa1bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.298ex; height:3.009ex;" alt="{\displaystyle 1:1:{\sqrt {2}}}"></span>. The "30-60-90 triangle" has sides in the ratio of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1:{\sqrt {3}}:2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>:</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:{\sqrt {3}}:2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec862f06a8911b1c99423e4ff89a5d26d28c22e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.298ex; height:2.843ex;" alt="{\displaystyle 1:{\sqrt {3}}:2}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Points,_lines_and_circles_associated_with_a_triangle"><span id="Points.2C_lines_and_circles_associated_with_a_triangle"></span>Points, lines and circles associated with a triangle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=3" title="खण्ड सम्पादन: Points, lines and circles associated with a triangle"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is <a href="/w/index.php?title=Ceva%27s_theorem&action=edit&redlink=1" class="new" title="Ceva's theorem (पौ मदु)">Ceva's theorem</a>, which gives a criterion for determining when three such lines are <a href="/w/index.php?title=Concurrent_lines&action=edit&redlink=1" class="new" title="Concurrent lines (पौ मदु)">concurrent</a>. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are <a href="/w/index.php?title=Collinear&action=edit&redlink=1" class="new" title="Collinear (पौ मदु)">collinear</a>: here <a href="/w/index.php?title=Menelaus%27_theorem&action=edit&redlink=1" class="new" title="Menelaus' theorem (पौ मदु)">Menelaus' theorem</a> gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained. </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Circumcenter.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Triangle.Circumcenter.svg/198px-Triangle.Circumcenter.svg.png" decoding="async" width="198" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Triangle.Circumcenter.svg/297px-Triangle.Circumcenter.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Triangle.Circumcenter.svg/396px-Triangle.Circumcenter.svg.png 2x" data-file-width="198" data-file-height="198" /></a><figcaption>The <a href="/w/index.php?title=Circumcenter&action=edit&redlink=1" class="new" title="Circumcenter (पौ मदु)">circumcenter</a> is the centre of a circle passing through the three vertices of the triangle.</figcaption></figure> <p>A <a href="/w/index.php?title=Bisection&action=edit&redlink=1" class="new" title="Bisection (पौ मदु)">perpendicular bisector</a> of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's <a href="/w/index.php?title=Circumcenter&action=edit&redlink=1" class="new" title="Circumcenter (पौ मदु)">circumcenter</a>; this point is the center of the <a href="/w/index.php?title=Circumcircle&action=edit&redlink=1" class="new" title="Circumcircle (पौ मदु)">circumcircle</a>, the <a href="/w/index.php?title=Circle&action=edit&redlink=1" class="new" title="Circle (पौ मदु)">circle</a> passing through all three vertices. The diameter of this circle can be found from the law of sines stated above. </p><p><a href="/w/index.php?title=Thales%27_theorem&action=edit&redlink=1" class="new" title="Thales' theorem (पौ मदु)">Thales' theorem</a> states that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. </p> <figure class="mw-halign-left" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Orthocenter.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Orthocenter.svg/182px-Triangle.Orthocenter.svg.png" decoding="async" width="182" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Orthocenter.svg/273px-Triangle.Orthocenter.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Orthocenter.svg/364px-Triangle.Orthocenter.svg.png 2x" data-file-width="182" data-file-height="146" /></a><figcaption>The intersection of the altitudes is the <a href="/w/index.php?title=Orthocenter&action=edit&redlink=1" class="new" title="Orthocenter (पौ मदु)">orthocenter</a>.</figcaption></figure> <p>An <a href="/w/index.php?title=Altitude_(triangle)&action=edit&redlink=1" class="new" title="Altitude (triangle) (पौ मदु)">altitude</a> of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the <i>base</i> of the altitude, and the point where the altitude intersects the base (or its extension) is called the <i>foot</i> of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the <a href="/w/index.php?title=Orthocenter&action=edit&redlink=1" class="new" title="Orthocenter (पौ मदु)">orthocenter</a> of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. The three vertices together with the orthocenter are said to form an <a href="/w/index.php?title=Orthocentric_system&action=edit&redlink=1" class="new" title="Orthocentric system (पौ मदु)">orthocentric system</a>. </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Incircle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle.Incircle.svg/182px-Triangle.Incircle.svg.png" decoding="async" width="182" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle.Incircle.svg/273px-Triangle.Incircle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle.Incircle.svg/364px-Triangle.Incircle.svg.png 2x" data-file-width="182" data-file-height="157" /></a><figcaption>The intersection of the angle bisectors finds the center of the <a href="/w/index.php?title=Incircle&action=edit&redlink=1" class="new" title="Incircle (पौ मदु)">incircle</a>.