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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">(Top)</div> </a> </li> <li id="toc-Etymology_and_trapezium_versus_trapezoid" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Etymology_and_trapezium_versus_trapezoid"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Etymology and <i>trapezium</i> versus <i>trapezoid</i></span> </div> </a> <ul id="toc-Etymology_and_trapezium_versus_trapezoid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inclusive_versus_exclusive_definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Inclusive_versus_exclusive_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Inclusive versus exclusive definition</span> </div> </a> <ul id="toc-Inclusive_versus_exclusive_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_cases" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_cases"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Special cases</span> </div> </a> <ul id="toc-Special_cases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Condition_of_existence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Condition_of_existence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Condition of existence</span> </div> </a> <ul id="toc-Condition_of_existence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characterizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Characterizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Characterizations</span> </div> </a> <ul id="toc-Characterizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Midsegment_and_height" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Midsegment_and_height"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Midsegment and height</span> </div> </a> <ul id="toc-Midsegment_and_height-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Area" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Area"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Area</span> </div> </a> <ul id="toc-Area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diagonals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diagonals"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Diagonals</span> </div> </a> <ul id="toc-Diagonals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Other properties</span> </div> </a> <ul id="toc-Other_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Architecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Architecture"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Architecture</span> </div> </a> <ul id="toc-Architecture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Geometry</span> </div> </a> <ul id="toc-Geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biology"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Biology</span> </div> </a> <ul id="toc-Biology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.4</span> <span>Computer engineering</span> </div> </a> <ul id="toc-Computer_engineering-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <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">Trapezoid</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 99 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-99" 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">99 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Trapesium" title="Trapesium – Afrikaans" lang="af" hreflang="af" data-title="Trapesium" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><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" title="شبه منحرف – Arabic" lang="ar" hreflang="ar" data-title="شبه منحرف" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Trapeciu_(xeometr%C3%ADa)" title="Trapeciu (xeometría) – Asturian" lang="ast" hreflang="ast" data-title="Trapeciu (xeometría)" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Trapesiya" title="Trapesiya – Azerbaijani" lang="az" hreflang="az" data-title="Trapesiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9F%E0%A7%8D%E0%A6%B0%E0%A6%BE%E0%A6%AA%E0%A6%BF%E0%A6%9C%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%AE" title="ট্রাপিজিয়াম – Bangla" lang="bn" hreflang="bn" data-title="ট্রাপিজিয়াম" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D0%B8%D1%8F" title="Трапеция – Bashkir" lang="ba" hreflang="ba" data-title="Трапеция" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D1%8B%D1%8F" title="Трапецыя – Belarusian" lang="be" hreflang="be" data-title="Трапецыя" 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%D0%B0%D0%BF%D1%8D%D1%86%D1%8B%D1%8F" title="Трапэцыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Трапэцыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Trapesoyd" title="Trapesoyd – Central Bikol" lang="bcl" hreflang="bcl" data-title="Trapesoyd" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86" title="Трапец – Bulgarian" lang="bg" hreflang="bg" data-title="Трапец" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A6%E0%BE%90%E0%BD%A6%E0%BC%8B%E0%BD%91%E0%BD%96%E0%BE%B1%E0%BD%B2%E0%BD%96%E0%BD%A6%E0%BC%8B" title="སྐས་དབྱིབས་ – Tibetan" lang="bo" hreflang="bo" data-title="སྐས་དབྱིབས་" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Trapez" title="Trapez – Bosnian" lang="bs" hreflang="bs" data-title="Trapez" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Trapez" title="Trapez – Breton" lang="br" hreflang="br" data-title="Trapez" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Trapezi" title="Trapezi – Catalan" lang="ca" hreflang="ca" data-title="Trapezi" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D0%B8" title="Трапеци – Chuvash" lang="cv" hreflang="cv" data-title="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/Lichob%C4%9B%C5%BEn%C3%ADk" title="Lichoběžník – Czech" lang="cs" hreflang="cs" data-title="Lichoběžník" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Gonyoina_sambambiri" title="Gonyoina sambambiri – Shona" lang="sn" hreflang="sn" data-title="Gonyoina sambambiri" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Trapesiwm" title="Trapesiwm – Welsh" lang="cy" hreflang="cy" data-title="Trapesiwm" 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/Trapez_(matematik)" title="Trapez (matematik) – Danish" lang="da" hreflang="da" data-title="Trapez (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Trapesa" title="Trapesa – Northern Sami" lang="se" hreflang="se" data-title="Trapesa" data-language-autonym="Davvisámegiella" data-language-local-name="Northern Sami" class="interlanguage-link-target"><span>Davvisámegiella</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Trapez_(Geometrie)" title="Trapez (Geometrie) – German" lang="de" hreflang="de" data-title="Trapez (Geometrie)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Trapets" title="Trapets – Estonian" lang="et" hreflang="et" data-title="Trapets" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%B1%CF%80%CE%AD%CE%B6%CE%B9%CE%BF" title="Τραπέζιο – Greek" lang="el" hreflang="el" data-title="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_(geometr%C3%ADa)" title="Trapecio (geometría) – Spanish" lang="es" hreflang="es" data-title="Trapecio (geometría)" 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/Trapezo" title="Trapezo – Esperanto" lang="eo" hreflang="eo" data-title="Trapezo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Trapezio" title="Trapezio – Basque" lang="eu" hreflang="eu" data-title="Trapezio" 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/%D8%B0%D9%88%D8%B2%D9%86%D9%82%D9%87" title="ذوزنقه – Persian" lang="fa" hreflang="fa" data-title="ذوزنقه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Trap%C3%A8ze" title="Trapèze – French" lang="fr" hreflang="fr" data-title="Trapèze" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Traip%C3%A9isiam" title="Traipéisiam – Irish" lang="ga" hreflang="ga" data-title="Traipéisiam" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Trapecio" title="Trapecio – Galician" lang="gl" hreflang="gl" data-title="Trapecio" 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%B8%E0%AA%AE%E0%AA%BE%E0%AA%82%E0%AA%A4%E0%AA%B0%E0%AA%AC%E0%AA%BE%E0%AA%9C%E0%AB%81_%E0%AA%9A%E0%AA%A4%E0%AB%81%E0%AA%B7%E0%AB%8D%E0%AA%95%E0%AB%8B%E0%AA%A3" title="સમાંતરબાજુ ચતુષ્કોણ – Gujarati" lang="gu" hreflang="gu" data-title="સમાંતરબાજુ ચતુષ્કોણ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%AC%EB%8B%A4%EB%A6%AC%EA%BC%B4" title="사다리꼴 – Korean" lang="ko" hreflang="ko" data-title="사다리꼴" 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%8D%D5%A5%D5%B2%D5%A1%D5%B6_(%D5%A5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6)" title="Սեղան (երկրաչափություն) – Armenian" lang="hy" hreflang="hy" data-title="Սեղան (երկրաչափություն)" 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%B2%E0%A4%AE%E0%A5%8D%E0%A4%AC_%E0%A4%9A%E0%A4%A4%E0%A5%81%E0%A4%B0%E0%A5%8D%E0%A4%AD%E0%A5%81%E0%A4%9C" title="समलम्ब चतुर्भुज – Hindi" lang="hi" hreflang="hi" data-title="समलम्ब चतुर्भुज" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Trapec" title="Trapec – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Trapec" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Trapez_(geometrija)" title="Trapez (geometrija) – Croatian" lang="hr" hreflang="hr" data-title="Trapez (geometrija)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Trapezoido" title="Trapezoido – Ido" lang="io" hreflang="io" data-title="Trapezoido" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Trapesium_(geometri)" title="Trapesium (geometri) – Indonesian" lang="id" hreflang="id" data-title="Trapesium (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Trapezio" title="Trapezio – Interlingua" lang="ia" hreflang="ia" data-title="Trapezio" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Trapisa" title="Trapisa – Icelandic" lang="is" hreflang="is" data-title="Trapisa" 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/Trapezio" title="Trapezio – Italian" lang="it" hreflang="it" data-title="Trapezio" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%A8%D7%A4%D7%96" title="טרפז – Hebrew" lang="he" hreflang="he" data-title="טרפז" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Ngapan-apan" title="Ngapan-apan – Javanese" lang="jv" hreflang="jv" data-title="Ngapan-apan" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A2%E1%83%A0%E1%83%90%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90" title="ტრაპეცია – Georgian" lang="ka" hreflang="ka" data-title="ტრაპეცია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D0%B8%D1%8F" title="Трапеция – Kazakh" lang="kk" hreflang="kk" data-title="Трапеция" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D0%B8%D1%8F" title="Трапеция – Kyrgyz" lang="ky" hreflang="ky" data-title="Трапеция" 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/Trapezium" title="Trapezium – Latin" lang="la" hreflang="la" data-title="Trapezium" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Trapece" title="Trapece – Latvian" lang="lv" hreflang="lv" data-title="Trapece" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trapecija" title="Trapecija – Lithuanian" lang="lt" hreflang="lt" data-title="Trapecija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Trap%C3%A9se_(geometr%C3%ACa)" title="Trapése (geometrìa) – Lombard" lang="lmo" hreflang="lmo" data-title="Trapése (geometrìa)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Trap%C3%A9z" title="Trapéz – Hungarian" lang="hu" hreflang="hu" data-title="Trapéz" 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%A2%D1%80%D0%B0%D0%BF%D0%B5%D0%B7" title="Трапез – Macedonian" lang="mk" hreflang="mk" data-title="Трапез" 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%B2%E0%B4%82%E0%B4%AC%E0%B4%95%E0%B4%82" title="ലംബകം – Malayalam" lang="ml" hreflang="ml" data-title="ലംബകം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%B2%E0%A4%82%E0%A4%AC_%E0%A4%9A%E0%A5%8C%E0%A4%95%E0%A5%8B%E0%A4%A8" title="समलंब चौकोन – Marathi" lang="mr" hreflang="mr" data-title="समलंब चौकोन" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A2%E1%83%A0%E1%83%90%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90" title="ტრაპეცია – Mingrelian" lang="xmf" hreflang="xmf" data-title="ტრაპეცია" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Trapezium" title="Trapezium – Malay" lang="ms" hreflang="ms" data-title="Trapezium" 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-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86" title="Трапец – Mongolian" lang="mn" hreflang="mn" data-title="Трапец" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Trapezium" title="Trapezium – Dutch" lang="nl" hreflang="nl" data-title="Trapezium" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%B0%E5%BD%A2" title="台形 – Japanese" lang="ja" hreflang="ja" data-title="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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Trapeets" title="Trapeets – Northern Frisian" lang="frr" hreflang="frr" data-title="Trapeets" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Trapes_(geometri)" title="Trapes (geometri) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Trapes (geometri)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Trapes_i_geometri" title="Trapes i geometri – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Trapes i geometri" 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-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D0%B8%D0%B9" title="Трапеций – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Трапеций" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%9F%E0%AD%8D%E0%AC%B0%E0%AC%BE%E0%AC%AA%E0%AC%BF%E0%AC%9C%E0%AC%BF%E0%AC%85%E0%AC%AE" title="ଟ୍ରାପିଜିଅମ – Odia" lang="or" hreflang="or" data-title="ଟ୍ରାପିଜିଅମ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Trapetsiya" title="Trapetsiya – Uzbek" lang="uz" hreflang="uz" data-title="Trapetsiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%AE%E0%A8%B2%E0%A9%B0%E0%A8%AC_%E0%A8%9A%E0%A8%A4%E0%A9%81%E0%A8%B0%E0%A8%AD%E0%A9%81%E0%A8%9C" title="ਸਮਲੰਬ ਚਤੁਰਭੁਜ – Punjabi" lang="pa" hreflang="pa" data-title="ਸਮਲੰਬ ਚਤੁਰਭੁਜ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9E%8F%E1%9E%BB%E1%9E%80%E1%9F%84%E1%9E%8E%E1%9E%96%E1%9F%92%E1%9E%93%E1%9E%B6%E1%9E%99" title="ចតុកោណព្នាយ – Khmer" lang="km" hreflang="km" data-title="ចតុកោណព្នាយ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Trapessi" title="Trapessi – Piedmontese" lang="pms" hreflang="pms" data-title="Trapessi" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Trapez" title="Trapez – Polish" lang="pl" hreflang="pl" data-title="Trapez" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Trap%C3%A9zio_(geometria)" title="Trapézio (geometria) – Portuguese" lang="pt" hreflang="pt" data-title="Trapézio (geometria)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Trapez" title="Trapez – Romanian" lang="ro" hreflang="ro" data-title="Trapez" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Putuq" title="Putuq – Quechua" lang="qu" hreflang="qu" data-title="Putuq" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%BF%D0%B5%D1%86%D0%B8%D1%8F" title="Трапеция – Russian" lang="ru" hreflang="ru" data-title="Трапеция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Trapezoid" title="Trapezoid – Simple English" lang="en-simple" hreflang="en-simple" data-title="Trapezoid" 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/Lichobe%C5%BEn%C3%ADk" title="Lichobežník – Slovak" lang="sk" hreflang="sk" data-title="Lichobežník" 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/Trapez" title="Trapez – Slovenian" lang="sl" hreflang="sl" data-title="Trapez" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Trapez" title="Trapez – Silesian" lang="szl" hreflang="szl" data-title="Trapez" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Koor" title="Koor – Somali" lang="so" hreflang="so" data-title="Koor" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%86%DB%8C%D9%85%DA%86%DB%95%D9%84%D8%A7%D8%AA%DB%95%D8%B1%DB%8C%D8%A8" title="نیمچەلاتەریب – Central Kurdish" lang="ckb" hreflang="ckb" data-title="نیمچەلاتەریب" 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%A2%D1%80%D0%B0%D0%BF%D0%B5%D0%B7_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Трапез (геометрија) – Serbian" lang="sr" hreflang="sr" data-title="Трапез (геометрија)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Trapez_(geometrija)" title="Trapez (geometrija) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Trapez (geometrija)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Trap%C3%A9sium" title="Trapésium – Sundanese" lang="su" hreflang="su" data-title="Trapésium" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Puolisuunnikas" title="Puolisuunnikas – Finnish" lang="fi" hreflang="fi" data-title="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/Parallelltrapets" title="Parallelltrapets – Swedish" lang="sv" hreflang="sv" data-title="Parallelltrapets" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Trapesoid" title="Trapesoid – Tagalog" lang="tl" hreflang="tl" data-title="Trapesoid" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%B0%E0%AE%BF%E0%AE%B5%E0%AE%95%E0%AE%AE%E0%AF%8D" title="சரிவகம் – Tamil" lang="ta" hreflang="ta" data-title="சரிவகம்" 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%B8%E0%B0%AE%E0%B0%B2%E0%B0%82%E0%B0%AC_%E0%B0%9A%E0%B0%A4%E0%B1%81%E0%B0%B0%E0%B1%8D%E0%B0%AD%E0%B1%81%E0%B0%9C%E0%B0%82" title="సమలంబ చతుర్భుజం – Telugu" lang="te" hreflang="te" data-title="సమలంబ చతుర్భుజం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%AA%E0%B8%B5%E0%B9%88%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1%E0%B8%84%E0%B8%B2%E0%B8%87%E0%B8%AB%E0%B8%A1%E0%B8%B9" title="รูปสี่เหลี่ยมคางหมู – Thai" lang="th" hreflang="th" data-title="รูปสี่เหลี่ยมคางหมู" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Yamuk" title="Yamuk – Turkish" lang="tr" 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منحرف – Urdu" lang="ur" hreflang="ur" data-title="شکل منحرف" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_thang" title="Hình thang – Vietnamese" lang="vi" hreflang="vi" data-title="Hình thang" 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/Trapezium" title="Trapezium – West Flemish" lang="vls" hreflang="vls" data-title="Trapezium" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Trapesoyd" title="Trapesoyd – Waray" lang="war" 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<div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> 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.