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class="vector-toc-numb">2</span> <span>Characterizations</span> </div> </a> <ul id="toc-Characterizations-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">3</span> <span>Other properties</span> </div> </a> <ul id="toc-Other_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Area_formula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Area_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Area formula</span> </div> </a> <button aria-controls="toc-Area_formula-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 Area formula subsection</span> </button> <ul id="toc-Area_formula-sublist" class="vector-toc-list"> <li id="toc-From_vertex_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#From_vertex_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>From vertex coordinates</span> </div> </a> <ul id="toc-From_vertex_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proof_that_diagonals_bisect_each_other" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof_that_diagonals_bisect_each_other"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Proof that diagonals bisect each other</span> </div> </a> <ul id="toc-Proof_that_diagonals_bisect_each_other-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lattice_of_parallelograms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lattice_of_parallelograms"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Lattice of parallelograms</span> </div> </a> <ul id="toc-Lattice_of_parallelograms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parallelograms_arising_from_other_figures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Parallelograms_arising_from_other_figures"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Parallelograms arising from other figures</span> </div> </a> <button aria-controls="toc-Parallelograms_arising_from_other_figures-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 Parallelograms arising from other figures subsection</span> </button> <ul id="toc-Parallelograms_arising_from_other_figures-sublist" class="vector-toc-list"> <li id="toc-Automedian_triangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Automedian_triangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Automedian triangle</span> </div> </a> <ul id="toc-Automedian_triangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Varignon_parallelogram" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Varignon_parallelogram"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Varignon parallelogram</span> </div> </a> <ul id="toc-Varignon_parallelogram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangent_parallelogram_of_an_ellipse" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tangent_parallelogram_of_an_ellipse"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Tangent parallelogram of an ellipse</span> </div> </a> <ul id="toc-Tangent_parallelogram_of_an_ellipse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Faces_of_a_parallelepiped" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Faces_of_a_parallelepiped"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Faces of a parallelepiped</span> </div> </a> <ul id="toc-Faces_of_a_parallelepiped-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">8</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">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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">Parallelogram</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 101 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-101" 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">101 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/Parallelogram" title="Parallelogram – Afrikaans" lang="af" hreflang="af" data-title="Parallelogram" 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/%D9%85%D8%AA%D9%88%D8%A7%D8%B2%D9%8A_%D8%A3%D8%B6%D9%84%D8%A7%D8%B9" title="متوازي أضلاع – Arabic" lang="ar" hreflang="ar" data-title="متوازي أضلاع" 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/Paralelogramu" title="Paralelogramu – Asturian" lang="ast" hreflang="ast" data-title="Paralelogramu" 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/Paraleloqram" title="Paraleloqram – Azerbaijani" lang="az" hreflang="az" data-title="Paraleloqram" 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%B8%E0%A6%BE%E0%A6%AE%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A6%B0%E0%A6%BF%E0%A6%95" title="সামান্তরিক – Bangla" lang="bn" hreflang="bn" data-title="সামান্তরিক" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC" title="Параллелограмм – Bashkir" lang="ba" hreflang="ba" data-title="Параллелограмм" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%B0%D0%B3%D1%80%D0%B0%D0%BC" title="Паралелаграм – Belarusian" lang="be" hreflang="be" data-title="Паралелаграм" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%B0%D0%B3%D1%80%D0%B0%D0%BC" title="Паралелаграм – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Паралелаграм" 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/Paralelogramo" title="Paralelogramo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Paralelogramo" 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%A3%D1%81%D0%BF%D0%BE%D1%80%D0%B5%D0%B4%D0%BD%D0%B8%D0%BA" title="Успоредник – Bulgarian" lang="bg" hreflang="bg" data-title="Успоредник" 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%98%E0%BD%89%E0%BD%98%E0%BC%8B%E0%BD%A0%E0%BD%82%E0%BE%B2%E0%BD%BC%E0%BC%8B%E0%BD%98%E0%BD%90%E0%BD%A0%E0%BC%8B%E0%BD%96%E0%BD%9E%E0%BD%B2%E0%BC%8B%E0%BD%96%E0%BE%B1%E0%BD%B2%E0%BD%A6%E0%BD%96%E0%BD%91%E0%BC%8D" title="མཉམ་འགྲོ་མཐའ་བཞི་བྱིསབད། – Tibetan" lang="bo" hreflang="bo" data-title="མཉམ་འགྲོ་མཐའ་བཞི་བྱིསབད།" 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/Paralelogram" title="Paralelogram – Bosnian" lang="bs" hreflang="bs" data-title="Paralelogram" 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/Kensturieg" title="Kensturieg – Breton" lang="br" hreflang="br" data-title="Kensturieg" 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/Paral%C2%B7lelogram" title="Paral·lelogram – Catalan" lang="ca" hreflang="ca" data-title="Paral·lelogram" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC" title="Параллелограмм – Chuvash" lang="cv" hreflang="cv" data-title="Параллелограмм" 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/Rovnob%C4%9B%C5%BEn%C3%ADk" title="Rovnoběžník – Czech" lang="cs" hreflang="cs" data-title="Rovnoběžník" 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_sambamba" title="Gonyoina sambamba – Shona" lang="sn" hreflang="sn" data-title="Gonyoina sambamba" 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/Paralelogram" title="Paralelogram – Welsh" lang="cy" hreflang="cy" data-title="Paralelogram" 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/Parallelogram" title="Parallelogram – Danish" lang="da" hreflang="da" data-title="Parallelogram" 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/Parallellogr%C3%A1mma" title="Parallellográmma – Northern Sami" lang="se" hreflang="se" data-title="Parallellográmma" 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/Parallelogramm" title="Parallelogramm – German" lang="de" hreflang="de" data-title="Parallelogramm" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Paralelogram" title="Paralelogram – Lower Sorbian" lang="dsb" hreflang="dsb" data-title="Paralelogram" data-language-autonym="Dolnoserbski" data-language-local-name="Lower Sorbian" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/R%C3%B6%C3%B6pk%C3%BClik" title="Rööpkülik – Estonian" lang="et" hreflang="et" data-title="Rööpkülik" 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%A0%CE%B1%CF%81%CE%B1%CE%BB%CE%BB%CE%B7%CE%BB%CF%8C%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%BF" title="Παραλληλόγραμμο – Greek" lang="el" hreflang="el" data-title="Παραλληλόγραμμο" 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/Paralelogramo" title="Paralelogramo – Spanish" lang="es" hreflang="es" data-title="Paralelogramo" 