</figcaption></figure> <p>An <a href="/w/index.php?title=Angle_bisector&action=edit&redlink=1" class="new" title="Angle bisector (पौ मदु)">angle bisector</a> of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the <a href="/w/index.php?title=Incenter&action=edit&redlink=1" class="new" title="Incenter (पौ मदु)">incenter</a>, the center of the triangle's <a href="/w/index.php?title=Incircle&action=edit&redlink=1" class="new" title="Incircle (पौ मदु)">incircle</a>. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the <a href="/w/index.php?title=Excircle&action=edit&redlink=1" class="new" title="Excircle (पौ मदु)">excircles</a>; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an <a href="/w/index.php?title=Orthocentric_system&action=edit&redlink=1" class="new" title="Orthocentric system (पौ मदु)">orthocentric system</a>. </p><p><br clear="left" /> </p> <figure class="mw-halign-left" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.Centroid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Triangle.Centroid.svg/809px-Triangle.Centroid.svg.png" decoding="async" width="809" height="654" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Triangle.Centroid.svg/1214px-Triangle.Centroid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Triangle.Centroid.svg/1618px-Triangle.Centroid.svg.png 2x" data-file-width="809" data-file-height="654" /></a><figcaption>The <a href="/w/index.php?title=Centroid&action=edit&redlink=1" class="new" title="Centroid (पौ मदु)">centroid</a> is the center of gravity.</figcaption></figure> <p>A <a href="/w/index.php?title=Median_(geometry)&action=edit&redlink=1" class="new" title="Median (geometry) (पौ मदु)">median</a> of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's <a href="/w/index.php?title=Centroid&action=edit&redlink=1" class="new" title="Centroid (पौ मदु)">centroid</a>. This is also the triangle's <a href="/w/index.php?title=Center_of_gravity&action=edit&redlink=1" class="new" title="Center of gravity (पौ मदु)">center of gravity</a>: if the triangle were made out of wood, say, you could balance it on its centroid, or on any line through the centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice as large as the distance between the centroid and the midpoint of the opposite side. </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.NinePointCircle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Triangle.NinePointCircle.svg/182px-Triangle.NinePointCircle.svg.png" decoding="async" width="182" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Triangle.NinePointCircle.svg/273px-Triangle.NinePointCircle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Triangle.NinePointCircle.svg/364px-Triangle.NinePointCircle.svg.png 2x" data-file-width="182" data-file-height="147" /></a><figcaption><a href="/w/index.php?title=Nine-point_circle&action=edit&redlink=1" class="new" title="Nine-point circle (पौ मदु)">Nine-point circle</a> demonstrates a symmetry where six points lie on the same circle.</figcaption></figure> <p>The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's <a href="/w/index.php?title=Nine-point_circle&action=edit&redlink=1" class="new" title="Nine-point circle (पौ मदु)">nine-point circle</a>. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the <a href="/w/index.php?title=Orthocenter&action=edit&redlink=1" class="new" title="Orthocenter (पौ मदु)">orthocenter</a>. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the <a href="/w/index.php?title=Feuerbach_point&action=edit&redlink=1" class="new" title="Feuerbach point (पौ मदु)">Feuerbach point</a>) and the three <a href="/w/index.php?title=Excircle&action=edit&redlink=1" class="new" title="Excircle (पौ मदु)">excircles</a>. </p><p><br clear="left" /> </p> <figure class="mw-halign-left" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.EulerLine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle.EulerLine.svg/520px-Triangle.EulerLine.svg.png" decoding="async" width="520" height="420" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle.EulerLine.svg/780px-Triangle.EulerLine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle.EulerLine.svg/1040px-Triangle.EulerLine.svg.png 2x" data-file-width="520" data-file-height="420" /></a><figcaption><a href="/w/index.php?title=Euler%27s_line&action=edit&redlink=1" class="new" title="Euler's line (पौ मदु)">Euler's line</a> is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).