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id="toc-Regular_polygons-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Polyhedrons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polyhedrons"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Polyhedrons</span> </div> </a> <ul id="toc-Polyhedrons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Higher dimensions</span> </div> </a> <button aria-controls="toc-Higher_dimensions-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 Higher dimensions subsection</span> </button> <ul id="toc-Higher_dimensions-sublist" class="vector-toc-list"> <li id="toc-N-Cube" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#N-Cube"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>N-Cube</span> </div> </a> <ul id="toc-N-Cube-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Geometry</span> </div> </a> <ul id="toc-Geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">6</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" title="Table of Contents" > <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">Diagonal</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 66 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-66" 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">66 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B6%D9%84%D8%B9_%D9%82%D8%B7%D8%B1%D9%8A" title="ضلع قطري – Arabic" lang="ar" hreflang="ar" data-title="ضلع قطري" 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/Diagonal" title="Diagonal – Asturian" lang="ast" hreflang="ast" data-title="Diagonal" 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/Diaqonal" title="Diaqonal – Azerbaijani" lang="az" hreflang="az" data-title="Diaqonal" 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%95%E0%A6%B0%E0%A7%8D%E0%A6%A3_(%E0%A6%9C%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF)" title="কর্ণ (জ্যামিতি) – Bangla" lang="bn" hreflang="bn" data-title="কর্ণ (জ্যামিতি)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB" title="Диагонал – Bulgarian" lang="bg" hreflang="bg" data-title="Диагонал" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Diagonal" title="Diagonal – Catalan" lang="ca" hreflang="ca" data-title="Diagonal" 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%94%D0%B8%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C" title="Диагональ – Chuvash" lang="cv" hreflang="cv" data-title="Диагональ" 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/%C3%9Ahlop%C5%99%C3%AD%C4%8Dka" title="Úhlopříčka – Czech" lang="cs" hreflang="cs" data-title="Úhlopříčka" 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/Guramakonyo" title="Guramakonyo – Shona" lang="sn" hreflang="sn" data-title="Guramakonyo" 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/Croeslin" title="Croeslin – Welsh" lang="cy" hreflang="cy" data-title="Croeslin" 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/Diagonal" title="Diagonal – Danish" lang="da" hreflang="da" data-title="Diagonal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Diagonale_(Geometrie)" title="Diagonale (Geometrie) – German" lang="de" hreflang="de" data-title="Diagonale (Geometrie)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diagonaal" title="Diagonaal – Estonian" lang="et" hreflang="et" data-title="Diagonaal" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diagonal" title="Diagonal – Spanish" lang="es" hreflang="es" data-title="Diagonal" 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/Diagonalo" title="Diagonalo – Esperanto" lang="eo" hreflang="eo" data-title="Diagonalo" 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/Diagonal_(geometria)" title="Diagonal (geometria) – Basque" lang="eu" hreflang="eu" data-title="Diagonal (geometria)" 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%82%D8%B7%D8%B1_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="قطر (هندسه) – Persian" lang="fa" hreflang="fa" data-title="قطر (هندسه)" 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/Diagonale" title="Diagonale – French" lang="fr" hreflang="fr" data-title="Diagonale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Diagonal" title="Diagonal – Galician" lang="gl" hreflang="gl" data-title="Diagonal" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Tui-kok-sien" title="Tui-kok-sien – Hakka Chinese" lang="hak" hreflang="hak" data-title="Tui-kok-sien" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EA%B0%81%EC%84%A0" title="대각선 – Korean" lang="ko" hreflang="ko" data-title="대각선" 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%B1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6%D5%A1%D5%A3%D5%AB%D5%AE" title="Անկյունագիծ – Armenian" lang="hy" hreflang="hy" data-title="Անկյունագիծ" 