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/Paralelogramo" title="Paralelogramo – Esperanto" lang="eo" hreflang="eo" data-title="Paralelogramo" 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/Paralelogramo" title="Paralelogramo – Basque" lang="eu" hreflang="eu" data-title="Paralelogramo" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AA%D9%88%D8%A7%D8%B2%DB%8C%E2%80%8C%D8%A7%D9%84%D8%A7%D8%B6%D9%84%D8%A7%D8%B9" title="متوازی‌الاضلاع – Persian" lang="fa" hreflang="fa" data-title="متوازی‌الاضلاع" 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/Parall%C3%A9logramme" title="Parallélogramme – French" lang="fr" hreflang="fr" data-title="Parallélogramme" 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/Comhthreomhar%C3%A1n" title="Comhthreomharán – Irish" lang="ga" hreflang="ga" data-title="Comhthreomharán" 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/Paralelogramo" title="Paralelogramo – Galician" lang="gl" hreflang="gl" data-title="Paralelogramo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%89%ED%96%89%EC%82%AC%EB%B3%80%ED%98%95" title="평행사변형 – Korean" lang="ko" hreflang="ko" data-title="평행사변형" 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/%D4%B6%D5%B8%D6%82%D5%A3%D5%A1%D5%B0%D5%A5%D5%BC%D5%A1%D5%A3%D5%AB%D5%AE" title="Զուգահեռագիծ – Armenian" lang="hy" hreflang="hy" data-title="Զուգահեռագիծ" 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%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0_%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="समान्तर चतुर्भुज – Hindi" lang="hi" hreflang="hi" data-title="समान्तर 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/Runob%C4%9B%C5%BEnik" title="Runoběžnik – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Runoběžnik" 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/Paralelogram" title="Paralelogram – Croatian" lang="hr" hreflang="hr" data-title="Paralelogram" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Jajar_genjang" title="Jajar genjang – Indonesian" lang="id" hreflang="id" data-title="Jajar genjang" 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/Parallelogramma" title="Parallelogramma – Interlingua" lang="ia" hreflang="ia" data-title="Parallelogramma" 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/Sams%C3%AD%C3%B0ungur" title="Samsíðungur – Icelandic" lang="is" hreflang="is" data-title="Samsíðungur" 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/Parallelogramma" title="Parallelogramma – Italian" lang="it" hreflang="it" data-title="Parallelogramma" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A7%D7%91%D7%99%D7%9C%D7%99%D7%AA" title="מקבילית – Hebrew" lang="he" hreflang="he" data-title="מקבילית" 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/Jajaran_g%C3%A9njang" title="Jajaran génjang – Javanese" lang="jv" hreflang="jv" data-title="Jajaran génjang" 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%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%9A%E1%83%94%E1%83%9A%E1%83%9D%E1%83%92%E1%83%A0%E1%83%90%E1%83%9B%E1%83%98" title="პარალელოგრამი – Georgian" lang="ka" hreflang="ka" data-title="პარალელოგრამი" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC" title="Параллелограмм – Kazakh" lang="kk" hreflang="kk" data-title="Параллелограмм" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/%C3%87argo%C5%9Feya_yeksan" title="Çargoşeya yeksan – Kurdish" lang="ku" hreflang="ku" data-title="Çargoşeya yeksan" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC" title="Параллелограмм – Kyrgyz" lang="ky" hreflang="ky" data-title="Параллелограмм" 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/Parallelogrammum" title="Parallelogrammum – Latin" lang="la" hreflang="la" data-title="Parallelogrammum" 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/Paralelograms" title="Paralelograms – Latvian" lang="lv" hreflang="lv" data-title="Paralelograms" 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/Lygiagretainis" title="Lygiagretainis – Lithuanian" lang="lt" hreflang="lt" data-title="Lygiagretainis" 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/Paralelogr%C3%A0m" title="Paralelogràm – Lombard" lang="lmo" hreflang="lmo" data-title="Paralelogràm" 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/Paralelogramma" title="Paralelogramma – Hungarian" lang="hu" hreflang="hu" data-title="Paralelogramma" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC" title="Паралелограм – Macedonian" lang="mk" hreflang="mk" data-title="Паралелограм" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Roazodafy" title="Roazodafy – Malagasy" lang="mg" hreflang="mg" data-title="Roazodafy" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%BE%E0%B4%AE%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B4%82" title="സാമാന്തരികം – Malayalam" lang="ml" hreflang="ml" data-title="സാമാന്തരികം" 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%BE%E0%A4%82%E0%A4%A4%E0%A4%B0%E0%A4%AD%E0%A5%81%E0%A4%9C_%E0%A4%9A%E0%A5%8C%E0%A4%95%E0%A5%8B%E0%A4%A8" title="समांतरभुज चौकोन – Marathi" lang="mr" hreflang="mr" data-title="समांतरभुज 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%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%9A%E1%83%94%E1%83%9A%E1%83%9D%E1%83%92%E1%83%A0%E1%83%90%E1%83%9B%E1%83%98" title="პარალელოგრამი – Mingrelian" lang="xmf" hreflang="xmf" data-title="პარალელოგრამი" 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/Segi_empat_selari" title="Segi empat selari – Malay" lang="ms" hreflang="ms" data-title="Segi empat selari" 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-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC%D0%B0%D1%81%D1%8C" title="Параллелограммась – Moksha" lang="mdf" hreflang="mdf" data-title="Параллелограммась" data-language-autonym="Мокшень" data-language-local-name="Moksha" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC" title="Параллелограм – Mongolian" lang="mn" hreflang="mn" data-title="Параллелограм" 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/Parallellogram" title="Parallellogram – Dutch" lang="nl" hreflang="nl" data-title="Parallellogram" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0_%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="समानान्तर चतुर्भुज – Nepali" lang="ne" hreflang="ne" data-title="समानान्तर चतुर्भुज" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E8%A1%8C%E5%9B%9B%E8%BE%BA%E5%BD%A2" title="平行四辺形 – Japanese" lang="ja" hreflang="ja" data-title="平行四辺形" 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/Paraleelogram" title="Paraleelogram – Northern Frisian" lang="frr" hreflang="frr" data-title="Paraleelogram" 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/Parallellogram" title="Parallellogram – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Parallellogram" 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/Parallellogram" title="Parallellogram – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Parallellogram" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC" title="Параллелограмм – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Параллелограмм" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Parallelogramm" title="Parallelogramm – Uzbek" lang="uz" hreflang="uz" data-title="Parallelogramm" 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%BE%E0%A8%82%E0%A8%A4%E0%A8%B0_%E0%A8%9A%E0%A8%A4%E0%A9%81%E0%A8%B0%E0%A8%AD%E0%A9%81%E0%A8%9C" title="ਸਮਾਂਤਰ ਚਤੁਰਭੁਜ – Punjabi" lang="pa" hreflang="pa" data-title="ਸਮਾਂਤਰ ਚਤੁਰਭੁਜ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%AA%D9%88%D8%A7%D8%B2%DB%8C_%D8%A7%D9%84%D8%A7%D8%B6%D9%84%D8%A7%D8%B9" title="متوازی الاضلاع – Western Punjabi" lang="pnb" hreflang="pnb" data-title="متوازی الاضلاع" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%94%E1%9F%92%E1%9E%9A%E1%9E%9B%E1%9F%81%E1%9E%A1%E1%9E%BC%E1%9E%80%E1%9F%92%E1%9E%9A%E1%9E%B6%E1%9E%98" title="ប្រលេឡូក្រាម – Khmer" lang="km" hreflang="km" data-title="ប្រលេឡូក្រាម" 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/Paralelograma" title="Paralelograma – Piedmontese" lang="pms" hreflang="pms" data-title="Paralelograma" 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/R%C3%B3wnoleg%C5%82obok" title="Równoległobok – Polish" lang="pl" hreflang="pl" data-title="Równoległobok" 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/Paralelogramo" title="Paralelogramo – Portuguese" lang="pt" hreflang="pt" data-title="Paralelogramo" 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/Paralelogram" title="Paralelogram – Romanian" lang="ro" hreflang="ro" data-title="Paralelogram" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC" title="Параллелограмм – Russian" lang="ru" hreflang="ru" data-title="Параллелограмм" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC" title="Параллелограмм – Yakut" lang="sah" hreflang="sah" data-title="Параллелограмм" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Paralelogrami" title="Paralelogrami – Albanian" lang="sq" hreflang="sq" data-title="Paralelogrami" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Paralleluggramma" title="Paralleluggramma – Sicilian" lang="scn" hreflang="scn" data-title="Paralleluggramma" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Parallelogram" title="Parallelogram – Simple English" lang="en-simple" hreflang="en-simple" data-title="Parallelogram" 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/Rovnobe%C5%BEn%C3%ADk" title="Rovnobežník – Slovak" lang="sk" hreflang="sk" data-title="Rovnobežník" 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/Paralelogram" title="Paralelogram – Slovenian" lang="sl" hreflang="sl" data-title="Paralelogram" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Barbaroole" title="Barbaroole – Somali" lang="so" hreflang="so" data-title="Barbaroole" 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%84%D8%A7%DA%BE%D8%A7%D9%88%D8%A8%DB%95%D8%B1%DB%95" title="لاھاوبەرە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="لاھاوبەرە" 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%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC" title="Паралелограм – Serbian" lang="sr" hreflang="sr" data-title="Паралелограм" 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/Paralelogram" title="Paralelogram – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Paralelogram" 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/Pasagi_doyong" title="Pasagi doyong – Sundanese" lang="su" hreflang="su" data-title="Pasagi doyong" 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/Suunnikas" title="Suunnikas – Finnish" lang="fi" hreflang="fi" data-title="Suunnikas" 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/Parallellogram" title="Parallellogram – Swedish" lang="sv" hreflang="sv" data-title="Parallellogram" 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/Paralelogram" title="Paralelogram – Tagalog" lang="tl" hreflang="tl" data-title="Paralelogram" 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%87%E0%AE%A3%E0%AF%88%E0%AE%95%E0%AE%B0%E0%AE%AE%E0%AF%8D" title="இணைகரம் – Tamil" lang="ta" hreflang="ta" data-title="இணைகரம்" 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%BE%E0%B0%82%E0%B0%A4%E0%B0%B0_%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="సమాంతర చతుర్భుజం – Telugu" lang="te" hreflang="te" data-title="సమాంతర 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%94%E0%B9%89%E0%B8%B2%E0%B8%99%E0%B8%82%E0%B8%99%E0%B8%B2%E0%B8%99" title="รูปสี่เหลี่ยมด้านขนาน – Thai" lang="th" hreflang="th" data-title="รูปสี่เหลี่ยมด้านขนาน" 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/Paralelkenar" title="Paralelkenar – Turkish" lang="tr" hreflang="tr" data-title="Paralelkenar" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BE%D0%B3%D1%80%D0%B0%D0%BC" title="Паралелограм – Ukrainian" lang="uk" hreflang="uk" data-title="Паралелограм" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_b%C3%ACnh_h%C3%A0nh" title="Hình bình hành – Vietnamese" lang="vi" hreflang="vi" data-title="Hình bình hành" 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/Parallellogram" title="Parallellogram – West Flemish" lang="vls" hreflang="vls" data-title="Parallellogram" 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/Paralelogramo" title="Paralelogramo – Waray" lang="war" hreflang="war" data-title="Paralelogramo" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%B9%B3%E8%A1%8C%E5%9B%9B%E8%BE%B9%E5%BD%A2" title="平行四边形 – Wu" lang="wuu" hreflang="wuu" data-title="平行四边形" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B9%B3%E8%A1%8C%E5%9B%9B%E9%82%8A%E5%BD%A2" title="平行四邊形 – Cantonese" lang="yue" hreflang="yue" data-title="平行四邊形" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B3%E8%A1%8C%E5%9B%9B%E8%BE%B9%E5%BD%A2" title="平行四边形 – Chinese" lang="zh" hreflang="zh" data-title="平行四边形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B4%B0%E2%B5%8E%E2%B5%99%E2%B4%B7%E2%B5%96%E2%B5%89%E2%B4%B7%E2%B5%89%E2%B5%99" title="ⴰⵎⵙⴷⵖⵉⴷⵉⵙ – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⴰⵎⵙⴷⵖⵉⴷⵉⵙ" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q45867#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Parallelogram" title="View the 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Quadrilateral with two pairs of parallel sides</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the quadrilateral. For the album by Linda Perhacs, see <a href="/wiki/Parallelograms_(album)" title="Parallelograms (album)">Parallelograms (album)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3;">Parallelogram</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Parallelogram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Parallelogram.svg/220px-Parallelogram.svg.png" decoding="async" width="220" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Parallelogram.svg/330px-Parallelogram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Parallelogram.svg/440px-Parallelogram.svg.png 2x" data-file-width="255" data-file-height="175" /></a></span><div class="infobox-caption">This parallelogram is a <a href="/wiki/Rhomboid" title="Rhomboid">rhomboid</a> as it has unequal sides and no right angles.</div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral</a>, <a href="/wiki/Trapezoid" title="Trapezoid">Trapezium</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a> and <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a></th><td class="infobox-data">4</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_planar_symmetry_groups" title="List of planar symmetry groups">Symmetry group</a></th><td class="infobox-data"><a href="/wiki/Point_reflection" title="Point reflection">C<sub>2</sub></a>, [2]<sup>+</sup>,</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Area" title="Area">Area</a></th><td class="infobox-data"><i>bh</i> (base × height);<br /><i>ab</i> sin <i>θ</i> (product of adjacent sides and sine of the vertex angle determined by them)</td></tr><tr><th scope="row" class="infobox-label">Properties</th><td class="infobox-data"><a href="/wiki/Convex_polygon" title="Convex polygon">Convex polygon</a></td></tr></tbody></table> <p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, a <b>parallelogram</b> is a <a href="/wiki/Simple_polygon" title="Simple polygon">simple</a> (non-<a href="/wiki/List_of_self-intersecting_polygons" title="List of self-intersecting polygons">self-intersecting</a>) <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> with two pairs of <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruence</a> of opposite sides and opposite angles is a direct consequence of the Euclidean <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. </p><p>By comparison, a quadrilateral with at least one pair of parallel sides is a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a> in American English or a trapezium in British English. </p><p>The three-dimensional counterpart of a parallelogram is a <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a>. </p><p>The word "parallelogram" comes from the Greek παραλληλό-γραμμον, <i>parallēló-grammon</i>, which means "a shape of parallel lines". </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=1" title="Edit section: Special cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a> – A parallelogram with four angles of equal size (right angles).