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id="toc-Circumradius-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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 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Available in 26 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-26" 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">26 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D8%A8%D9%87_%D9%85%D9%86%D8%AD%D8%B1%D9%81_%D9%85%D8%AA%D8%B3%D8%A7%D9%88%D9%8A_%D8%A7%D9%84%D8%B3%D8%A7%D9%82%D9%8A%D9%86" title="شبه منحرف متساوي الساقين – Arabic" lang="ar" hreflang="ar" data-title="شبه منحرف متساوي الساقين" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/B%C9%99rab%C9%99ryanl%C4%B1_trapesiya" title="Bərabəryanlı trapesiya – Azerbaijani" lang="az" hreflang="az" data-title="Bərabəryanlı trapesiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B0%D0%BD%D0%B0%D1%8F%D0%BA%D0%BB%C4%83_%D1%82%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D0%B8" title="Танаяклă трапеци – Chuvash" lang="cv" hreflang="cv" data-title="Танаяклă трапеци" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rovnoramenn%C3%BD_lichob%C4%9B%C5%BEn%C3%ADk" title="Rovnoramenný lichoběžník – Czech" lang="cs" hreflang="cs" data-title="Rovnoramenný lichoběžník" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%99%CF%83%CE%BF%CF%83%CE%BA%CE%B5%CE%BB%CE%AD%CF%82_%CF%84%CF%81%CE%B1%CF%80%CE%AD%CE%B6%CE%B9%CE%BF" title="Ισοσκελές τραπέζιο – Greek" lang="el" hreflang="el" data-title="Ισοσκελές τραπέζιο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Trapecio_is%C3%B3sceles" title="Trapecio isósceles – Spanish" lang="es" hreflang="es" data-title="Trapecio isósceles" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Izocela_trapezo" title="Izocela trapezo – Esperanto" lang="eo" hreflang="eo" data-title="Izocela trapezo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B0%D9%88%D8%B2%D9%86%D9%82%D9%87_%D9%85%D8%AA%D8%B3%D8%A7%D9%88%DB%8C%E2%80%8C%D8%A7%D9%84%D8%B3%D8%A7%D9%82%DB%8C%D9%86" title="ذوزنقه متساوی‌الساقین – Persian" lang="fa" hreflang="fa" data-title="ذوزنقه متساوی‌الساقین" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%93%B1%EB%B3%80%EC%82%AC%EB%8B%A4%EB%A6%AC%EA%BC%B4" title="등변사다리꼴 – Korean" lang="ko" hreflang="ko" data-title="등변사다리꼴" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%A1%D5%BD%D6%80%D5%B8%D6%82%D5%B6_%D5%BD%D5%A5%D5%B2%D5%A1%D5%B6" title="Հավասարասրուն սեղան – Armenian" lang="hy" hreflang="hy" data-title="Հավասարասրուն սեղան" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%AC%E0%A4%BE%E0%A4%B9%E0%A5%81_%E0%A4%B8%E0%A4%AE%E0%A4%B2%E0%A4%AE%E0%A5%8D%E0%A4%AC" title="समद्विबाहु समलम्ब – Hindi" lang="hi" hreflang="hi" data-title="समद्विबाहु समलम्ब" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%A8%D7%A4%D7%96_%D7%A9%D7%95%D7%95%D7%94-%D7%A9%D7%95%D7%A7%D7%99%D7%99%D7%9D" title="טרפז שווה-שוקיים – Hebrew" lang="he" hreflang="he" data-title="טרפז שווה-שוקיים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/H%C3%BArtrap%C3%A9z" title="Húrtrapéz – Hungarian" lang="hu" hreflang="hu" data-title="Húrtrapéz" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%BC%D0%BD%D0%BE%D0%BA%D1%80%D0%B0%D0%BA_%D1%82%D1%80%D0%B0%D0%BF%D0%B5%D0%B7" title="Рамнокрак трапез – Macedonian" lang="mk" hreflang="mk" data-title="Рамнокрак трапез" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%AD%89%E8%84%9A%E5%8F%B0%E5%BD%A2" title="等脚台形 – Japanese" lang="ja" hreflang="ja" data-title="等脚台形" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B2%D0%BD%D0%BE%D0%B1%D0%B5%D0%B4%D1%80%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D1%82%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D0%B8%D1%8F" title="Равнобедренная трапеция – Russian" lang="ru" hreflang="ru" data-title="Равнобедренная трапеция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Enakokraki_trapez" title="Enakokraki trapez – Slovenian" lang="sl" hreflang="sl" data-title="Enakokraki trapez" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%86%DB%8C%D9%85%DA%86%DB%95%D9%84%D8%A7%D8%AA%DB%95%D8%B1%DB%8C%D8%A8%DB%8C_%D8%AF%D9%88%D9%88%D9%84%D8%A7%DB%8C%DB%95%DA%A9%D8%B3%D8%A7%D9%86" title="نیمچەلاتەریبی دوولایەکسان – Central Kurdish" lang="ckb" hreflang="ckb" data-title="نیمچەلاتەریبی