</figcaption></figure> <p>The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red point) all lie on a single line, known as <a href="/w/index.php?title=Euler%27s_line&action=edit&redlink=1" class="new" title="Euler's line (पौ मदु)">Euler's line</a> (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. </p><p>The center of the incircle is not in general located on Euler's line. </p><p>If one reflects a median at the angle bisector that passes through the same vertex, one obtains a <a href="/w/index.php?title=Symmedian&action=edit&redlink=1" class="new" title="Symmedian (पौ मदु)">symmedian</a>. The three symmedians intersect in a single point, the <a href="/w/index.php?title=Symmedian_point&action=edit&redlink=1" class="new" title="Symmedian point (पौ मदु)">symmedian point</a> of the triangle. </p><p><br clear="all" /> </p> <div class="mw-heading mw-heading2"><h2 id="Computing_the_area_of_a_triangle">Computing the area of a triangle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=4" title="खण्ड सम्पादन: Computing the area of a triangle"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>४<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Using_geometry">Using geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=5" title="खण्ड सम्पादन: Using geometry"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/w/index.php?title=Surface_area&action=edit&redlink=1" class="new" title="Surface area (पौ मदु)">area</a> <i>S</i> of a triangle is <i>S</i> = ½<i>bh</i>, where <i>b</i> is the length of any side of the triangle (the <i>base</i>) and <i>h</i> (the <i>altitude</i>) is the perpendicular distance between the base and the vertex not on the base. This can be shown with the following geometric construction. </p> <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.GeometryArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Triangle.GeometryArea.svg/504px-Triangle.GeometryArea.svg.png" decoding="async" width="504" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Triangle.GeometryArea.svg/756px-Triangle.GeometryArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Triangle.GeometryArea.svg/1008px-Triangle.GeometryArea.svg.png 2x" data-file-width="504" data-file-height="126" /></a><figcaption>The triangle is first transformed into a <a href="/w/index.php?title=Parallelogram&action=edit&redlink=1" class="new" title="Parallelogram (पौ मदु)">parallelogram</a> with twice the area of the triangle, then into a rectangle.</figcaption></figure> <p>To find the area of a given triangle (green), first make an exact copy of the triangle (blue), rotate it 180°, and join it to the given triangle along one side to obtain a <a href="/w/index.php?title=Parallelogram&action=edit&redlink=1" class="new" title="Parallelogram (पौ मदु)">parallelogram</a>. Cut off a part and join it at the other side of the parallelogram to form a rectangle. Because the area of the rectangle is <i>bh</i>, the area of the given triangle must be ½<i>bh</i>. </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.VectorArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Triangle.VectorArea.svg/200px-Triangle.VectorArea.svg.png" decoding="async" width="200" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Triangle.VectorArea.svg/300px-Triangle.VectorArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Triangle.VectorArea.svg/400px-Triangle.VectorArea.svg.png 2x" data-file-width="200" data-file-height="108" /></a><figcaption>The area of the parallelogram is the magnitude of the cross product of the two vectors.</figcaption></figure> <p>The product of the <a href="/w/index.php?title=Inradius&action=edit&redlink=1" class="new" title="Inradius (पौ मदु)">inradius</a> and the <a href="/w/index.php?title=Semiperimeter&action=edit&redlink=1" class="new" title="Semiperimeter (पौ मदु)">semiperimeter</a> of a triangle also gives its area. </p> <div class="mw-heading mw-heading3"><h3 id="Using_vectors">Using vectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=6" title="खण्ड सम्पादन: Using vectors"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The area of a parallelogram can also be calculated by the use of <a href="/w/index.php?title=Vector_(spatial)&action=edit&redlink=1" class="new" title="Vector (spatial) (पौ मदु)">vectors</a>. If <i>AB</i> and <i>AC</i> are vectors pointing from A to B and from A to C, respectively, the area of parallelogram ABDC is |<i>AB</i> × <i>AC</i>|, the magnitude of the <a href="/w/index.php?title=Cross_product&action=edit&redlink=1" class="new" title="Cross product (पौ मदु)">cross product</a> of vectors <i>AB</i> and <i>AC</i>. |<i>AB</i> × <i>AC</i>| is also equal to |<i>h</i> × <i>AC</i>|, where <i>h</i> represents the altitude <i>h</i> as a vector. </p><p>The area of triangle ABC is half of