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;">Trapezoid (AmE)<br />Trapezium (BrE)</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Trapezoid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Trapezoid.svg/220px-Trapezoid.svg.png" decoding="async" width="220" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Trapezoid.svg/330px-Trapezoid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Trapezoid.svg/440px-Trapezoid.svg.png 2x" data-file-width="292" data-file-height="172" /></a></span><div class="infobox-caption">Trapezoid or trapezium</div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</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/Area" title="Area">Area</a></th><td class="infobox-data"><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 {\tfrac {a+b}{2}}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a+b}{2}}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff4c5cc5c122812856900d504c35d645e06b485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.029ex; height:3.676ex;" alt="{\displaystyle {\tfrac {a+b}{2}}h}"></span></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></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/trapezoid" class="extiw" title="wiktionary:trapezoid"> trapezoid</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>trapezoid</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'t' in 'tie'">t</span><span title="'r' in 'rye'">r</span><span title="/æ/: 'a' in 'bad'">æ</span><span title="'p' in 'pie'">p</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'z' in 'zoom'">z</span><span title="/ɔɪ/: 'oi' in 'choice'">ɔɪ</span><span title="'d' in 'dye'">d</span></span>/</a></span></span>) in <a href="/wiki/North_American_English" title="North American English">North American English</a>, or <b>trapezium</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="'t' in 'tie'">t</span><span title="'r' in 'rye'">r</span><span title="/ə/: 'a' in 'about'">ə</span><span title="/ˈ/: primary stress follows">ˈ</span><span title="'p' in 'pie'">p</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="'z' in 'zoom'">z</span><span title="/i/: 'y' in 'happy'">i</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'m' in 'my'">m</span></span>/</a></span></span>) in <a href="/wiki/British_English" title="British English">British English</a>,<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><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> is a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> that has one pair of <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> sides. </p><p>The parallel sides are called the <i>bases</i> of the trapezoid. The other two sides are called the <i>legs</i> (or the <i>lateral sides</i>) if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases. A <i>scalene trapezoid</i> is a trapezoid with no sides of <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equal</a> measure,<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> in contrast with the <a href="#Special_cases">special cases</a> below. </p><p>A trapezoid is usually considered to be a <a href="/wiki/Convex_polygon" title="Convex polygon">convex</a> quadrilateral in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, but there are also <a href="/wiki/Crossed_polygon" title="Crossed polygon">crossed</a> cases. If <i>ABCD</i> is a convex trapezoid, then <i>ABDC</i> is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology_and_trapezium_versus_trapezoid">Etymology and <i>trapezium</i> versus <i>trapezoid</i></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=1" title="Edit section: Etymology and trapezium versus trapezoid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trapezium_and_Trapezoid,_Hutton%E2%80%99s_mistake_in_1795.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Trapezium_and_Trapezoid%2C_Hutton%E2%80%99s_mistake_in_1795.png/220px-Trapezium_and_Trapezoid%2C_Hutton%E2%80%99s_mistake_in_1795.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Trapezium_and_Trapezoid%2C_Hutton%E2%80%99s_mistake_in_1795.png/330px-Trapezium_and_Trapezoid%2C_Hutton%E2%80%99s_mistake_in_1795.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Trapezium_and_Trapezoid%2C_Hutton%E2%80%99s_mistake_in_1795.png/440px-Trapezium_and_Trapezoid%2C_Hutton%E2%80%99s_mistake_in_1795.png 2x" data-file-width="1220" data-file-height="436" /></a><figcaption>Hutton's definitions in 1795<sup id="cite_ref-oed_4-0" class="reference"><a href="#cite_note-oed-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The ancient Greek mathematician <a href="/wiki/Euclid" title="Euclid">Euclid</a> defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια (<i>trapezia</i><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> literally 'table', itself from τετράς (<i>tetrás</i>) 'four' + πέζα (<i>péza</i>) 'foot; end, border, edge').<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Two types of <i>trapezia</i> were introduced by <a href="/wiki/Proclus" title="Proclus">Proclus</a> (AD 412 to 485) in his commentary on the first book of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>:<sup id="cite_ref-oed_4-1" class="reference"><a href="#cite_note-oed-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ConwayBurgiel2016_7-0" class="reference"><a href="#cite_note-ConwayBurgiel2016-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <ul><li>one pair of parallel sides – a <i>trapezium</i> (τραπέζιον), divided into isosceles (equal legs) and scalene (unequal) trapezia</li> <li>no parallel sides – <i>trapezoid</i> (τραπεζοειδή, <i>trapezoeidé</i>, literally 'trapezium-like' (<a href="https://en.wiktionary.org/wiki/%CE%B5%E1%BC%B6%CE%B4%CE%BF%CF%82" class="extiw" title="wikt:εἶδος">εἶδος</a> means 'resembles'), in the same way as <i><a href="/wiki/Cuboid" title="Cuboid">cuboid</a></i> means '<a href="/wiki/Cube" title="Cube">cube</a>-like' and <i><a href="/wiki/Rhomboid" title="Rhomboid">rhomboid</a></i> means '<a href="/wiki/Rhombus" title="Rhombus">rhombus</a>-like')</li></ul> <p>All European languages follow Proclus's structure<sup id="cite_ref-ConwayBurgiel2016_7-1" class="reference"><a href="#cite_note-ConwayBurgiel2016-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> as did English until the late 18th century, until an influential mathematical dictionary published by <a href="/wiki/Charles_Hutton" title="Charles Hutton">Charles Hutton</a> in 1795 supported without explanation a transposition of the terms. This was reversed in British English in about 1875, but it has been retained in American English to the present.<sup id="cite_ref-oed_4-2" class="reference"><a href="#cite_note-oed-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The following table compares usages, with the most specific definitions at the top to the most general at the bottom. </p> <table class="wikitable"> <tbody><tr> <th>Type </th> <th>Sets of parallel sides</th> <th>Image</th> <th colspan="3">Original terminology</th> <th colspan="2">Modern terminology </th></tr> <tr> <td> </td> <td> </td> <td> </td> <td><b>Euclid</b> (Definition 22) </td> <td><b>Proclus</b> (Definitions 30–34, quoting Posidonius) </td> <td><b>Euclid / Proclus definition</b> </td> <td><b>British English</b> </td> <td><b>American English</b> </td></tr> <tr> <td rowspan="2"><b>Parallelogram</b> </td> <td rowspan="2"><b>2</b></td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Rhombus_2_(PSF).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Rhombus_2_%28PSF%29.png/100px-Rhombus_2_%28PSF%29.png" decoding="async" width="100" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Rhombus_2_%28PSF%29.png/150px-Rhombus_2_%28PSF%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Rhombus_2_%28PSF%29.png/200px-Rhombus_2_%28PSF%29.png 2x" data-file-width="547" data-file-height="483" /></a></span></td> <td colspan="2">ῥόμβος (rhombos)</td> <td>equilateral but not right-angled</td> <td colspan="2">Rhombus/Parallelogram </td></tr> <tr> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Rhomboid_2_(PSF).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Rhomboid_2_%28PSF%29.png/100px-Rhomboid_2_%28PSF%29.png" decoding="async" width="100" height="52" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Rhomboid_2_%28PSF%29.png/150px-Rhomboid_2_%28PSF%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Rhomboid_2_%28PSF%29.png/200px-Rhomboid_2_%28PSF%29.png 2x" data-file-width="566" data-file-height="295" /></a></span></td> <td colspan="2">ῥομβοειδὲς (rhomboides)</td> <td>opposite sides and angles equal to one another but not equilateral nor right-angled</td> <td colspan="2">Rhomboid/Parallelogram </td></tr> <tr> <td rowspan="3"><b>Non-parallelogram</b> </td> <td rowspan="2"><b>1</b></td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Trapezoid_2_(PSF).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Trapezoid_2_%28PSF%29.png/100px-Trapezoid_2_%28PSF%29.png" decoding="async" width="100" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Trapezoid_2_%28PSF%29.png/150px-Trapezoid_2_%28PSF%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Trapezoid_2_%28PSF%29.png/200px-Trapezoid_2_%28PSF%29.png 2x" data-file-width="1279" data-file-height="426" /></a></span></td> <td rowspan="3">τραπέζια (trapezia)</td> <td>τραπέζιον ἰσοσκελὲς (<b>trapez<u>ion</u></b> isoskelés)</td> <td>Two parallel sides, and a line of symmetry</td> <td><b>Isosceles Trapez<u>ium</u></b> </td> <td><b>Isosceles Trapez<u>oid</u></b> </td></tr> <tr> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Trapezoid_3_(PSF).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Trapezoid_3_%28PSF%29.png/100px-Trapezoid_3_%28PSF%29.png" decoding="async" width="100" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Trapezoid_3_%28PSF%29.png/150px-Trapezoid_3_%28PSF%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Trapezoid_3_%28PSF%29.png/200px-Trapezoid_3_%28PSF%29.png 2x" data-file-width="564" data-file-height="621" /></a></span></td> <td>τραπέζιον σκαληνὸν (<b>trapez<u>ion</u></b> skalinón)</td> <td>Two parallel sides, and no line of symmetry </td> <td><b>Trapez<u>ium</u></b> </td> <td><b>Trapez<u>oid</u></b> </td></tr> <tr> <td><b>0</b> </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Trapezium_(PSF).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Trapezium_%28PSF%29.png/100px-Trapezium_%28PSF%29.png" decoding="async" width="100" height="57" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Trapezium_%28PSF%29.png/150px-Trapezium_%28PSF%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Trapezium_%28PSF%29.png/200px-Trapezium_%28PSF%29.png 2x" data-file-width="1151" data-file-height="657" /></a></span></td> <td>τραπέζοειδὲς (<b>trapez<u>oides</u></b>)</td> <td>No parallel sides</td> <td>Irregular quadrilateral/<b>Trapez<u>oid</u> <sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></b> </td> <td><b>Trapez<u>ium</u></b> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Inclusive_versus_exclusive_definition">Inclusive versus exclusive definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=2" title="Edit section: Inclusive versus exclusive definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is some disagreement whether <a href="/wiki/Parallelogram" title="Parallelogram">parallelograms</a>, which have two pairs of parallel sides, should be regarded as trapezoids. </p><p>Some define a trapezoid as a quadrilateral having <i>only</i> one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Some sources use the term <i>proper trapezoid</i> to describe trapezoids under the exclusive definition, analogous to uses of the word <i>proper</i> in some other mathematical objects.