دوولایەکسان" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%88%D0%B5%D0%B4%D0%BD%D0%B0%D0%BA%D0%BE%D0%BA%D1%80%D0%B0%D0%BA%D0%B8_%D1%82%D1%80%D0%B0%D0%BF%D0%B5%D0%B7" title="Једнакокраки трапез – Serbian" lang="sr" hreflang="sr" data-title="Једнакокраки трапез" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tasakylkinen_puolisuunnikas" title="Tasakylkinen puolisuunnikas – Finnish" lang="fi" hreflang="fi" data-title="Tasakylkinen puolisuunnikas" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Likbent_parallelltrapets" title="Likbent parallelltrapets – Swedish" lang="sv" hreflang="sv" data-title="Likbent parallelltrapets" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B0%E0%AF%81%E0%AE%9A%E0%AE%AE%E0%AE%AA%E0%AE%95%E0%AF%8D%E0%AE%95_%E0%AE%9A%E0%AE%B0%E0%AE%BF%E0%AE%B5%E0%AE%95%E0%AE%AE%E0%AF%8D" title="இருசமபக்க சரிவகம் – Tamil" lang="ta" hreflang="ta" data-title="இருசமபக்க சரிவகம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0kizkenar_yamuk" title="İkizkenar yamuk – Turkish" lang="tr" hreflang="tr" data-title="İkizkenar yamuk" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D0%BE%D0%B1%D1%96%D1%87%D0%BD%D0%B0_%D1%82%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D1%96%D1%8F" title="Рівнобічна трапеція – Ukrainian" lang="uk" hreflang="uk" data-title="Рівнобічна трапеція" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_thang_c%C3%A2n" title="Hình thang cân – Vietnamese" lang="vi" hreflang="vi" data-title="Hình thang cân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%AD%89%E8%85%B0%E6%A2%AF%E5%BD%A2" title="等腰梯形 – Chinese" lang="zh" hreflang="zh" data-title="等腰梯形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1194115#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Isosceles_trapezoid" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Isosceles_trapezoid" rel="discussion" title="Discuss 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from <a href="/w/index.php?title=Isosceles_trapezium&redirect=no" class="mw-redirect" title="Isosceles trapezium">Isosceles trapezium</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Trapezoid symmetrical about an axis</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3;">Isosceles trapezoid</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Isosceles_trapezoid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Isosceles_trapezoid.svg/220px-Isosceles_trapezoid.svg.png" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Isosceles_trapezoid.svg/330px-Isosceles_trapezoid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Isosceles_trapezoid.svg/440px-Isosceles_trapezoid.svg.png 2x" data-file-width="290" data-file-height="237" /></a></span><div class="infobox-caption">Isosceles trapezoid with axis of symmetry</div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a>, <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a> and <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a></th><td class="infobox-data">4</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_planar_symmetry_groups" title="List of planar symmetry groups">Symmetry group</a></th><td class="infobox-data"><a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">Dih<sub>1</sub></a>, [ ], (*), order 1</td></tr><tr><th scope="row" class="infobox-label">Properties</th><td class="infobox-data"><a href="/wiki/Convex_polygon" title="Convex polygon">convex</a>, <a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">cyclic</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dual_polygon" title="Dual polygon">Dual polygon</a></th><td class="infobox-data"><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></td></tr></tbody></table> <p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, an <b>isosceles trapezoid</b> (<b>isosceles trapezium</b> in <a href="/wiki/British_English" title="British English">British English</a>) is a <a href="/wiki/Convex_polygon" title="Convex polygon">convex</a> <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> with a line of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> bisecting one pair of opposite sides. It is a special case of a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a>. Alternatively, it can be defined as a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a> in which both legs and both base angles are of equal measure,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or as a trapezoid whose diagonals have equal length.