</li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a> – A parallelogram with four sides of equal length. Any parallelogram that is neither a rectangle nor a rhombus was traditionally called a <a href="/wiki/Rhomboid" title="Rhomboid">rhomboid</a> but this term is not used in modern mathematics.<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></li> <li><a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">Square</a> – A parallelogram with four sides of equal length and angles of equal size (right angles).</li></ul> <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=Parallelogram&action=edit&section=2" title="Edit section: Characterizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Simple_polygon" title="Simple polygon">simple</a> (non-self-intersecting) <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> is a parallelogram <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> any one of the following statements is true:<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><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Two pairs of opposite sides are parallel (by definition).</li> <li>Two pairs of opposite sides are equal in length.</li> <li>Two pairs of opposite angles are equal in measure.</li> <li>The <a href="/wiki/Diagonal" title="Diagonal">diagonals</a> bisect each other.</li> <li>One pair of opposite sides is <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> and equal in length.</li> <li><a href="/wiki/Adjacent_angles" class="mw-redirect" title="Adjacent angles">Adjacent angles</a> are <a href="/wiki/Supplementary_angles" class="mw-redirect" title="Supplementary angles">supplementary</a>.</li> <li>Each diagonal divides the quadrilateral into two <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> <a href="/wiki/Triangle" title="Triangle">triangles</a>.</li> <li>The sum of the <a href="/wiki/Square_number" title="Square number">squares</a> of the sides equals the sum of the squares of the diagonals. (This is the <a href="/wiki/Parallelogram_law" title="Parallelogram law">parallelogram law</a>.)</li> <li>It has <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> of order 2.</li> <li>The sum of the distances from any interior point to the sides is independent of the location of the point.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (This is an extension of <a href="/wiki/Viviani%27s_theorem" title="Viviani's theorem">Viviani's theorem</a>.)</li> <li>There is a point <i>X</i> in the plane of the quadrilateral with the property that every straight line through <i>X</i> divides the quadrilateral into two regions of equal area.<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></li></ul> <p>Thus, all parallelograms have all the properties listed above, and <a href="/wiki/Converse_(logic)" title="Converse (logic)">conversely</a>, if just any one of these statements is true in a simple quadrilateral, then it is considered a parallelogram. </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=Parallelogram&action=edit&section=3" title="Edit section: Other properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.</li> <li>The area of a parallelogram is twice the area of a triangle created by one of its diagonals.</li> <li>The area of a parallelogram is also equal to the magnitude of the <a href="/wiki/Vector_cross_product" class="mw-redirect" title="Vector cross product">vector cross product</a> of two <a href="/wiki/Adjacent_side_(polygon)" class="mw-redirect" title="Adjacent side (polygon)">adjacent</a> sides.</li> <li>Any line through the midpoint of a parallelogram bisects the area.<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></li> <li>Any non-degenerate <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> takes a parallelogram to another parallelogram.</li> <li>A parallelogram has <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of <a href="/wiki/Reflectional_symmetry" class="mw-redirect" title="Reflectional symmetry">reflectional symmetry</a> then it must be a rhombus or an oblong (a non-square rectangle). If it has four lines of reflectional symmetry, it is a <a href="/wiki/Square" title="Square">square</a>.</li> <li>The perimeter of a parallelogram is 2(<i>a</i> + <i>b</i>) where <i>a</i> and <i>b</i> are the lengths of adjacent sides.</li> <li>Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square.<sup id="cite_ref-Weisstein_8-0" class="reference"><a href="#cite_note-Weisstein-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li>If two lines parallel to sides of a parallelogram are constructed <a href="/wiki/Concurrent_lines" title="Concurrent lines">concurrent</a> to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area.<sup id="cite_ref-Weisstein_8-1" class="reference"><a href="#cite_note-Weisstein-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li>The diagonals of a parallelogram divide it into four triangles of equal area.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Area_formula">Area formula<span class="anchor" id="Area"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=4" title="Edit section: Area formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:ParallelogramArea.svg" class="mw-file-description"><img alt="A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/180px-ParallelogramArea.svg.png" decoding="async" width="180" height="236" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/270px-ParallelogramArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/ParallelogramArea.svg/360px-ParallelogramArea.svg.png 2x" data-file-width="204" data-file-height="268" /></a><figcaption>A parallelogram can be rearranged into a rectangle with the same area.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Parallelogram_area_animated.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Parallelogram_area_animated.gif/180px-Parallelogram_area_animated.gif" decoding="async" width="180" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Parallelogram_area_animated.gif/270px-Parallelogram_area_animated.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/2/27/Parallelogram_area_animated.gif 2x" data-file-width="300" data-file-height="200" /></a><figcaption>Animation for the area formula <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=bh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>b</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=bh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a8b110c6ce0b30cf7d53619c3ebaf0a8865227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.501ex; height:2.176ex;" alt="{\displaystyle K=bh}"></span>.</figcaption></figure> <p>All of the <a href="/wiki/Quadrilateral#Area_of_a_convex_quadrilateral" title="Quadrilateral">area formulas for general convex quadrilaterals</a> apply to parallelograms. Further formulas are specific to parallelograms: </p><p>A parallelogram with base <i>b</i> and height <i>h</i> can be divided into a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a> and a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a>, and rearranged into a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>, as shown in the figure to the left. This means that the <a href="/wiki/Area" title="Area">area</a> of a parallelogram is the same as that of a rectangle with the same base and height: </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=bh.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>b</mi> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=bh.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aafb35db413cd4e1e428c61e0564f097692f1a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.148ex; height:2.176ex;" alt="{\displaystyle K=bh.}"></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Parallelogram_area.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Parallelogram_area.svg/220px-Parallelogram_area.svg.