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Others<sup id="cite_ref-Mathworld_13-0" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="Eric Weisstein says nothing about trapezoids having at least one pair of parallel sides (April 2023)">failed verification</span></a></i>]</sup> define a trapezoid as a quadrilateral with <i>at least</i> one pair of parallel sides (the inclusive definition<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as <a href="/wiki/Calculus" title="Calculus">calculus</a>. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the <a href="/wiki/Quadrilateral#Taxonomy" title="Quadrilateral">taxonomy of quadrilaterals</a>. </p><p>Under the inclusive definition, all parallelograms (including <a href="/wiki/Rhombus" title="Rhombus">rhombuses</a>, <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">squares</a> and non-square <a href="/wiki/Rectangle" title="Rectangle">rectangles</a>) are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. </p> <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=Trapezoid&action=edit&section=3" 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:Trapezoid_special_cases.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Trapezoid_special_cases.png/280px-Trapezoid_special_cases.png" decoding="async" width="280" height="242" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Trapezoid_special_cases.png/420px-Trapezoid_special_cases.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Trapezoid_special_cases.png/560px-Trapezoid_special_cases.png 2x" data-file-width="849" data-file-height="734" /></a><figcaption>Trapezoid special cases. The orange figures also qualify as parallelograms.</figcaption></figure> <p>A <b>right trapezoid</b> (also called <i>right-angled trapezoid</i>) has two adjacent <a href="/wiki/Right_angles" class="mw-redirect" title="Right angles">right angles</a>.<sup id="cite_ref-Mathworld_13-1" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Right trapezoids are used in the <a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">trapezoidal rule</a> for estimating areas under a curve. </p><p>An <b>acute trapezoid</b> has two adjacent acute angles on its longer <i>base</i> edge. </p><p>An <b>obtuse trapezoid</b> on the other hand has one acute and one obtuse angle on each <i>base</i>. </p><p>An <b><a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">isosceles trapezoid</a></b> is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a>. This is possible for acute trapezoids or right trapezoids (as rectangles). </p><p>A <b><a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a></b> is (under the inclusive definition) a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> (or <a href="/wiki/Point_reflection" title="Point reflection">point reflection</a> symmetry). It is possible for obtuse trapezoids or right trapezoids (rectangles). </p><p>A <b><a href="/wiki/Tangential_trapezoid" title="Tangential trapezoid">tangential trapezoid</a></b> is a trapezoid that has an <a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">incircle</a>. </p><p>A <a href="/wiki/Saccheri_quadrilateral" title="Saccheri quadrilateral">Saccheri quadrilateral</a> is similar to a trapezoid in the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic</a> plane, with two adjacent right angles, while it is a rectangle in the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>. A <a href="/wiki/Lambert_quadrilateral" title="Lambert quadrilateral">Lambert quadrilateral</a> in the hyperbolic plane has 3 right angles. </p> <div class="mw-heading mw-heading2"><h2 id="Condition_of_existence">Condition of existence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=4" title="Edit section: Condition of existence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Four lengths <i>a</i>, <i>c</i>, <i>b</i>, <i>d</i> can constitute the consecutive sides of a non-parallelogram trapezoid with <i>a</i> and <i>b</i> parallel only when<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<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 \displaystyle |d-c|<|b-a|<d+c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>d</mi> <mo>+</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle |d-c|<|b-a|<d+c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c79086b3eb011e748d4e61d9effa16a3813d46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.625ex; height:2.843ex;" alt="{\displaystyle \displaystyle |d-c|<|b-a|<d+c.}"></span></dd></dl> <p>The quadrilateral is a parallelogram when <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-c=b-a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>−</mo> <mi>c</mi> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d-c=b-a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3b7dc4a3b62b946eb25be0fd20a2e14aee2b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.49ex; height:2.343ex;" alt="{\displaystyle d-c=b-a=0}"></span>, but it is an <a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">ex-tangential quadrilateral</a> (which is not a trapezoid) when <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-c|=|b-a|\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |d-c|=|b-a|\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/375eb698787fafbe97082b7bfe0bd473c1e5f86b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.078ex; height:2.843ex;" alt="{\displaystyle |d-c|=|b-a|\neq 0}"></span>.<sup id="cite_ref-Josefsson_16-0" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 35">: p. 35 </span></sup> </p> <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=Trapezoid&action=edit&section=5" title="Edit section: Characterizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trapez_mittellinie_en_labels.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Trapez_mittellinie_en_labels.svg/220px-Trapez_mittellinie_en_labels.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Trapez_mittellinie_en_labels.svg/330px-Trapez_mittellinie_en_labels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Trapez_mittellinie_en_labels.svg/440px-Trapez_mittellinie_en_labels.svg.png 2x" data-file-width="461" data-file-height="457" /></a><figcaption><b>general trapezoid/trapezium:</b><br />parallel sides: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,\,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,\,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12ceaff955e1a98e48db7fa8fa43121630fa209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.648ex; height:2.509ex;" alt="{\displaystyle a,\,b}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span> <br />legs: <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,\,d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c,\,d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a52598dce2dc7796bed7fbebbe1f0dba75e65de7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.644ex; height:2.509ex;" alt="{\displaystyle c,\,d}"></span><br />diagonals: <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 q,\,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q,\,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/255872e3f28f3239dd66aad58e9d3be23402c038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.66ex; height:2.009ex;" alt="{\displaystyle q,\,p}"></span><br /> midsegment: <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span><br /> height/altitude: <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trapez_mittellinie_en_labels_areas.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Trapez_mittellinie_en_labels_areas.svg/220px-Trapez_mittellinie_en_labels_areas.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Trapez_mittellinie_en_labels_areas.svg/330px-Trapez_mittellinie_en_labels_areas.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Trapez_mittellinie_en_labels_areas.svg/440px-Trapez_mittellinie_en_labels_areas.svg.png 2x" data-file-width="461" data-file-height="457" /></a><figcaption>trapezoid/trapezium with opposing triangles <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,\,T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S,\,T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94489f0f78e18acf328476a2b51b48d91ccfd668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.557ex; height:2.509ex;" alt="{\displaystyle S,\,T}"></span> formed by the diagonals</figcaption></figure> <p>Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid: </p> <ul><li>It has two adjacent <a href="/wiki/Angle" title="Angle">angles</a> that are <a href="/wiki/Supplementary_angles" class="mw-redirect" title="Supplementary angles">supplementary</a>, that is, they add up to 180 <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>.</li> <li>The angle between a side and a <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> is equal to the angle between the opposite side and the same diagonal.</li> <li>The diagonals cut each other in mutually the same <a href="/wiki/Ratio" title="Ratio">ratio</a> (this ratio is the same as that between the lengths of the parallel sides).</li> <li>The diagonals cut the quadrilateral into four <a href="/wiki/Triangle" title="Triangle">triangles</a> of which one opposite pair have equal areas.<sup id="cite_ref-Josefsson_16-1" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Prop.5">: Prop.5 </span></sup></li> <li>The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.