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Note that a non-rectangular <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides (the bases) are <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a>, and the two other sides (the legs) are of equal length (properties shared with the <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>), and the diagonals have equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the <a href="/wiki/Supplementary_angle" class="mw-redirect" title="Supplementary angle">supplementary angle</a> of a base angle at the other base). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=1" title="Edit section: Special cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Isosceles_trapezoid_special_cases.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Isosceles_trapezoid_special_cases.png/280px-Isosceles_trapezoid_special_cases.png" decoding="async" width="280" height="288" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Isosceles_trapezoid_special_cases.png/420px-Isosceles_trapezoid_special_cases.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/31/Isosceles_trapezoid_special_cases.png 2x" data-file-width="534" data-file-height="550" /></a><figcaption>Special cases of isosceles <a href="/wiki/Trapezoid" title="Trapezoid">trapezoids</a></figcaption></figure> <p><a href="/wiki/Rectangle" title="Rectangle">Rectangles</a> and <a href="/wiki/Square" title="Square">squares</a> are usually considered to be special cases of isosceles trapezoids though some sources would exclude them.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Another special case is a <i>3-equal side trapezoid</i>, sometimes known as a <i>trilateral trapezoid</i><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> or a <i>trisosceles trapezoid</i>. They can also be seen dissected from <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> of 5 sides or more as a truncation of 4 sequential vertices. </p> <div class="mw-heading mw-heading3"><h3 id="Self-intersections">Self-intersections</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=2" title="Edit section: Self-intersections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any non-self-crossing <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> with exactly one axis of symmetry must be either an isosceles trapezoid or a <a href="/wiki/Kite_(geometry)" title="Kite (geometry)">kite</a>.<sup id="cite_ref-esg_5-0" class="reference"><a href="#cite_note-esg-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the <a href="/wiki/Antiparallelogram" title="Antiparallelogram">antiparallelograms</a>, crossed quadrilaterals in which opposite sides have equal length. </p><p>Every <a href="/wiki/Antiparallelogram" title="Antiparallelogram">antiparallelogram</a> has an isosceles trapezoid as its <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a>, and may be formed from the diagonals and non-parallel sides (or either pair of opposite sides in the case of a rectangle) of an isosceles trapezoid.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Isosceles_trapezoid_example.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Isosceles_trapezoid_example.png/100px-Isosceles_trapezoid_example.png" decoding="async" width="100" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Isosceles_trapezoid_example.png/150px-Isosceles_trapezoid_example.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Isosceles_trapezoid_example.png/200px-Isosceles_trapezoid_example.png 2x" data-file-width="332" data-file-height="606" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Crossed_isosceles_trapezoid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Crossed_isosceles_trapezoid.png/100px-Crossed_isosceles_trapezoid.png" decoding="async" width="100" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Crossed_isosceles_trapezoid.png/150px-Crossed_isosceles_trapezoid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Crossed_isosceles_trapezoid.png/200px-Crossed_isosceles_trapezoid.png 