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Parallelogram_area.svg/330px-Parallelogram_area.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Parallelogram_area.svg/440px-Parallelogram_area.svg.png 2x" data-file-width="520" data-file-height="320" /></a><figcaption>The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram</figcaption></figure> <p>The base × height area formula can also be derived using the figure to the right. The area <i>K</i> of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\text{rect}}=(B+A)\times H\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rect</mtext> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>H</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\text{rect}}=(B+A)\times H\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/624d8b3c282546b2da21eebbf5230221831ae316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.496ex; height:2.843ex;" alt="{\displaystyle K_{\text{rect}}=(B+A)\times H\,}"></span></dd></dl> <p>and the area of a single triangle is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\text{tri}}={\frac {A}{2}}\times H.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tri</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mn>2</mn> </mfrac> </mrow> <mo>×</mo> <mi>H</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\text{tri}}={\frac {A}{2}}\times H.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a207b19540ff507ebf5aea92b2216b934baa939d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.563ex; height:5.343ex;" alt="{\displaystyle K_{\text{tri}}={\frac {A}{2}}\times H.\,}"></span></dd></dl> <p>Therefore, the area of the parallelogram is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=K_{\text{rect}}-2\times K_{\text{tri}}=((B+A)\times H)-(A\times H)=B\times H.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rect</mtext> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo>×</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tri</mtext> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>×</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>B</mi> <mo>×</mo> <mi>H</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=K_{\text{rect}}-2\times K_{\text{tri}}=((B+A)\times H)-(A\times H)=B\times H.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74aba8c98bfea059c67c19e4c6bedd1f50b135d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.583ex; height:2.843ex;" alt="{\displaystyle K=K_{\text{rect}}-2\times K_{\text{tri}}=((B+A)\times H)-(A\times H)=B\times H.}"></span></dd></dl> <p>Another area formula, for two sides <i>B</i> and <i>C</i> and angle θ, is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=B\cdot C\cdot \sin \theta .\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>B</mi> <mo>⋅</mo> <mi>C</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=B\cdot C\cdot \sin \theta .\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f761ff8f6540b6a058436e66ee822609c8f7d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.42ex; height:2.176ex;" alt="{\displaystyle K=B\cdot C\cdot \sin \theta .\,}"></span></dd></dl> <p>Provided that the parallelogram is not a rhombus, the area can be expressed using sides <i>B</i> and <i>C</i> and angle <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> at the intersection of the diagonals:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <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 {|\tan \gamma |}{2}}\cdot \left|B^{2}-C^{2}\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow> <mo>|</mo> <mrow> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {|\tan \gamma |}{2}}\cdot \left|B^{2}-C^{2}\right|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67c66072c35ad604d43cd4becc4b1287f18c9476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.208ex; height:5.676ex;" alt="{\displaystyle K={\frac {|\tan \gamma |}{2}}\cdot \left|B^{2}-C^{2}\right|.}"></span></dd></dl> <p>When the parallelogram is specified from the lengths <i>B</i> and <i>C</i> of two adjacent sides together with the length <i>D</i><sub>1</sub> of either diagonal, then the area can be found from <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a>. Specifically it is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=2{\sqrt {S(S-B)(S-C)(S-D_{1})}}={\frac {1}{2}}{\sqrt {(B+C+D_{1})(-B+C+D_{1})(B-C+D_{1})(B+C-D_{1})}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>−</mo> <mi>C</mi> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=2{\sqrt {S(S-B)(S-C)(S-D_{1})}}={\frac {1}{2}}{\sqrt {(B+C+D_{1})(-B+C+D_{1})(B-C+D_{1})(B+C-D_{1})}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a719f89fd8dcafa6268d084859adc97ed8eca14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:100.977ex; height:5.176ex;" alt="{\displaystyle K=2{\sqrt {S(S-B)(S-C)(S-D_{1})}}={\frac {1}{2}}{\sqrt {(B+C+D_{1})(-B+C+D_{1})(B-C+D_{1})(B+C-D_{1})}},}"></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=(B+C+D_{1})/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=(B+C+D_{1})/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13090983ab2505b560086c26b020ddd1feab6932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.922ex; height:2.843ex;" alt="{\displaystyle S=(B+C+D_{1})/2}"></span> and the leading factor 2 comes from the fact that the chosen diagonal divides the parallelogram into <i>two</i> congruent triangles. </p> <div class="mw-heading mw-heading3"><h3 id="From_vertex_coordinates">From vertex coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=5" title="Edit section: From vertex coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let vectors <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 \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98efc77fcc0696625eb4dc597fa365b611269ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.392ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{2}}"></span> and let <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={\begin{bmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{bmatrix}}\in \mathbb {R} ^{2\times 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\begin{bmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{bmatrix}}\in \mathbb {R} ^{2\times 2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a933b8f3dc284db8e64464c93f09365e525b4b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.656ex; height:6.176ex;" alt="{\displaystyle V={\begin{bmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{bmatrix}}\in \mathbb {R} ^{2\times 2}}"></span> denote the matrix with elements of <b>a</b> and <b>b</b>. Then the area of the parallelogram generated by <b>a</b> and <b>b</b> is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\det(V)|=|a_{1}b_{2}-a_{2}b_{1}|\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\det(V)|=|a_{1}b_{2}-a_{2}b_{1}|\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d4d45a7960c88424fa90befaaf70a70e41a5ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.798ex; height:2.843ex;" alt="{\displaystyle |\det(V)|=|a_{1}b_{2}-a_{2}b_{1}|\,}"></span>. </p><p>Let vectors <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 \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f47b03efec8aaa368a671bd60936c05d5385445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.556ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}}"></span> and let <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={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\\b_{1}&b_{2}&\dots &b_{n}\end{bmatrix}}\in \mathbb {R} ^{2\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\\b_{1}&b_{2}&\dots &b_{n}\end{bmatrix}}\in \mathbb {R} ^{2\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eba21fc083a518af12de294b662182251daf121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.637ex; height:6.176ex;" alt="{\displaystyle V={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\\b_{1}&b_{2}&\dots &b_{n}\end{bmatrix}}\in \mathbb {R} ^{2\times n}}"></span>. Then the area of the parallelogram generated by <b>a</b> and <b>b</b> is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\det(VV^{\mathrm {T} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>V</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\det(VV^{\mathrm {T} })}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20242172ff954885af682ddc428033f4f7e512c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.486ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\det(VV^{\mathrm {T} })}}}"></span>. </p><p>Let points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0964054bb701d9a51030dd1d4c21ff5d9371a881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.875ex; height:3.009ex;" alt="{\displaystyle a,b,c\in \mathbb {R} ^{2}}"></span>. Then the <a href="/wiki/Signed_area" title="Signed area">signed area</a> of the parallelogram with vertices at <i>a</i>, <i>b</i> and <i>c</i> is equivalent to the determinant of a matrix built using <i>a</i>, <i>b</i> and <i>c</i> as rows with the last column padded using ones as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\left|{\begin{matrix}a_{1}&a_{2}&1\\b_{1}&b_{2}&1\\c_{1}&c_{2}&1\end{matrix}}\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\left|{\begin{matrix}a_{1}&a_{2}&1\\b_{1}&b_{2}&1\\c_{1}&c_{2}&1\end{matrix}}\right|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f4f90e3c2fe95f38f069988909511f7c5158cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.619ex; height:9.176ex;" alt="{\displaystyle K=\left|{\begin{matrix}a_{1}&a_{2}&1\\b_{1}&b_{2}&1\\c_{1}&c_{2}&1\end{matrix}}\right|.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Proof_that_diagonals_bisect_each_other">Proof that diagonals bisect each other</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=6" title="Edit section: Proof that diagonals bisect each other"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/File:Parallelogram1.svg" class="mw-file-description" title="Parallelogram ABCD"><img alt="Parallelogram ABCD" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Parallelogram1.svg/200px-Parallelogram1.svg.png" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Parallelogram1.svg/300px-Parallelogram1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Parallelogram1.svg/400px-Parallelogram1.svg.png 2x" data-file-width="200" data-file-height="150" /></a><figcaption>Parallelogram ABCD</figcaption></figure> <p>To prove that the diagonals of a parallelogram bisect each other, we will use <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> <a href="/wiki/Triangle#Basic_facts" title="Triangle">triangles</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle ABE\cong \angle CDE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>E</mi> <mo>≅</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>D</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ABE\cong \angle CDE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab9e09a52dea0bf091179212defbbf6035ff086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.204ex; height:2.176ex;" alt="{\displaystyle \angle ABE\cong \angle CDE}"></span> <i>(alternate interior angles are equal in measure)</i></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 \angle BAE\cong \angle DCE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>E</mi> <mo>≅</mo> <mi mathvariant="normal">∠</mi> <mi>D</mi> <mi>C</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle BAE\cong \angle DCE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6807434bee15ed8506eb916def45f584f398fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.204ex; height:2.176ex;" alt="{\displaystyle \angle BAE\cong \angle DCE}"></span> <i>(alternate interior angles are equal in measure)</i>.</dd></dl> <p>(since these are angles that a transversal makes with <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel lines</a> <i>AB</i> and <i>DC</i>). </p><p>Also, side <i>AB</i> is equal in length to side <i>DC</i>, since opposite sides of a parallelogram are equal in length. </p><p>Therefore, triangles <i>ABE</i> and <i>CDE</i> are congruent (ASA postulate, <i>two corresponding angles and the included side</i>). </p><p>Therefore, </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 AE=CE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>E</mi> <mo>=</mo> <mi>C</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AE=CE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d80254dff1b2250ab27448227c0257efc43fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.159ex; height:2.176ex;" alt="{\displaystyle AE=CE}"></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 BE=DE.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>E</mi> <mo>=</mo> <mi>D</mi> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BE=DE.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff2cf29d8b7a94f92079ccca0cfbf611369228c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.985ex; height:2.176ex;" alt="{\displaystyle BE=DE.}"></span></dd></dl> <p>Since the diagonals <i>AC</i> and <i>BD</i> divide each other into segments of equal length, the diagonals bisect each other. </p><p>Separately, since the diagonals <i>AC</i> and <i>BD</i> bisect each other at point <i>E</i>, point <i>E</i> is the midpoint of each diagonal. </p> <div class="mw-heading mw-heading2"><h2 id="Lattice_of_parallelograms">Lattice of parallelograms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=7" title="Edit section: Lattice of parallelograms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Parallelograms can tile the plane by translation. If edges are equal, or angles are right, the symmetry of the lattice is higher. These represent the four <a href="/wiki/Bravais_lattice#In_2_dimensions" title="Bravais lattice">Bravais lattices in 2 dimensions</a>. </p> <table class="wikitable"> <caption>Lattices </caption> <tbody><tr> <th>Form</th> <th>Square</th> <th>Rectangle</th> <th>Rhombus</th> <th>Rhomboid </th></tr> <tr> <th>System </th> <th>Square<br />(tetragonal) </th> <th>Rectangular<br />(orthorhombic) </th> <th>Centered rectangular<br />(orthorhombic) </th> <th>Oblique<br />(monoclinic) </th></tr> <tr align="center"> <th>Constraints </th> <td>α=90°, a=b </td> <td>α=90° </td> <td>a=b </td> <td>None </td></tr> <tr align="center"> <th><a href="/wiki/List_of_planar_symmetry_groups#Wallpaper_groups" title="List of planar symmetry groups">Symmetry</a> </th> <td>p4m, [4,4], order 8<i>n</i></td> <td colspan="2">pmm, [∞,2,∞], order 4<i>n</i></td> <td>p1, [∞<sup>+</sup>,2,∞<sup>+</sup>], order 2<i>n</i> </td></tr> <tr align="center"> <th>Form </th> <td><span typeof="mw:File"><a href="/wiki/File:Lattice_of_squares.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lattice_of_squares.svg/160px-Lattice_of_squares.svg.png" decoding="async" width="160" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lattice_of_squares.svg/240px-Lattice_of_squares.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lattice_of_squares.svg/320px-Lattice_of_squares.svg.png 2x" data-file-width="718" data-file-height="416" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Lattice_of_rectangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Lattice_of_rectangles.svg/160px-Lattice_of_rectangles.svg.png" decoding="async" width="160" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Lattice_of_rectangles.svg/240px-Lattice_of_rectangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Lattice_of_rectangles.svg/320px-Lattice_of_rectangles.svg.png 2x" data-file-width="718" data-file-height="416" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Lattice_of_rhombuses.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Lattice_of_rhombuses.svg/160px-Lattice_of_rhombuses.svg.png" decoding="async" width="160" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Lattice_of_rhombuses.svg/240px-Lattice_of_rhombuses.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Lattice_of_rhombuses.svg/320px-Lattice_of_rhombuses.svg.png 2x" data-file-width="729" data-file-height="416" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Lattice_of_rhomboids.