<sup id="cite_ref-Josefsson_16-2" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Thm.6">: Thm.6 </span></sup></li> <li>The areas <i>S</i> and <i>T</i> of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation</li></ul> <dl><dd><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 {\sqrt {K}}={\sqrt {S}}+{\sqrt {T}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>K</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>S</mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>T</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {K}}={\sqrt {S}}+{\sqrt {T}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0367ca89df3e78cb408b3237eaa88ceeb2b92c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.595ex; height:3.009ex;" alt="{\displaystyle {\sqrt {K}}={\sqrt {S}}+{\sqrt {T}},}"></span></dd></dl></dd></dl> <dl><dd>where <i>K</i> is the area of the quadrilateral.<sup id="cite_ref-Josefsson_16-3" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Thm.8">: Thm.8 </span></sup></dd></dl> <ul><li>The midpoints of two opposite sides of the trapezoid and the intersection of the diagonals are <a href="/wiki/Collinear" class="mw-redirect" title="Collinear">collinear</a>.<sup id="cite_ref-Josefsson_16-4" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Thm.15">: Thm.15 </span></sup></li> <li>The angles in the quadrilateral <i>ABCD</i> satisfy <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 A\sin C=\sin B\sin D.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mi>C</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin A\sin C=\sin B\sin D.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b40ec5bdb936df1b37368d8cb1c8f88afc96a52c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.688ex; height:2.176ex;" alt="{\displaystyle \sin A\sin C=\sin B\sin D.}"></span><sup id="cite_ref-Josefsson_16-5" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 25">: p. 25 </span></sup></li> <li>The cosines of two adjacent angles <a href="/wiki/Addition" title="Addition">sum</a> to 0, as do the cosines of the other two angles.<sup id="cite_ref-Josefsson_16-6" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 25">: p. 25 </span></sup></li> <li>The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.<sup id="cite_ref-Josefsson_16-7" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 26">: p. 26 </span></sup></li> <li>One bimedian divides the quadrilateral into two quadrilaterals of equal areas.<sup id="cite_ref-Josefsson_16-8" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 26">: p. 26 </span></sup></li> <li>Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.<sup id="cite_ref-Josefsson_16-9" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 31">: p. 31 </span></sup></li></ul> <p>Additionally, the following properties are equivalent, and each implies that opposite sides <i>a</i> and <i>b</i> are parallel: </p> <ul><li>The consecutive sides <i>a</i>, <i>c</i>, <i>b</i>, <i>d</i> and the diagonals <i>p</i>, <i>q</i> satisfy the equation<sup id="cite_ref-Josefsson_16-10" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Cor.11">: Cor.11 </span></sup></li></ul> <dl><dd><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^{2}+q^{2}=c^{2}+d^{2}+2ab.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</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> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}+q^{2}=c^{2}+d^{2}+2ab.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07da0d022eac82de43d4baf8e198f010eccb8c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:24.436ex; height:3.009ex;" alt="{\displaystyle p^{2}+q^{2}=c^{2}+d^{2}+2ab.}"></span></dd></dl></dd></dl> <ul><li>The distance <i>v</i> between the midpoints of the diagonals satisfies the equation<sup id="cite_ref-Josefsson_16-11" class="reference"><a href="#cite_note-Josefsson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Thm.12">: Thm.12 </span></sup></li></ul> <dl><dd><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 v={\frac {|a-b|}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {|a-b|}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150350d92c4eb1c3fbae3b6d0cf248d8a413d19a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.07ex; height:5.676ex;" alt="{\displaystyle v={\frac {|a-b|}{2}}.}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Midsegment_and_height">Midsegment and height</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=6" title="Edit section: Midsegment and height"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>midsegment</i> of a trapezoid is the segment that joins the <a href="/wiki/Midpoint" title="Midpoint">midpoints</a> of the legs. It is parallel to the bases. Its length <i>m</i> is equal to the average of the lengths of the bases <i>a</i> and <i>b</i> of the trapezoid,<sup id="cite_ref-Mathworld_13-2" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<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 m={\frac {a+b}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {a+b}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd41c4bddaa33acbbf99add0b0cf94cb317bd5af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.69ex; height:5.343ex;" alt="{\displaystyle m={\frac {a+b}{2}}.}"></span></dd></dl> <p>The midsegment of a trapezoid is one of the two <a href="/wiki/Quadrilateral#Special_line_segments" title="Quadrilateral">bimedians</a> (the other bimedian divides the trapezoid into equal areas). </p><p>The <i>height</i> (or altitude) is the <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> distance between the bases. In the case that the two bases have different lengths (<i>a</i> ≠ <i>b</i>), the height of a trapezoid <i>h</i> can be determined by the length of its four sides using the formula<sup id="cite_ref-Mathworld_13-3" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<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 h={\frac {\sqrt {(p-a)(p-b)(p-b-d)(p-b-c)}}{2|b-a|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </msqrt> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\frac {\sqrt {(p-a)(p-b)(p-b-d)(p-b-c)}}{2|b-a|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5a83d3a9447016fa949cb298b529cb95e702ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43ex; height:7.009ex;" alt="{\displaystyle h={\frac {\sqrt {(p-a)(p-b)(p-b-d)(p-b-c)}}{2|b-a|}}}"></span></dd></dl> <p>where <i>c</i> and <i>d</i> are the lengths of the legs and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=a+b+c+d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=a+b+c+d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a65171a2c73ba67252f46b1896eb312f8a5176cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.329ex; height:2.509ex;" alt="{\displaystyle p=a+b+c+d}"></span>. </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=Trapezoid&action=edit&section=7" title="Edit section: Area"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The area <i>K</i> of a trapezoid is given by<sup id="cite_ref-Mathworld_13-4" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<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 K={\frac {a+b}{2}}\cdot h=mh}"> <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>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>h</mi> <mo>=</mo> <mi>m</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {a+b}{2}}\cdot h=mh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4eaf3ce47528c8a56c893d5aa6d9b02517006c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.564ex; height:5.343ex;" alt="{\displaystyle K={\frac {a+b}{2}}\cdot h=mh}"></span></dd></dl> <p>where <i>a</i> and <i>b</i> are the lengths of the parallel sides, <i>h</i> is the height (the perpendicular distance between these sides), and <i>m</i> is the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> of the lengths of the two parallel sides. In 499 AD <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>, a great <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>-<a href="/wiki/Astronomer" title="Astronomer">astronomer</a> from the classical age of <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a> and <a href="/wiki/Indian_astronomy" title="Indian astronomy">Indian astronomy</a>, used this method in the <i><a href="/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a></i> (section 2.8). This yields as a <a href="/wiki/Special_case" title="Special case">special case</a> the well-known formula for the area of a <a href="/wiki/Triangle" title="Triangle">triangle</a>, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point. </p><p>The 7th-century Indian mathematician <a href="/wiki/Bh%C4%81skara_I" title="Bhāskara I">Bhāskara I</a> derived the following formula for the area of a trapezoid with consecutive sides <i>a</i>, <i>c</i>, <i>b</i>, <i>d</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {1}{2}}(a+b){\sqrt {c^{2}-{\frac {1}{4}}\left((b-a)+{\frac {c^{2}-d^{2}}{b-a}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {1}{2}}(a+b){\sqrt {c^{2}-{\frac {1}{4}}\left((b-a)+{\frac {c^{2}-d^{2}}{b-a}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56bb0374008c35784adbdefd5f24f0fca81f3c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.467ex; height:7.676ex;" alt="{\displaystyle K={\frac {1}{2}}(a+b){\sqrt {c^{2}-{\frac {1}{4}}\left((b-a)+{\frac {c^{2}-d^{2}}{b-a}}\right)^{2}}}}"></span></dd></dl> <p>where <i>a</i> and <i>b</i> are parallel and <i>b</i> > <i>a</i>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This formula can be factored into a more symmetric version<sup id="cite_ref-Mathworld_13-5" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<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 K={\frac {a+b}{4|b-a|}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}.}"> <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> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {a+b}{4|b-a|}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3862ad1a5eb04c67e7dd2ae67fc95fcec4b5a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:77.425ex; height:6.176ex;" alt="{\displaystyle K={\frac {a+b}{4|b-a|}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}.