2x" data-file-width="328" data-file-height="595" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Antiparallelogram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Antiparallelogram.svg/100px-Antiparallelogram.svg.png" decoding="async" width="100" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Antiparallelogram.svg/150px-Antiparallelogram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Antiparallelogram.svg/200px-Antiparallelogram.svg.png 2x" data-file-width="267" data-file-height="496" /></a></span> </td></tr> <tr> <th>Convex isosceles<br />trapezoid </th> <th>Crossed isosceles<br />trapezoid </th> <th><a href="/wiki/Antiparallelogram" title="Antiparallelogram">antiparallelogram</a> </th></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Characterizations">Characterizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=3" title="Edit section: Characterizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a quadrilateral is known to be a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a>, it is <i>not</i> sufficient just to check that the legs have the same length in order to know that it is an isosceles trapezoid, since a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a> is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry through the midpoints of opposite sides. </p><p>Any one of the following properties distinguishes an isosceles trapezoid from other trapezoids: </p> <ul><li>The diagonals have the same length.</li> <li>The base angles have the same measure.</li> <li>The segment that joins the midpoints of the parallel sides is perpendicular to them.</li> <li>Opposite angles are supplementary, which in turn implies that isosceles trapezoids are <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilaterals</a>.</li> <li>The diagonals divide each other into segments with lengths that are pairwise equal; in terms of the picture below, <span class="nowrap"><i>AE</i> = <i>DE</i></span>, <span class="nowrap"><i>BE</i> = <i>CE</i></span> (and <span class="nowrap"><i>AE</i> ≠ <i>CE</i></span> if one wishes to exclude rectangles).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Angles">Angles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=4" title="Edit section: Angles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In an isosceles trapezoid, the base angles have the same measure pairwise. In the picture below, angles ∠<i>ABC</i> and ∠<i>DCB</i> are <a href="/wiki/Angle#Types_of_angles" title="Angle">obtuse</a> angles of the same measure, while angles ∠<i>BAD</i> and ∠<i>CDA</i> are <a href="/wiki/Angle#Types_of_angles" title="Angle">acute angles</a>, also of the same measure. </p><p>Since the lines <i>AD</i> and <i>BC</i> are parallel, angles adjacent to opposite bases are <a href="/wiki/Supplementary_angles" class="mw-redirect" title="Supplementary angles">supplementary</a>, that is, angles <span class="nowrap">∠<i>ABC</i> + ∠<i>BAD</i> = 180°.</span> </p> <div class="mw-heading mw-heading2"><h2 id="Diagonals_and_height">Diagonals and height</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=5" title="Edit section: Diagonals and height"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Isoscelestriangle2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Isoscelestriangle2.svg/350px-Isoscelestriangle2.svg.png" decoding="async" width="350" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Isoscelestriangle2.svg/525px-Isoscelestriangle2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Isoscelestriangle2.svg/700px-Isoscelestriangle2.svg.png 2x" data-file-width="575" data-file-height="288" /></a><figcaption>Another isosceles trapezoid.</figcaption></figure> <p>The <a href="/wiki/Diagonal" title="Diagonal">diagonals</a> of an isosceles trapezoid have the same length; that is, every isosceles trapezoid is an <a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">equidiagonal quadrilateral</a>. Moreover, the diagonals divide each other in the same proportions. As pictured, the diagonals <i>AC</i> and <i>BD</i> have the same length (<span class="nowrap"><i>AC</i> = <i>BD</i></span>) and divide each other into segments of the same length (<span class="nowrap"><i>AE</i> = <i>DE</i></span> and <span class="nowrap"><i>BE</i> = <i>CE</i></span>). </p><p>The <a href="/wiki/Ratio" title="Ratio">ratio</a> in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, that is, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {AE}{EC}}={\frac {DE}{EB}}={\frac {AD}{BC}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mrow> <mi>E</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>E</mi> </mrow> <mrow> <mi>E</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>D</mi> </mrow> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {AE}{EC}}={\frac {DE}{EB}}={\frac {AD}{BC}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13431892ac581078654bb36021adaea08da0d3c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.261ex; height:5.509ex;" alt="{\displaystyle {\frac {AE}{EC}}={\frac {DE}{EB}}={\frac {AD}{BC}}.