this, or <i>S</i> = ½|<i>AB</i> × <i>AC</i>|. </p><p>The area of triangle ABC can also be expressed in term of <a href="/w/index.php?title=Dot_product&action=edit&redlink=1" class="new" title="Dot product (पौ मदु)">dot products</a> as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}{\sqrt {(\mathbf {AB} \cdot \mathbf {AB} )(\mathbf {AC} \cdot \mathbf {AC} )-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}={\frac {1}{2}}{\sqrt {|\mathbf {AB} |^{2}|\mathbf {AC} |^{2}-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">C</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">C</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">C</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">C</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}{\sqrt {(\mathbf {AB} \cdot \mathbf {AB} )(\mathbf {AC} \cdot \mathbf {AC} )-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}={\frac {1}{2}}{\sqrt {|\mathbf {AB} |^{2}|\mathbf {AC} |^{2}-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/385950bec5acdd2c573b848f3890d61892cb528c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:77.538ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}{\sqrt {(\mathbf {AB} \cdot \mathbf {AB} )(\mathbf {AC} \cdot \mathbf {AC} )-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}={\frac {1}{2}}{\sqrt {|\mathbf {AB} |^{2}|\mathbf {AC} |^{2}-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}}"></span></dd></dl> <figure class="mw-halign-left" typeof="mw:File/Frame"><a href="/wiki/%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE:Triangle.TrigArea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Triangle.TrigArea.svg/165px-Triangle.TrigArea.svg.png" decoding="async" width="165" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Triangle.TrigArea.svg/248px-Triangle.TrigArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Triangle.TrigArea.svg/330px-Triangle.TrigArea.svg.png 2x" data-file-width="165" data-file-height="148" /></a><figcaption>Applying trigonometry to find the altitude <i>h</i>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Using_trigonometry">Using trigonometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=7" title="खण्ड सम्पादन: Using trigonometry"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The altitude of a triangle can be found through an application of <a href="/w/index.php?title=Trigonometry&action=edit&redlink=1" class="new" title="Trigonometry (पौ मदु)">trigonometry</a>. Using the labelling as in the image on the left, the altitude is <i>h</i> = <i>a</i> sin γ. Substituting this in the formula <i>S</i> = ½<i>bh</i> derived above, the area of the triangle can be expressed as <i>S</i> = ½<i>ab</i> sin γ. </p><p>It is of course no coincidence that the area of a parallelogram is <i>ab</i> sin γ. </p><p>If one uses </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d013f8ae80e83f4fe4a843989cb3bab8bfc5e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.277ex; height:5.843ex;" alt="{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin C={\sqrt {1-\cos ^{2}C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mi>C</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin C={\sqrt {1-\cos ^{2}C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb11895c6d0a767c5f6a771687dd9112128b322f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.753ex; height:3.509ex;" alt="{\displaystyle \sin C={\sqrt {1-\cos ^{2}C}}}"></span></dd></dl> <p>and also the formula shown above, then one arrives at the following formula for area </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db729c535b55abdccbaa80a45b83e9207f14bb95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.496ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}"></span></dd></dl> <p>[Note that, this is a multiplied out form of Heron's formula] </p> <div class="mw-heading mw-heading3"><h3 id="Using_coordinates">Using coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=8" title="खण्ड सम्पादन: Using coordinates"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If vertex A is located at the origin (0, 0) of a <a href="/w/index.php?title=Cartesian_coordinate_system&action=edit&redlink=1" class="new" title="Cartesian coordinate system (पौ मदु)">Cartesian coordinate system</a> and the coordinates of the other two vertices are given by B = (<i>x</i><sub>B</sub>, <i>y</i><sub>B</sub>) and C = (<i>x</i><sub>C</sub>, <i>y</i><sub>C</sub>), then the area <i>S</i> can be computed as ½ times the <a href="/w/index.php?title=Absolute_value&action=edit&redlink=1" class="new" title="Absolute value (पौ मदु)">absolute value</a> of the <a href="/w/index.php?title=Determinant&action=edit&redlink=1" class="new" title="Determinant (पौ मदु)">determinant</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{pmatrix}}\right|={\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{pmatrix}}\right|={\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfda1b14c04f853eb6cfcd12f949b2ab3c29a865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.747ex; height:6.176ex;" alt="{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{pmatrix}}\right|={\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.