}"></span></dd></dl> <p>When one of the parallel sides has shrunk to a point (say <i>a</i> = 0), this formula reduces to <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> for the area of a triangle. </p><p>Another equivalent formula for the area, which more closely resembles Heron's formula, is<sup id="cite_ref-Mathworld_13-6" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<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 K={\frac {a+b}{|b-a|}}{\sqrt {(s-b)(s-a)(s-b-c)(s-b-d)}}}"> <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> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {a+b}{|b-a|}}{\sqrt {(s-b)(s-a)(s-b-c)(s-b-d)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105d7291d81a6f80e06b0b70354d3dcd07ec45f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.772ex; height:6.176ex;" alt="{\displaystyle K={\frac {a+b}{|b-a|}}{\sqrt {(s-b)(s-a)(s-b-c)(s-b-d)}}}"></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+c+d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\tfrac {1}{2}}(a+b+c+d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801b31ad5295e801b48d56fc44c4c899e7dacff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.628ex; height:3.509ex;" alt="{\displaystyle s={\tfrac {1}{2}}(a+b+c+d)}"></span> is the <a href="/wiki/Semiperimeter" title="Semiperimeter">semiperimeter</a> of the trapezoid. (This formula is similar to <a href="/wiki/Brahmagupta%27s_formula" title="Brahmagupta's formula">Brahmagupta's formula</a>, but it differs from it, in that a trapezoid might not be <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic</a> (inscribed in a circle). The formula is also a special case of <a href="/wiki/Bretschneider%27s_formula" title="Bretschneider's formula">Bretschneider's formula</a> for a general <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a>). </p><p>From Bretschneider's formula, it follows that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\sqrt {{\frac {(ab^{2}-a^{2}b-ad^{2}+bc^{2})(ab^{2}-a^{2}b-ac^{2}+bd^{2})}{4(b-a)^{2}}}-\left({\frac {c^{2}+d^{2}-a^{2}-b^{2}}{4}}\right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <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> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <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> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</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> </mrow> <mn>4</mn> </mfrac> </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 K={\sqrt {{\frac {(ab^{2}-a^{2}b-ad^{2}+bc^{2})(ab^{2}-a^{2}b-ac^{2}+bd^{2})}{4(b-a)^{2}}}-\left({\frac {c^{2}+d^{2}-a^{2}-b^{2}}{4}}\right)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1066a446784c1ba7fd04ff722751820dd1eac0e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:81.222ex; height:7.509ex;" alt="{\displaystyle K={\sqrt {{\frac {(ab^{2}-a^{2}b-ad^{2}+bc^{2})(ab^{2}-a^{2}b-ac^{2}+bd^{2})}{4(b-a)^{2}}}-\left({\frac {c^{2}+d^{2}-a^{2}-b^{2}}{4}}\right)^{2}}}.}"></span></dd></dl> <p>The <a href="/wiki/Bimedian" class="mw-redirect" title="Bimedian">bimedian</a> connecting the parallel sides bisects the area. </p> <div class="mw-heading mw-heading2"><h2 id="Diagonals">Diagonals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=8" title="Edit section: Diagonals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Trapezium.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Trapezium.svg/200px-Trapezium.svg.png" decoding="async" width="200" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Trapezium.svg/300px-Trapezium.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Trapezium.svg/400px-Trapezium.svg.png 2x" data-file-width="770" data-file-height="470" /></a><figcaption></figcaption></figure> <p>The lengths of the diagonals are<sup id="cite_ref-Mathworld_13-7" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<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 p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>a</mi> <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> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f046deab8a64b3ec066337318ed70f074021bfdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:29.809ex; height:7.509ex;" alt="{\displaystyle p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},}"></span></dd> <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 q={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>a</mi> <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> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a743276ec66764453665963e004d2385a90fa16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.973ex; height:7.509ex;" alt="{\displaystyle q={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}}}"></span></dd></dl> <p>where <i>a</i> is the short base, <i>b</i> is the long base, and <i>c</i> and <i>d</i> are the trapezoid legs. </p><p>If the trapezoid is divided into four triangles by its diagonals <i>AC</i> and <i>BD</i> (as shown on the right), intersecting at <i>O</i>, then the area of <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> <i>AOD</i></span> is equal to that of <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> <i>BOC</i></span>, and the product of the areas of <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> <i>AOD</i></span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> <i>BOC</i></span> is equal to that of <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> <i>AOB</i></span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> <i>COD</i></span>. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.<sup id="cite_ref-Mathworld_13-8" class="reference"><a href="#cite_note-Mathworld-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Let the trapezoid have vertices <i>A</i>, <i>B</i>, <i>C</i>, and <i>D</i> in sequence and have parallel sides <i>AB</i> and <i>DC</i>. Let <i>E</i> be the intersection of the diagonals, and let <i>F</i> be on side <i>DA</i> and <i>G</i> be on side <i>BC</i> such that <i>FEG</i> is parallel to <i>AB</i> and <i>CD</i>. Then <i>FG</i> is the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> of <i>AB</i> and <i>DC</i>:<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<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 {\frac {1}{FG}}={\frac {1}{2}}\left({\frac {1}{AB}}+{\frac {1}{DC}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>F</mi> <mi>G</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>D</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{FG}}={\frac {1}{2}}\left({\frac {1}{AB}}+{\frac {1}{DC}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffffe1ad7eb34faf819d099c38736c4f82182b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.053ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{FG}}={\frac {1}{2}}\left({\frac {1}{AB}}+{\frac {1}{DC}}\right).}"></span></dd></dl> <p>The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.<sup id="cite_ref-Byer_19-0" class="reference"><a href="#cite_note-Byer-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_properties">Other properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=9" title="Edit section: Other properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The center of area (center of mass for a uniform <a href="/wiki/Planar_lamina" title="Planar lamina">lamina</a>) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance <i>x</i> from the longer side <i>b</i> given by<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<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 x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mn>3</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727771a57ac805ad09107885a21951523a785c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.512ex; height:6.176ex;" alt="{\displaystyle x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).}"></span></dd></dl> <p>The center of area divides this segment in the ratio (when taken from the short to the long side)<sup id="cite_ref-AM_21-0" class="reference"><a href="#cite_note-AM-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 862">: p. 862 </span></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 {\frac {a+2b}{2a+b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>b</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+2b}{2a+b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054b8620ee51fd6797c2c89ca5a7d5600033054d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.713ex; height:5.676ex;" alt="{\displaystyle {\frac {a+2b}{2a+b}}.}"></span></dd></dl> <p>If the angle bisectors to angles <i>A</i> and <i>B</i> intersect at <i>P</i>, and the angle bisectors to angles <i>C</i> and <i>D</i> intersect at <i>Q</i>, then<sup id="cite_ref-Byer_19-1" class="reference"><a href="#cite_note-Byer-19"><span class="cite-bracket">[</span>19<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 PQ={\frac {|AD+BC-AB-CD|}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>D</mi> <mo>+</mo> <mi>B</mi> <mi>C</mi> <mo>−</mo> <mi>A</mi> <mi>B</mi> <mo>−</mo> <mi>C</mi> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PQ={\frac {|AD+BC-AB-CD|}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7768db6f0b5b8e7671b45d5fbb903921c1fa7553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.375ex; height:5.676ex;" alt="{\displaystyle PQ={\frac {|AD+BC-AB-CD|}{2}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=10" title="Edit section: Applications"><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:Temple_of_Dendur-_night.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Temple_of_Dendur-_night.jpg/250px-Temple_of_Dendur-_night.jpg" decoding="async" width="250" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Temple_of_Dendur-_night.jpg/375px-Temple_of_Dendur-_night.