}"></span></dd></dl> <p>The length of each diagonal is, according to <a href="/wiki/Ptolemy%27s_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a>, given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\sqrt {ab+c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\sqrt {ab+c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f3a9e2341d822fc777a578a011cb5f68ac91d8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.81ex; height:3.509ex;" alt="{\displaystyle p={\sqrt {ab+c^{2}}}}"></span></dd></dl> <p>where <i>a</i> and <i>b</i> are the lengths of the parallel sides <i>AD</i> and <i>BC</i>, and <i>c</i> is the length of each leg <i>AB</i> and <i>CD</i>. </p><p>The height is, according to the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\sqrt {p^{2}-\left({\frac {a+b}{2}}\right)^{2}}}={\tfrac {1}{2}}{\sqrt {4c^{2}-(a-b)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\sqrt {p^{2}-\left({\frac {a+b}{2}}\right)^{2}}}={\tfrac {1}{2}}{\sqrt {4c^{2}-(a-b)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/902eedaecc34d1abe242b053508ae8460e76dbbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.927ex; height:7.676ex;" alt="{\displaystyle h={\sqrt {p^{2}-\left({\frac {a+b}{2}}\right)^{2}}}={\tfrac {1}{2}}{\sqrt {4c^{2}-(a-b)^{2}}}.}"></span></dd></dl> <p>The distance from point <i>E</i> to base <i>AD</i> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {ah}{a+b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>h</mi> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {ah}{a+b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8e5522aa12ca62bea9f27f60dd4b1e19785cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.218ex; height:5.676ex;" alt="{\displaystyle d={\frac {ah}{a+b}}}"></span></dd></dl> <p>where <i>a</i> and <i>b</i> are the lengths of the parallel sides <i>AD</i> and <i>BC</i>, and <i>h</i> is the height of the trapezoid. </p> <div class="mw-heading mw-heading2"><h2 id="Area">Area</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=6" title="Edit section: Area"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The area of an isosceles (or any) trapezoid is equal to the average of the lengths of the base and top (<i>the parallel sides</i>) times the height. In the adjacent diagram, if we write <span class="nowrap"><i>AD</i> = <i>a</i></span>, and <span class="nowrap"><i>BC</i> = <i>b</i></span>, and the height <i>h</i> is the length of a line segment between <i>AD</i> and <i>BC</i> that is perpendicular to them, then the area <i>K</i> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\tfrac {1}{2}}\left(a+b\right)h.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\tfrac {1}{2}}\left(a+b\right)h.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d1bb3d99d810824c59118c946835ca6b1d0051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.46ex; height:3.509ex;" alt="{\displaystyle K={\tfrac {1}{2}}\left(a+b\right)h.}"></span></dd></dl> <p>If instead of the height of the trapezoid, the common length of the legs <i>AB</i> =<i>CD</i> = <i>c</i> is known, then the area can be computed using <a href="/wiki/Brahmagupta%27s_formula" title="Brahmagupta's formula">Brahmagupta's formula</a> for the area of a cyclic quadrilateral, which with two sides equal simplifies to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=(s-c){\sqrt {(s-a)(s-b)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=(s-c){\sqrt {(s-a)(s-b)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1e5944e79670d45f9bae1b272262630a39f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.59ex; height:4.843ex;" alt="{\displaystyle K=(s-c){\sqrt {(s-a)(s-b)}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\tfrac {1}{2}}(a+b+2c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\tfrac {1}{2}}(a+b+2c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/805ce9fee90280862b7719a4e8ecb70ebaad2035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.734ex; height:3.509ex;" alt="{\displaystyle s={\tfrac {1}{2}}(a+b+2c)}"></span> is the semi-perimeter of the trapezoid. This formula is analogous to <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> to compute the area of a triangle. The previous formula for area can also be written 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 K={\frac {a+b}{4}}{\sqrt {(a-b+2c)(b-a+2c)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {a+b}{4}}{\sqrt {(a-b+2c)(b-a+2c)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93db2d9d76048d4b17c263bcd8b73880bd9b2ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.812ex; height:5.343ex;" alt="{\displaystyle K={\frac {a+b}{4}}{\sqrt {(a-b+2c)(b-a+2c)}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Circumradius">Circumradius</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=7" title="Edit section: Circumradius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The radius in the circumscribed circle is given by<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <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 R=c{\sqrt {\frac {ab+c^{2}}{4c^{2}-(a-b)^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=c{\sqrt {\frac {ab+c^{2}}{4c^{2}-(a-b)^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b823d5a810d225d120a6d5d0b57d0a192c834cec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.671ex; height:7.676ex;" alt="{\displaystyle R=c{\sqrt {\frac {ab+c^{2}}{4c^{2}-(a-b)^{2}}}}.}"></span></dd></dl> <p>In a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> where <i>a</i> = <i>b</i> this is simplified to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\tfrac {1}{2}}{\sqrt {a^{2}+c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\tfrac {1}{2}}{\sqrt {a^{2}+c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d674c44de5a9be4860142abebffa440161d67ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.03ex; height:3.843ex;" alt="{\displaystyle R={\tfrac {1}{2}}{\sqrt {a^{2}+c^{2}}}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Tangential_trapezoid#Isosceles_tangential_trapezoid" title="Tangential trapezoid">Isosceles tangential trapezoid</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathopenref.com/trapezoid.html">"Trapezoid - math word definition - Math Open Reference"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Trapezoid+-+math+word+definition+-+Math+Open+Reference&rft_id=http%3A%2F%2Fwww.mathopenref.com%2Ftrapezoid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsosceles+trapezoid" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyoti1967" class="citation journal cs1">Ryoti, Don E. (1967). "What is an Isosceles Trapezoid?". <i>The Mathematics Teacher</i>. <b>60</b> (7): 729–730. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2FMT.60.7.0729">10.5951/MT.60.7.0729</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27957671">27957671</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Teacher&rft.atitle=What+is+an+Isosceles+Trapezoid%3F&rft.volume=60&rft.issue=7&rft.pages=729-730&rft.date=1967&rft_id=info%3Adoi%2F10.5951%2FMT.60.7.0729&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27957671%23id-name%3DJSTOR&rft.aulast=Ryoti&rft.aufirst=Don+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsosceles+trapezoid" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarsonBoswell2016" class="citation book cs1">Larson, Ron; Boswell, Laurie (2016). <i>Big Ideas MATH, Geometry, Texas Edition</i>. Big Ideas Learning, LLC (2016). p. 398. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1608408153" title="Special:BookSources/978-1608408153"><bdi>978-1608408153</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Big+Ideas+MATH%2C+Geometry%2C+Texas+Edition&rft.pages=398&rft.pub=Big+Ideas+Learning%2C+LLC+%282016%29&rft.date=2016&rft.isbn=978-1608408153&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Boswell%2C+Laurie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsosceles+trapezoid" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://dynamicmathematicslearning.com/quad-tree-web.html">"A Hierarchical Classification of Quadrilaterals"</a>. <i>dynamicmathematicslearning.