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Lattice_of_rhomboids.svg/160px-Lattice_of_rhomboids.svg.png" decoding="async" width="160" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Lattice_of_rhomboids.svg/240px-Lattice_of_rhomboids.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Lattice_of_rhomboids.svg/320px-Lattice_of_rhomboids.svg.png 2x" data-file-width="737" data-file-height="427" /></a></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Parallelograms_arising_from_other_figures">Parallelograms arising from other figures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=8" title="Edit section: Parallelograms arising from other figures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Automedian_triangle">Automedian triangle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=9" title="Edit section: Automedian triangle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Automedian_triangle" title="Automedian triangle">automedian triangle</a> is one whose <a href="/wiki/Median_(geometry)" title="Median (geometry)">medians</a> are in the same proportions as its sides (though in a different order). If <i>ABC</i> is an automedian triangle in which vertex <i>A</i> stands opposite the side <i>a</i>, <i>G</i> is the <a href="/wiki/Centroid" title="Centroid">centroid</a> (where the three medians of <i>ABC</i> intersect), and <i>AL</i> is one of the extended medians of <i>ABC</i> with <i>L</i> lying on the circumcircle of <i>ABC</i>, then <i>BGCL</i> is a parallelogram. </p> <div class="mw-heading mw-heading3"><h3 id="Varignon_parallelogram">Varignon parallelogram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=10" title="Edit section: Varignon parallelogram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Varignon%27s_theorem" title="Varignon's theorem">Varignon's theorem</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Varignon_parallelogram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Varignon_parallelogram.svg/220px-Varignon_parallelogram.svg.png" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Varignon_parallelogram.svg/330px-Varignon_parallelogram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Varignon_parallelogram.svg/440px-Varignon_parallelogram.svg.png 2x" data-file-width="512" data-file-height="683" /></a><figcaption>Proof without words of Varignon's theorem </figcaption></figure> <p><a href="/wiki/Varignon%27s_theorem" title="Varignon's theorem">Varignon's theorem</a> holds that the <a href="/wiki/Midpoint" title="Midpoint">midpoints</a> of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its <i>Varignon parallelogram</i>. If the quadrilateral is <a href="/wiki/Convex_polygon" title="Convex polygon">convex</a> or <a href="/wiki/Concave_polygon" title="Concave polygon">concave</a> (that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral. </p><p><a href="/wiki/Proof_without_words" title="Proof without words">Proof without words</a> (see figure): </p> <ol><li>An arbitrary quadrilateral and its diagonals.</li> <li>Bases of similar triangles are parallel to the blue diagonal.</li> <li>Ditto for the red diagonal.</li> <li>The base pairs form a parallelogram with half the area of the quadrilateral, <i>A<sub>q</sub></i>, as the sum of the areas of the four large triangles, <i>A<sub>l</sub></i> is 2 <i>A<sub>q</sub></i> (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, <i>A<sub>s</sub></i> is a quarter of <i>A<sub>l</sub></i> (half linear dimensions yields quarter area), and the area of the parallelogram is <i>A<sub>q</sub></i> minus <i>A<sub>s</sub></i>.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Tangent_parallelogram_of_an_ellipse">Tangent parallelogram of an ellipse</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=11" title="Edit section: Tangent parallelogram of an ellipse"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>, two diameters are said to be <a href="/wiki/Conjugate_diameters" title="Conjugate diameters">conjugate</a> if and only if the <a href="/wiki/Tangent_line" class="mw-redirect" title="Tangent line">tangent line</a> to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding <a href="/wiki/Tangent_parallelogram" class="mw-redirect" title="Tangent parallelogram">tangent parallelogram</a>, sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. All tangent parallelograms for a given ellipse have the same area. </p><p>It is possible to <a href="/wiki/Compass_and_straightedge_constructions" class="mw-redirect" title="Compass and straightedge constructions">reconstruct</a> an ellipse from any pair of conjugate diameters, or from any tangent parallelogram. </p> <div class="mw-heading mw-heading3"><h3 id="Faces_of_a_parallelepiped">Faces of a parallelepiped</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=12" title="Edit section: Faces of a parallelepiped"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a> is a three-dimensional figure whose six <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> are parallelograms. </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=Parallelogram&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a></li> <li><a href="/wiki/Levi-Civita_parallelogramoid" title="Levi-Civita parallelogramoid">Levi-Civita parallelogramoid</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output 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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://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf">"CIMT - Page no longer available at Plymouth University servers"</a> <span class="cs1-format">(PDF)</span>. <i>www.cimt.plymouth.ac.uk</i>. Archived from <a rel="nofollow" class="external text" href="http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2014-05-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.cimt.plymouth.ac.uk&rft.atitle=CIMT+-+Page+no+longer+available+at+Plymouth+University+servers&rft_id=http%3A%2F%2Fwww.cimt.plymouth.ac.uk%2Fresources%2Ftopics%2Fart002.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelogram" 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">Owen Byer, Felix Lazebnik and <a href="/wiki/Deirdre_Smeltzer" title="Deirdre Smeltzer">Deirdre Smeltzer</a>, <i>Methods for Euclidean Geometry</i>, Mathematical Association of America, 2010, pp. 51-52.</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">Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", <i><a href="/wiki/The_College_Mathematics_Journal" title="The College Mathematics Journal">The College Mathematics Journal</a></i> 37(5), 2006, pp. 390–391.</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">Problem 5, <i>2006 British Mathematical Olympiad</i>, <a rel="nofollow" class="external autonumber" href="http://artofproblemsolving.com/downloads/printable_post_collections/5063">[1]</a>.</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">Dunn, J.A., and J.E. Pretty, "Halving a triangle", <i>Mathematical Gazette</i> 56, May 1972, p. 105.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/TriangleCircumscribing.html">"Triangle Circumscribing"</a>. <i>Wolfram Math World</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+Math+World&rft.atitle=Triangle+Circumscribing&rft.aulast=Weisstein&rft.aufirst=Eric+W&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FTriangleCircumscribing.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelogram" class="Z3988"></span></span> </li> <li id="cite_note-Weisstein-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Weisstein_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Weisstein_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Weisstein, Eric W. "Parallelogram." From MathWorld--A Wolfram Web Resource. <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/Parallelogram.html">http://mathworld.wolfram.com/Parallelogram.html</a></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">Mitchell, Douglas W., "The area of a quadrilateral", <i>Mathematical Gazette</i>, July 2009.