}"></span></dd></dl> <p>For three general vertices, the equation is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right|={\frac {1}{2}}{\big |}x_{A}y_{C}-x_{A}y_{B}+x_{B}y_{A}-x_{B}y_{C}+x_{C}y_{B}-x_{C}y_{A}{\big |}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <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-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right|={\frac {1}{2}}{\big |}x_{A}y_{C}-x_{A}y_{B}+x_{B}y_{A}-x_{B}y_{C}+x_{C}y_{B}-x_{C}y_{A}{\big |}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10e2b0c994e2554c41659964111ade45d660fc26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:83.528ex; height:9.509ex;" alt="{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right|={\frac {1}{2}}{\big |}x_{A}y_{C}-x_{A}y_{B}+x_{B}y_{A}-x_{B}y_{C}+x_{C}y_{B}-x_{C}y_{A}{\big |}.}"></span></dd></dl> <p>In three dimensions, the area of a general triangle {A = (<i>x</i><sub>A</sub>, <i>y</i><sub>A</sub>, <i>z</i><sub>A</sub>), B = (<i>x</i><sub>B</sub>, <i>y</i><sub>B</sub>, <i>z</i><sub>B</sub>) and C = (<i>x</i><sub>C</sub>, <i>y</i><sub>C</sub>, <i>z</i><sub>C</sub>)} is the 'Pythagorean' sum of the areas of the respective projections on the three principal planes (i.e. <i>x</i>=0, <i>y</i>=0 and <i>z</i>=0): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {1}{2}}{\sqrt {\left(\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{pmatrix}}\right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2}}{\sqrt {\left(\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{pmatrix}}\right)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36bc7fae76befe8915a06456760485f56a836c14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:94.654ex; height:10.509ex;" alt="{\displaystyle S={\frac {1}{2}}{\sqrt {\left(\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{pmatrix}}\right)^{2}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Using_Heron's_formula"><span id="Using_Heron.27s_formula"></span>Using Heron's formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=9" title="खण्ड सम्पादन: Using Heron's formula"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Yet another way to compute <i>S</i> is <a href="/w/index.php?title=Heron%27s_Formula&action=edit&redlink=1" class="new" title="Heron's Formula (पौ मदु)">Heron's Formula</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6596e9344ada420302022229b1d8122cc7a3141c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.467ex; height:4.843ex;" alt="{\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}"></span></dd></dl> <p>where <i>s</i> = ½ (<i>a</i> + <i>b</i> + <i>c</i>) is the <b>semiperimeter</b>, or half of the triangle's perimeter. </p><p>Multiplied out form of Heron's formula (see above for proof) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db729c535b55abdccbaa80a45b83e9207f14bb95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.496ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Non-planar_triangles">Non-planar triangles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=10" title="खण्ड सम्पादन: Non-planar triangles"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in noneuclidean geometries are <a href="/w/index.php?title=Spherical_triangle&action=edit&redlink=1" class="new" title="Spherical triangle (पौ मदु)">spherical triangles</a> in <a href="/w/index.php?title=Spherical_geometry&action=edit&redlink=1" class="new" title="Spherical geometry (पौ मदु)">spherical geometry</a> and <a href="/w/index.php?title=Hyperbolic_triangle&action=edit&redlink=1" class="new" title="Hyperbolic triangle (पौ मदु)">hyperbolic triangles</a> in <a href="/w/index.php?title=Hyperbolic_geometry&action=edit&redlink=1" class="new" title="Hyperbolic geometry (पौ मदु)">hyperbolic geometry</a>. </p><p>While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, one would find that the sum of its angles were greater than 180°. </p> <div class="mw-heading mw-heading2"><h2 id="स्रोत"><span id=".E0.A4.B8.E0.A5.8D.E0.A4.B0.E0.A5.8B.E0.A4.A4"></span>स्रोत</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=11" title="खण्ड सम्पादन: स्रोत"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/EquilateralTriangle.html">http://mathworld.wolfram.com/EquilateralTriangle.html</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/IsoscelesTriangle.html">http://mathworld.wolfram.com/IsoscelesTriangle.html</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/ScaleneTriangle.html">http://mathworld.wolfram.com/ScaleneTriangle.html</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/TriangleArea.html">http://mathworld.wolfram.com/TriangleArea.html</a></span> </li> </ol> <div class="mw-heading mw-heading2"><h2 id="पिनेयागु_स्वापूतः"><span id=".E0.A4.AA.E0.A4.BF.E0.A4.A8.E0.A5.87.E0.A4.AF.E0.A4.BE.E0.A4.97.E0.A5.81_.E0.A4.B8.E0.A5.8D.E0.A4.B5.E0.A4.BE.E0.A4.AA.E0.A5.82.E0.A4.A4.E0.A4.83"></span>पिनेयागु स्वापूतः</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82&action=edit&section=12" title="खण्ड सम्पादन: पिनेयागु स्वापूतः"><span>सम्पादन</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://ostermiller.org/calc/triangle.html">Triangle Calculator</a> - solves for remaining sides and angles when given three sides or angles, supports degrees and radians.