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">February 10,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=dynamicmathematicslearning.com&rft.atitle=A+Hierarchical+Classification+of+Quadrilaterals&rft_id=http%3A%2F%2Fdynamicmathematicslearning.com%2Fquad-tree-web.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsosceles+trapezoid" class="Z3988"></span></span> </li> <li id="cite_note-esg-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-esg_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalsted1896" class="citation cs2"><a href="/wiki/G._B._Halsted" title="G. B. Halsted">Halsted, George Bruce</a> (1896), "Chapter XIV. Symmetrical Quadrilaterals", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=H3ALAAAAYAAJ&pg=PA49"><i>Elementary Synthetic Geometry</i></a>, J. Wiley & sons, pp. 49–53</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+XIV.+Symmetrical+Quadrilaterals&rft.btitle=Elementary+Synthetic+Geometry&rft.pages=49-53&rft.pub=J.+Wiley+%26+sons&rft.date=1896&rft.aulast=Halsted&rft.aufirst=George+Bruce&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DH3ALAAAAYAAJ%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsosceles+trapezoid" class="Z3988"></span>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitneySmith1911" class="citation cs2">Whitney, William Dwight; Smith, Benjamin Eli (1911), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ownpAAAAMAAJ&pg=PA1547"><i>The Century Dictionary and Cyclopedia</i></a>, The Century co., p. 1547</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Century+Dictionary+and+Cyclopedia&rft.pages=1547&rft.pub=The+Century+co.&rft.date=1911&rft.aulast=Whitney&rft.aufirst=William+Dwight&rft.au=Smith%2C+Benjamin+Eli&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DownpAAAAMAAJ%26pg%3DPA1547&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsosceles+trapezoid" class="Z3988"></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Trapezoid at Math24.net: Formulas and Tables <a rel="nofollow" class="external autonumber" href="http://www.math24.net/trapezoid.html">[1]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180628153032/https://www.math24.net/trapezoid/">Archived</a> June 28, 2018, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Accessed 1 July 2014.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isosceles_trapezoid&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.efunda.com/math/areas/IsosTrapazoid.cfm">Some engineering formulas involving isosceles trapezoids</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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title="Kepler triangle">Kepler</a></li> <li><a href="/wiki/Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Obtuse</a></li> <li><a href="/wiki/Right_triangle" title="Right triangle">Right</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilaterals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a></li> <li><a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">Bicentric</a></li> <li><a href="/wiki/Crossed_quadrilateral" class="mw-redirect" title="Crossed quadrilateral">Crossed</a></li> <li><a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">Cyclic</a></li> <li><a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">Equidiagonal</a></li> <li><a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">Ex-tangential</a></li> <li><a href="/wiki/Harmonic_quadrilateral" title="Harmonic quadrilateral">Harmonic</a></li> <li><a class="mw-selflink selflink">Isosceles trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li> <li><a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">Orthodiagonal</a></li> <li><a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Right_kite" title="Right kite">Right kite</a></li> <li><a href="/wiki/Right_trapezoid" class="mw-redirect" title="Right trapezoid">Right trapezoid</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">Tangential</a></li> <li><a href="/wiki/Tangential_trapezoid" title="Tangential trapezoid">Tangential trapezoid</a></li> <li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By number <br />of sides</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">1–10 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Monogon" title="Monogon">Monogon (1)</a></li> <li><a href="/wiki/Digon" title="Digon">Digon (2)</a></li> <li><a href="/wiki/Triangle" title="Triangle">Triangle (3)</a></li> <li><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral (4)</a></li> <li><a href="/wiki/Pentagon" title="Pentagon">Pentagon (5)</a></li> <li><a href="/wiki/Hexagon" title="Hexagon">Hexagon (6)</a></li> <li><a href="/wiki/Heptagon" title="Heptagon">Heptagon (7)</a></li> <li><a href="/wiki/Octagon" title="Octagon">Octagon (8)</a></li> <li><a href="/wiki/Nonagon" title="Nonagon">Nonagon/Enneagon (9)</a></li> <li><a href="/wiki/Decagon" title="Decagon">Decagon (10)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11–20 sides</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hendecagon" title="Hendecagon">Hendecagon (11)</a></li> <li><a href="/wiki/Dodecagon" title="Dodecagon">Dodecagon (12)</a></li> <li><a href="/wiki/Tridecagon" title="Tridecagon">Tridecagon (13)</a></li> <li><a href="/wiki/Tetradecagon" title="Tetradecagon">Tetradecagon (14)</a></li> <li><a href="/wiki/Pentadecagon" title="Pentadecagon">Pentadecagon (15)</a></li> <li><a href="/wiki/Hexadecagon" title="Hexadecagon">Hexadecagon (16)</a></li> <li><a href="/wiki/Heptadecagon" title="Heptadecagon">Heptadecagon (17)</a></li> <li><a href="/wiki/Octadecagon" title="Octadecagon">Octadecagon (18)</a></li> <li><a href="/wiki/Icosagon" title="Icosagon">Icosagon (20)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">>20 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Icositrigon" title="Icositrigon">Icositrigon (23)</a></li> <li><a href="/wiki/Icositetragon" title="Icositetragon">Icositetragon (24)</a></li> <li><a href="/wiki/Triacontagon" title="Triacontagon">Triacontagon (30)</a></li> <li><a href="/wiki/257-gon" title="257-gon">257-gon</a></li> <li><a href="/wiki/Chiliagon" title="Chiliagon">Chiliagon (1000)</a></li> <li><a href="/wiki/Myriagon" title="Myriagon">Myriagon (10,000)</a></li> <li><a href="/wiki/65537-gon" title="65537-gon">65537-gon</a></li> <li><a href="/wiki/Megagon" title="Megagon">Megagon (1,000,000)</a></li> <li><a href="/wiki/Apeirogon" title="Apeirogon">Apeirogon (∞)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Star_polygon" title="Star polygon">Star polygons</a><br /></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentagram" title="Pentagram">Pentagram</a></li> <li><a href="/wiki/Hexagram" title="Hexagram">Hexagram</a></li> <li><a href="/wiki/Heptagram" title="Heptagram">Heptagram</a></li> <li><a href="/wiki/Octagram" title="Octagram">Octagram</a></li> <li><a href="/wiki/Enneagram_(geometry)" title="Enneagram (geometry)">Enneagram</a></li> <li><a href="/wiki/Decagram_(geometry)" title="Decagram (geometry)">Decagram</a></li> <li><a href="/wiki/Hendecagram" title="Hendecagram">Hendecagram</a></li> <li><a href="/wiki/Dodecagram" title="Dodecagram">Dodecagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Concave_polygon" title="Concave polygon">Concave</a></li> <li><a href="/wiki/Convex_polygon" title="Convex polygon">Convex</a></li> <li><a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">Cyclic</a></li> <li><a href="/wiki/Equiangular_polygon" title="Equiangular polygon">Equiangular</a></li> <li><a href="/wiki/Equilateral_polygon" title="Equilateral polygon">Equilateral</a></li> <li><a href="/wiki/Infinite_skew_polygon" title="Infinite skew polygon">Infinite skew</a></li> <li><a href="/wiki/Isogonal_figure" title="Isogonal figure">Isogonal</a></li> <li><a href="/wiki/Isotoxal_figure" title="Isotoxal figure">Isotoxal</a></li> <li><a href="/wiki/Magic_polygon" title="Magic polygon">Magic</a></li> <li><a href="/wiki/Pseudotriangle" title="Pseudotriangle">Pseudotriangle</a></li> <li><a href="/wiki/Rectilinear_polygon" title="Rectilinear polygon">Rectilinear</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular</a></li> <li><a href="/wiki/Reinhardt_polygon" title="Reinhardt polygon">Reinhardt</a></li> <li><a href="/wiki/Simple_polygon" title="Simple polygon">Simple</a></li> <li><a href="/wiki/Skew_polygon" title="Skew polygon">Skew</a></li> <li><a href="/wiki/Star-shaped_polygon" title="Star-shaped polygon">Star-shaped</a></li> <li><a href="/wiki/Tangential_polygon" title="Tangential polygon">Tangential</a></li> <li><a href="/wiki/Weakly_simple_polygon" title="Weakly simple polygon">Weakly simple</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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