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelogram&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Parallelograms" class="extiw" title="commons:Category:Parallelograms">Parallelograms</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.elsy.at/kurse/index.php?kurs=Parallelogram+and+Rhombus&status=public">Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Parallelogram"><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/Parallelogram.html">"Parallelogram"</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=Parallelogram&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FParallelogram.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelogram" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/index.php">Interactive Parallelogram --sides, angles and slope</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Geometry/AreaOfParallelogram.shtml">Area of Parallelogram</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Geometry/EquiTriOnPara.shtml">Equilateral Triangles On Sides of a Parallelogram</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/parallelogram.html">Definition and properties of a parallelogram</a> with animated applet</li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/parallelogramarea.html">Interactive applet showing parallelogram area calculation</a> interactive applet</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 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href="/wiki/Equilateral_triangle" title="Equilateral triangle">Equilateral</a></li> <li><a href="/wiki/Ideal_triangle" title="Ideal triangle">Ideal</a></li> <li><a href="/wiki/Isosceles_triangle" title="Isosceles triangle">Isosceles</a></li> <li><a href="/wiki/Kepler_triangle" title="Kepler triangle">Kepler</a></li> <li><a href="/wiki/Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Obtuse</a></li> <li><a href="/wiki/Right_triangle" title="Right triangle">Right</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilaterals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a></li> <li><a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">Bicentric</a></li> <li><a href="/wiki/Crossed_quadrilateral" class="mw-redirect" title="Crossed quadrilateral">Crossed</a></li> <li><a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">Cyclic</a></li> <li><a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">Equidiagonal</a></li> <li><a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">Ex-tangential</a></li> <li><a href="/wiki/Harmonic_quadrilateral" title="Harmonic quadrilateral">Harmonic</a></li> <li><a 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 class="mw-selflink selflink">Parallelogram</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Right_kite" title="Right kite">Right kite</a></li> <li><a href="/wiki/Right_trapezoid" class="mw-redirect" title="Right trapezoid">Right trapezoid</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">Tangential</a></li> <li><a href="/wiki/Tangential_trapezoid" title="Tangential trapezoid">Tangential trapezoid</a></li> <li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By number <br />of sides</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">1–10 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Monogon" title="Monogon">Monogon (1)</a></li> <li><a href="/wiki/Digon" title="Digon">Digon (2)</a></li> <li><a href="/wiki/Triangle" title="Triangle">Triangle (3)</a></li> <li><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral (4)</a></li> <li><a href="/wiki/Pentagon" title="Pentagon">Pentagon (5)</a></li> <li><a href="/wiki/Hexagon" title="Hexagon">Hexagon (6)</a></li> <li><a href="/wiki/Heptagon" title="Heptagon">Heptagon (7)</a></li> <li><a href="/wiki/Octagon" title="Octagon">Octagon (8)</a></li> <li><a href="/wiki/Nonagon" title="Nonagon">Nonagon/Enneagon (9)</a></li> <li><a href="/wiki/Decagon" title="Decagon">Decagon (10)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11–20 sides</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hendecagon" title="Hendecagon">Hendecagon (11)</a></li> <li><a href="/wiki/Dodecagon" title="Dodecagon">Dodecagon (12)</a></li> <li><a href="/wiki/Tridecagon" title="Tridecagon">Tridecagon (13)</a></li> <li><a href="/wiki/Tetradecagon" title="Tetradecagon">Tetradecagon (14)</a></li> <li><a href="/wiki/Pentadecagon" title="Pentadecagon">Pentadecagon (15)</a></li> <li><a href="/wiki/Hexadecagon" title="Hexadecagon">Hexadecagon (16)</a></li> <li><a href="/wiki/Heptadecagon" title="Heptadecagon">Heptadecagon (17)</a></li> <li><a href="/wiki/Octadecagon" title="Octadecagon">Octadecagon (18)</a></li> <li><a href="/wiki/Icosagon" title="Icosagon">Icosagon (20)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">>20 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Icositrigon" title="Icositrigon">Icositrigon (23)</a></li> <li><a href="/wiki/Icositetragon" title="Icositetragon">Icositetragon (24)</a></li> <li><a href="/wiki/Triacontagon" title="Triacontagon">Triacontagon (30)</a></li> <li><a href="/wiki/257-gon" title="257-gon">257-gon</a></li> <li><a href="/wiki/Chiliagon" title="Chiliagon">Chiliagon (1000)</a></li> <li><a href="/wiki/Myriagon" title="Myriagon">Myriagon (10,000)</a></li> <li><a href="/wiki/65537-gon" title="65537-gon">65537-gon</a></li> <li><a href="/wiki/Megagon" title="Megagon">Megagon (1,000,000)</a></li> <li><a href="/wiki/Apeirogon" title="Apeirogon">Apeirogon (∞)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Star_polygon" title="Star polygon">Star polygons</a><br /></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentagram" title="Pentagram">Pentagram</a></li> <li><a href="/wiki/Hexagram" title="Hexagram">Hexagram</a></li> <li><a href="/wiki/Heptagram" title="Heptagram">Heptagram</a></li> <li><a href="/wiki/Octagram" title="Octagram">Octagram</a></li> <li><a href="/wiki/Enneagram_(geometry)" title="Enneagram (geometry)">Enneagram</a></li> <li><a href="/wiki/Decagram_(geometry)" title="Decagram (geometry)">Decagram</a></li> <li><a href="/wiki/Hendecagram" title="Hendecagram">Hendecagram</a></li> <li><a href="/wiki/Dodecagram" title="Dodecagram">Dodecagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Concave_polygon" title="Concave polygon">Concave</a></li> <li><a href="/wiki/Convex_polygon" title="Convex polygon">Convex</a></li> <li><a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">Cyclic</a></li> <li><a href="/wiki/Equiangular_polygon" title="Equiangular polygon">Equiangular</a></li> <li><a href="/wiki/Equilateral_polygon" title="Equilateral polygon">Equilateral</a></li> <li><a href="/wiki/Infinite_skew_polygon" title="Infinite skew polygon">Infinite skew</a></li> <li><a href="/wiki/Isogonal_figure" title="Isogonal figure">Isogonal</a></li> <li><a href="/wiki/Isotoxal_figure" title="Isotoxal figure">Isotoxal</a></li> <li><a href="/wiki/Magic_polygon" title="Magic polygon">Magic</a></li> <li><a href="/wiki/Pseudotriangle" title="Pseudotriangle">Pseudotriangle</a></li> <li><a href="/wiki/Rectilinear_polygon" title="Rectilinear polygon">Rectilinear</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular</a></li> <li><a href="/wiki/Reinhardt_polygon" title="Reinhardt polygon">Reinhardt</a></li> <li><a href="/wiki/Simple_polygon" title="Simple polygon">Simple</a></li> <li><a href="/wiki/Skew_polygon" title="Skew polygon">Skew</a></li> <li><a href="/wiki/Star-shaped_polygon" title="Star-shaped polygon">Star-shaped</a></li> <li><a href="/wiki/Tangential_polygon" title="Tangential polygon">Tangential</a></li> <li><a href="/wiki/Weakly_simple_polygon" title="Weakly simple polygon">Weakly simple</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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 dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Parallelogram&oldid=1253833543">https://en.wikipedia.org/w/index.php?title=Parallelogram&oldid=1253833543</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Types_of_quadrilaterals" title="Category:Types of quadrilaterals">Types of quadrilaterals</a></li><li><a href="/wiki/Category:Elementary_shapes" title="Category:Elementary shapes">Elementary 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