</li> <li><a rel="nofollow" class="external text" href="http://agutie.homestead.com/files/Napoleon0.htm">Napoleon's theorem</a> A triangle with three equilateral triangles. A purely geometric proof. It uses the Fermat point to prove Napoleon's theorem without transformations by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"</li> <li><a href="/w/index.php?title=William_Kahan&action=edit&redlink=1" class="new" title="William Kahan (पौ मदु)">William Kahan</a>: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061110144003/http://http.cs.berkeley.edu/~wkahan/Triangle.pdf">Miscalculating Area and Angles of a Needle-like Triangle</a>.</li> <li>Clark Kimberling: <a rel="nofollow" class="external text" href="http://faculty.evansville.edu/ck6/encyclopedia/ETC.html">Encyclopedia of triangle centers</a>. Lists some 1600 interesting points associated with any triangle.</li> <li>Christian Obrecht: <a rel="nofollow" class="external text" href="http://perso.wanadoo.fr/obrecht/">Eukleides</a>. Software package for creating illustrations of facts about triangles and other theorems in Euclidean geometry.</li> <li><a rel="nofollow" class="external text" href="http://www.apronus.com/geometry/triangle.htm">Proof that the sum of the angles in a triangle is 180 degrees</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/triangle">Triangle constructions, remarkable points and lines, and metric relations in a triangle</a> at <a href="/w/index.php?title=Cut-the-knot&action=edit&redlink=1" class="new" title="Cut-the-knot (पौ मदु)">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1469&CurriculumID=4">Printable Worksheet on Types of Triangles</a></li> <li><a rel="nofollow" class="external text" href="http://www.vias.org/comp_geometry/geom_triangle.html">Compendium Geometry</a> Analytical Geometry of Triangles</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061110032901/http://www.btinternet.com/~se16/hgb/triangle.htm">Area of a triangle - 7 different ways</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/tocs/triangletoc.html">Triangle definition pages</a> with interactive applets that are also useful in a classroom setting. Math Open Reference</li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/constequilateral.html">Constructing an equilateral triangle </a> , <a rel="nofollow" class="external text" href="http://www.mathopenref.com/constisosceles.html">Isosceles triangle </a> and <a rel="nofollow" class="external text" href="http://www.mathopenref.com/constcopytriangle.html">Copying a Triangle</a> with only a compass and straightedge, interactive animation.</li></ul> <table class="metadata mbox-small plainlinks" style="border:1px solid var(--border-color-base, #a2a9b1); background-color:var(--background-color-neutral-subtle, #f8f9fa); color:inherit;"> <tbody><tr> <td class="mbox-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></td> <td class="mbox-text" style=""> <a href="https://commons.wikimedia.org/wiki/Main_Page" class="extiw" title="commons:Main Page">विकिमिडिया मंका</a> य् थ्व विषय नाप स्वापु दुगु मिडिया दु: <i><b><a href="https://commons.wikimedia.org/wiki/Category:Triangles" class="extiw" title="commons:Category:Triangles">Triangles</a></b></i> </td> </tr> </tbody></table> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐85db4f49b5‐8dlhn Cached time: 20241113235002 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.069 seconds Real time usage: 0.179 seconds Preprocessor visited node count: 371/1000000 Post‐expand include size: 2550/2097152 bytes Template argument size: 695/2097152 bytes Highest expansion depth: 16/100 Expensive parser function count: 0/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 2279/5000000 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 20.174 1 Template:Commonscat 100.00% 20.174 1 -total 86.64% 17.479 1 Template:Commons 73.74% 14.876 1 Template:Sister 60.35% 12.174 1 Template:Side_box 46.02% 9.284 1 Template:Sec_link_auto 19.52% 3.939 1 Template:Sec_link/normal_link 12.31% 2.483 1 Template:Sec_link/text --> <!-- Saved in parser cache with key newwiki:pcache:idhash:6123-0!canonical and timestamp 20241113235002 and revision id 867346. 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