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Temple_of_Dendur-_night.jpg/500px-Temple_of_Dendur-_night.jpg 2x" data-file-width="2206" data-file-height="1618" /></a><figcaption>The <a href="/wiki/Temple_of_Dendur" title="Temple of Dendur">Temple of Dendur</a> in the <a href="/wiki/Metropolitan_Museum_of_Art" title="Metropolitan Museum of Art">Metropolitan Museum of Art</a> in <a href="/wiki/New_York_City" title="New York City">New York City</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Architecture">Architecture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=11" title="Edit section: Architecture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually <a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">isosceles trapezoids</a>. This was the standard style for the doors and windows of the <a href="/wiki/Inca_architecture" title="Inca architecture">Inca</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Geometry">Geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=12" title="Edit section: Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Crossed_ladders_problem" title="Crossed ladders problem">crossed ladders problem</a> is the problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection. </p> <div class="mw-heading mw-heading3"><h3 id="Biology">Biology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=13" title="Edit section: Biology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Juanita_Vilas_Marchant_Stenocephalidae_Heteroptera_HemipteraP.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Juanita_Vilas_Marchant_Stenocephalidae_Heteroptera_HemipteraP.jpg/220px-Juanita_Vilas_Marchant_Stenocephalidae_Heteroptera_HemipteraP.jpg" decoding="async" width="220" height="313" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Juanita_Vilas_Marchant_Stenocephalidae_Heteroptera_HemipteraP.jpg/330px-Juanita_Vilas_Marchant_Stenocephalidae_Heteroptera_HemipteraP.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Juanita_Vilas_Marchant_Stenocephalidae_Heteroptera_HemipteraP.jpg/440px-Juanita_Vilas_Marchant_Stenocephalidae_Heteroptera_HemipteraP.jpg 2x" data-file-width="674" data-file-height="960" /></a><figcaption>Example of a trapeziform <a href="/wiki/Prothorax" title="Prothorax">pronotum</a> outlined on a <a href="/wiki/Stenocephalidae" class="mw-redirect" title="Stenocephalidae">spurge bug</a></figcaption></figure> <p>In <a href="/wiki/Morphology_(biology)" title="Morphology (biology)">morphology</a>, <a href="/wiki/Taxonomy_(biology)" title="Taxonomy (biology)">taxonomy</a> and other descriptive disciplines in which a term for such shapes is necessary, terms such as <i>trapezoidal</i> or <i>trapeziform</i> commonly are useful in descriptions of particular organs or forms.<sup id="cite_ref-Capinera2008_23-0" class="reference"><a href="#cite_note-Capinera2008-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Computer_engineering">Computer engineering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=14" title="Edit section: Computer engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In computer engineering, specifically digital logic and computer architecture, trapezoids are typically utilized to symbolize <a href="/wiki/Multiplexer" title="Multiplexer">multiplexors</a>. Multiplexors are logic elements that select between multiple elements and produce a single output based on a select signal. Typical designs will employ trapezoids without specifically stating they are multiplexors as they are universally equivalent. </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=Trapezoid&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Frustum" title="Frustum">Frustum</a>, a solid having trapezoidal faces</li> <li><a href="/wiki/Polite_number" title="Polite number">Polite number</a>, also known as a trapezoidal number</li> <li><a href="/wiki/Wedge_(geometry)" title="Wedge (geometry)">Wedge</a>, a polyhedron defined by two triangles and three trapezoid faces.</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=Trapezoid&action=edit&section=16" 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 mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="https://www.mathopenref.com/trapezoid.html">"Trapezoid – math word definition – Math Open Reference"</a>. <i>www.mathopenref.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-05-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathopenref.com&rft.atitle=Trapezoid+%E2%80%93+math+word+definition+%E2%80%93+Math+Open+Reference&rft_id=https%3A%2F%2Fwww.mathopenref.com%2Ftrapezoid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" 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">A. D. Gardiner & C. J. Bradley, <i>Plane Euclidean Geometry: Theory and Problems</i>, UKMT, 2005, p. 34.</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.basic-mathematics.com/types-of-quadrilaterals.html">"Types of Quadrilaterals"</a>. <i>Basic-mathematics.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Basic-mathematics.com&rft.atitle=Types+of+Quadrilaterals&rft_id=https%3A%2F%2Fwww.basic-mathematics.com%2Ftypes-of-quadrilaterals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-oed-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-oed_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-oed_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-oed_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_A._H._Murray1926" class="citation book cs1">James A. H. Murray (1926). <a rel="nofollow" class="external text" href="https://archive.org/details/oedxaarch/page/n296/mode/1up"><i>A New English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society</i></a>. Vol. X. Clarendon Press at Oxford. p. 286 (Trapezium). <q>With Euclid (c 300 B.C.) τραπέζιον included all quadrilateral figures except the square, rectangle, rhombus, and rhomboid; into the varieties of trapezia he did not enter. But Proclus, who wrote Commentaries on the First Book of Euclid's Elements A.D. 450, retained the name τραπέζιον only for quadrilaterals having two sides parallel, subdividing these into the τραπέζιον ἰσοσκελὲς, isosceles trapezium, having the two non-parallel sides (and the angles at their bases) equal, and σκαληνὸν τραπέζιον, scalene trapezium, in which these sides and angles are unequal. For quadrilaterals having no sides parallel, Proclus introduced the name τραπέζοειδὲς TRAPEZOID. This nomenclature is retained in all the continental languages, and was universal in England till late in the 18th century, when the application of the terms was transposed, so that the figure which Proclus and modern geometers of other nations call specifically a trapezium (F. trapèze, Ger. trapez, Du. trapezium, It. trapezio) became with most English writers a trapezoid, and the trapezoid of Proclus and other nations a trapezium. This changed sense of trapezoid is given in Hutton's Mathematical Dictionary, 1795, as 'sometimes' used – he does not say by whom; but he himself unfortunately adopted and used it, and his Dictionary was doubtless the chief agent in its diffusion. Some geometers however continued to use the terms in their original senses, and since c 1875 this is the prevalent use.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+New+English+Dictionary+on+Historical+Principles%3A+Founded+Mainly+on+the+Materials+Collected+by+the+Philological+Society&rft.pages=286+%28Trapezium%29&rft.pub=Clarendon+Press+at+Oxford&rft.date=1926&rft.au=James+A.+H.+Murray&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foedxaarch%2Fpage%2Fn296%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.perseus.tufts.edu/hopper/text?doc=urn:cts:greekLit:tlg1799.tlg001.perseus-grc1:1.def.22">"Euclid, Elements, book 1, type Def, number 22"</a>. <i>www.perseus.tufts.edu</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.perseus.tufts.edu&rft.atitle=Euclid%2C+Elements%2C+book+1%2C+type+Def%2C+number+22&rft_id=http%3A%2F%2Fwww.perseus.tufts.edu%2Fhopper%2Ftext%3Fdoc%3Durn%3Acts%3AgreekLit%3Atlg1799.tlg001.perseus-grc1%3A1.def.22&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" 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">πέζα is said to be the Doric and Arcadic form of πούς 'foot', but recorded only in the sense 'instep [of a human foot]', whence the meaning 'edge, border'. τράπεζα 'table' is Homeric. Henry George Liddell, Robert Scott, Henry Stuart Jones, <i>A Greek-English Lexicon</i>, Oxford, Clarendon Press (1940), s.v. <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/morph?l=peza&la=greek#lexicon">πέζα</a>, <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dtra%2Fpeza">τράπεζα</a>.</span> </li> <li id="cite_note-ConwayBurgiel2016-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-ConwayBurgiel2016_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ConwayBurgiel2016_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConwayBurgielGoodman-Strauss2016" class="citation book cs1">Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (5 April 2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Drj1CwAAQBAJ&pg=PA286"><i>The Symmetries of Things</i></a>. CRC Press. p. 286. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4398-6489-0" title="Special:BookSources/978-1-4398-6489-0"><bdi>978-1-4398-6489-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Symmetries+of+Things&rft.pages=286&rft.pub=CRC+Press&rft.date=2016-04-05&rft.isbn=978-1-4398-6489-0&rft.aulast=Conway&rft.aufirst=John+H.&rft.au=Burgiel%2C+Heidi&rft.au=Goodman-Strauss%2C+Chaim&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDrj1CwAAQBAJ%26pg%3DPA286&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">For example: French <i>trapèze</i>, Italian <i>trapezio</i>, Portuguese <i>trapézio</i>, Spanish <i>trapecio</i>, German <i>Trapez</i>, Ukrainian "трапеція", e.g. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.larousse.fr/dictionnaires/francais/trap%C3%A9zo%C3%AFde/79256">"Larousse definition for trapézoïde"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Larousse+definition+for+trap%C3%A9zo%C3%AFde&rft_id=http%3A%2F%2Fwww.larousse.fr%2Fdictionnaires%2Ffrancais%2Ftrap%25C3%25A9zo%25C3%25AFde%2F79256&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.chambersharrap.co.uk/chambers/features/chref/chref.py/main?xref=21C44644&title=21st&query=trapezoid">"chambersharrap.co.uk"</a>. <i>www.chambersharrap.co.uk</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.chambersharrap.co.uk&rft.atitle=chambersharrap.co.uk&rft_id=http%3A%2F%2Fwww.chambersharrap.co.uk%2Fchambers%2Ffeatures%2Fchref%2Fchref.py%2Fmain%3Fxref%3D21C44644%26title%3D21st%26query%3Dtrapezoid&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.merriam-webster.com/dictionary/trapezium">"1913 American definition of trapezium"</a>. <i>Merriam-Webster Online Dictionary</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2007-12-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Merriam-Webster+Online+Dictionary&rft.atitle=1913+American+definition+of+trapezium&rft_id=http%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Ftrapezium&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</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://www.math.com/school/glossary/defs/trapezoid.html">"American School definition from "math.com"<span class="cs1-kern-right"></span>"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2008-04-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=American+School+definition+from+%22math.com%22&rft_id=http%3A%2F%2Fwww.math.com%2Fschool%2Fglossary%2Fdefs%2Ftrapezoid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichon" class="citation web cs1">Michon, Gérard P. <a rel="nofollow" class="external text" href="http://www.numericana.com/answer/culture.htm">"History and Nomenclature"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-06-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=History+and+Nomenclature&rft.aulast=Michon&rft.aufirst=G%C3%A9rard+P.&rft_id=http%3A%2F%2Fwww.numericana.com%2Fanswer%2Fculture.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-Mathworld-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Mathworld_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Mathworld_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Mathworld_13-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Mathworld_13-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Mathworld_13-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Mathworld_13-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Mathworld_13-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Mathworld_13-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Mathworld_13-8"><sup><i><b>i</b></i></sup></a></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Trapezoid"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Trapezoid.html">"Trapezoid"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Trapezoid&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTrapezoid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Trapezoids, <a rel="nofollow" class="external autonumber" href="http://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html">[1]</a>. Retrieved 2012-02-24.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><i>Ask Dr. Math</i> (2008), <a rel="nofollow" class="external text" href="http://mathforum.org/library/drmath/view/72738.html">"Area of Trapezoid Given Only the Side Lengths"</a>.</span> </li> <li id="cite_note-Josefsson-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Josefsson_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Josefsson_16-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Josefsson_16-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Josefsson_16-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Josefsson_16-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Josefsson_16-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Josefsson_16-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Josefsson_16-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Josefsson_16-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-Josefsson_16-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-Josefsson_16-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-Josefsson_16-11"><sup><i><b>l</b></i></sup></a></span> <span class="reference-text">Martin Josefsson, <a rel="nofollow" class="external text" href="http://forumgeom.fau.edu/FG2013volume13/FG201305.pdf">"Characterizations of trapezoids"</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged November 2024">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup>, Forum Geometricorum, 13 (2013) 23–35.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">T. K. Puttaswamy, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=8oVRSu692qoC&q=Bhaskara">Mathematical achievements of pre-modern Indian mathematicians</a></i>, Elsevier, 2012, p. 156.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://gogeometry.com/problem/p747-trapezoid-diagonal-parallel-similarity-harmonic-mean-high-school-college.htm">"Math Education Geometry Problem 747: Trapezoid, Diagonals, Parallel, Bases, Midpoint, Similarity, Harmonic Mean. Level: High School, Honors Geometry, College, Mathematics Education. Distance learning"</a>. <i>gogeometry.com</i><span class="reference-accessdate">. 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Mnatsakanian (December 2004). <a rel="nofollow" class="external text" href="http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Apostol853-863.pdf">"Figures Circumscribing Circles"</a> <span class="cs1-format">(PDF)</span>. <i>American Mathematical Monthly</i>. <b>111</b> (10): 853–863. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F4145094">10.2307/4145094</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/4145094">4145094</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-04-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Figures+Circumscribing+Circles&rft.volume=111&rft.issue=10&rft.pages=853-863&rft.date=2004-12&rft_id=info%3Adoi%2F10.2307%2F4145094&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4145094%23id-name%3DJSTOR&rft.au=Tom+M.+Apostol+and+Mamikon+A.+Mnatsakanian&rft_id=http%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fimages%2Fupload_library%2F22%2FFord%2FApostol853-863.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://gogeometry.com/MachuPicchu.htm">"Machu Picchu Lost City of the Incas – Inca Geometry"</a>. <i>gogeometry.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-02-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=gogeometry.com&rft.atitle=Machu+Picchu+Lost+City+of+the+Incas+%E2%80%93+Inca+Geometry.&rft_id=http%3A%2F%2Fgogeometry.com%2FMachuPicchu.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> <li id="cite_note-Capinera2008-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Capinera2008_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_L._Capinera2008" class="citation book cs1">John L. Capinera (11 August 2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=i9ITMiiohVQC&pg=PA1247"><i>Encyclopedia of Entomology</i></a>. Springer Science & Business Media. pp. 386, 1062, 1247. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-6242-1" title="Special:BookSources/978-1-4020-6242-1"><bdi>978-1-4020-6242-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+Entomology&rft.pages=386%2C+1062%2C+1247&rft.pub=Springer+Science+%26+Business+Media&rft.date=2008-08-11&rft.isbn=978-1-4020-6242-1&rft.au=John+L.+Capinera&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Di9ITMiiohVQC%26pg%3DPA1247&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=17" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>D. Fraivert, A. Sigler and M. Stupel : <a rel="nofollow" class="external text" href="https://dx.doi.org/10.18642/jmsaa_7100121635"><i>Common properties of trapezoids and convex quadrilaterals</i></a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trapezoid&action=edit&section=18" 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="https://www.encyclopediaofmath.org/index.php/Trapezium">"Trapezium"</a> at the <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i></li> <li><span class="citation mathworld" id="Reference-Mathworld-Right_trapezoid"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RightTrapezoid.html">"Right trapezoid"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Right+trapezoid&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRightTrapezoid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrapezoid" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/trapezoid.html">Trapezoid definition</a>, <a rel="nofollow" class="external text" href="http://www.mathopenref.com/trapezoidarea.html">Area of a trapezoid</a>, <a rel="nofollow" class="external text" href="http://www.mathopenref.com/trapezoidmedian.html">Median of a trapezoid</a> (with interactive animations)</li> <li><a rel="nofollow" class="external text" href="http://www.elsy.at/kurse/index.php?kurs=Trapezoid+%28North+America%29&status=public">Trapezoid (North America)</a> at elsy.at: Animated course (construction, circumference, area)</li> <li><a rel="nofollow" class="external text" href="http://numericalmethods.eng.usf.edu/topics/trapezoidal_rule.html">Trapezoidal Rule</a> on <i>Numerical Methods for Stem Undergraduate</i></li> <li>Autar Kaw and E. Eric Kalu, <i><a rel="nofollow" class="external text" href="http://www.autarkaw.com/books/numericalmethods/index.html">Numerical Methods with Applications</a></i> (2008)</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 dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist 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</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 href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">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 class="mw-selflink selflink">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 src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a 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