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%B5%E0%A4%BF%E0%A4%95%E0%A4%B0%E0%A5%8D%E0%A4%A3" title="विकर्ण – Hindi" lang="hi" hreflang="hi" data-title="विकर्ण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Dijagonala" title="Dijagonala – Croatian" lang="hr" hreflang="hr" data-title="Dijagonala" 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/Diagonal" title="Diagonal – Indonesian" lang="id" hreflang="id" data-title="Diagonal" 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-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-diagonal" title="I-diagonal – Xhosa" lang="xh" hreflang="xh" data-title="I-diagonal" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diagonale" title="Diagonale – Italian" lang="it" hreflang="it" data-title="Diagonale" 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%90%D7%9C%D7%9B%D7%A1%D7%95%D7%9F" title="אלכסון – Hebrew" lang="he" hreflang="he" data-title="אלכסון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C" title="Диагональ – Kazakh" lang="kk" hreflang="kk" data-title="Диагональ" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Ulalo" title="Ulalo – Swahili" lang="sw" hreflang="sw" data-title="Ulalo" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B8%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C" title="Диагональ – Kyrgyz" lang="ky" hreflang="ky" data-title="Диагональ" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Diagon%C4%81le" title="Diagonāle – Latvian" lang="lv" hreflang="lv" data-title="Diagonāle" 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/%C4%AEstri%C5%BEain%C4%97" title="Įstrižainė – Lithuanian" lang="lt" hreflang="lt" data-title="Įstrižainė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%81tl%C3%B3" title="Átló – Hungarian" lang="hu" hreflang="hu" data-title="Átló" 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%94%D0%B8%D1%98%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%B0" title="Дијагонала – Macedonian" lang="mk" hreflang="mk" data-title="Дијагонала" 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/Savazoro" title="Savazoro – Malagasy" lang="mg" hreflang="mg" data-title="Savazoro" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pepenjuru" title="Pepenjuru – Malay" lang="ms" hreflang="ms" data-title="Pepenjuru" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diagonaal" title="Diagonaal – Dutch" lang="nl" hreflang="nl" data-title="Diagonaal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AF%BE%E8%A7%92%E7%B7%9A" title="対角線 – Japanese" lang="ja" hreflang="ja" data-title="対角線" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diagonal" title="Diagonal – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Diagonal" 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/Diagonal" title="Diagonal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Diagonal" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Diagonal_(geometriya)" title="Diagonal (geometriya) – Uzbek" lang="uz" hreflang="uz" data-title="Diagonal (geometriya)" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przek%C4%85tna" title="Przekątna – Polish" lang="pl" hreflang="pl" data-title="Przekątna" 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/Diagonais_de_um_pol%C3%ADgono" title="Diagonais de um polígono – Portuguese" lang="pt" hreflang="pt" data-title="Diagonais de um polígono" 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/Diagonal%C4%83" title="Diagonală – Romanian" lang="ro" hreflang="ro" data-title="Diagonală" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Ch%27iqan" title="Ch'iqan – Quechua" lang="qu" hreflang="qu" data-title="Ch'iqan" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C" title="Диагональ – Russian" lang="ru" hreflang="ru" data-title="Диагональ" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tn mw-list-item"><a href="https://tn.wikipedia.org/wiki/Daegonale" title="Daegonale – Tswana" lang="tn" hreflang="tn" data-title="Daegonale" data-language-autonym="Setswana" data-language-local-name="Tswana" class="interlanguage-link-target"><span>Setswana</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Diagonalja" title="Diagonalja – Albanian" lang="sq" hreflang="sq" data-title="Diagonalja" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AF%E0%B7%99%E0%B6%9A%E0%B7%9C%E0%B6%B1%E0%B7%90%E0%B6%BD%E0%B7%92%E0%B6%BA" title="දෙකොනැලිය – Sinhala" lang="si" hreflang="si" data-title="දෙකොනැලිය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Diagonal" title="Diagonal – Simple English" lang="en-simple" hreflang="en-simple" data-title="Diagonal" 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/Uhloprie%C4%8Dka" title="Uhlopriečka – Slovak" lang="sk" hreflang="sk" data-title="Uhlopriečka" 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/Diagonala" title="Diagonala – Slovenian" lang="sl" hreflang="sl" data-title="Diagonala" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%DB%86%D8%AA%DB%95%D8%B1%DB%95" title="کۆتەرە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="کۆتەرە" 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 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class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%82%E0%AE%B2%E0%AF%88%E0%AE%B5%E0%AE%BF%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="மூலைவிட்டம் – Tamil" lang="ta" hreflang="ta" data-title="மூலைவிட்டம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99%E0%B8%97%E0%B9%81%E0%B8%A2%E0%B8%87%E0%B8%A1%E0%B8%B8%E0%B8%A1" title="เส้นทแยงมุม – Thai" lang="th" hreflang="th" data-title="เส้นทแยงมุม" 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/K%C3%B6%C5%9Fegen" title="Köşegen – Turkish" 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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">In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron</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">For other uses, see <a href="/wiki/Diagonal_(disambiguation)" class="mw-disambig" title="Diagonal (disambiguation)">Diagonal (disambiguation)</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cube_diagonals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cube_diagonals.svg/220px-Cube_diagonals.svg.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cube_diagonals.svg/330px-Cube_diagonals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cube_diagonals.svg/440px-Cube_diagonals.svg.png 2x" data-file-width="275" data-file-height="265" /></a><figcaption>The diagonals of a <a href="/wiki/Cube" title="Cube">cube</a> with side length 1. AC' (shown in blue) is a <a href="/wiki/Space_diagonal" title="Space diagonal">space diagonal</a> with length <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 {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}" /></span>, while AC (shown in red) is a <a href="/wiki/Face_diagonal" title="Face diagonal">face diagonal</a> and has length <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 {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span>.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>diagonal</b> is a <a href="/wiki/Line_segment" title="Line segment">line segment</a> joining two <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> of a <a href="/wiki/Polygon" title="Polygon">polygon</a> or <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a>, when those vertices are not on the same <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edge</a>. Informally, any sloping line is called diagonal. The word <i>diagonal</i> derives from the <a href="/wiki/Ancient_Greek" title="Ancient Greek">ancient Greek</a> διαγώνιος <i>diagonios</i>,<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> "from corner to corner" (from διά- <i>dia-</i>, "through", "across" and γωνία <i>gonia</i>, "corner", related to <i>gony</i> "knee"); it was used by both <a href="/wiki/Strabo" title="Strabo">Strabo</a><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> and <a href="/wiki/Euclid" title="Euclid">Euclid</a><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> to refer to a line connecting two vertices of a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a> or <a href="/wiki/Cuboid" title="Cuboid">cuboid</a>,<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> and later adopted into Latin as <i>diagonus</i> ("slanting line"). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Polygons">Polygons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=1" title="Edit section: Polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As applied to a <a href="/wiki/Polygon" title="Polygon">polygon</a>, a diagonal is a <a href="/wiki/Line_segment" title="Line segment">line segment</a> joining any two non-consecutive vertices. Therefore, a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> has two diagonals, joining opposite pairs of vertices. For any <a href="/wiki/Convex_polygon" title="Convex polygon">convex polygon</a>, all the diagonals are inside the polygon, but for <a href="/wiki/Re-entrant_polygon" class="mw-redirect" title="Re-entrant polygon">re-entrant polygons</a>, some diagonals are outside of the polygon. </p><p>Any <i>n</i>-sided polygon (<i>n</i> ≥ 3), <a href="/wiki/Convex_polygon" title="Convex polygon">convex</a> or <a href="/wiki/Concave_polygon" title="Concave polygon">concave</a>, has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n(n-3)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n(n-3)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e631e8f3e69d09d8721ea3b4d4bf4cd5fa76bc1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.188ex; height:4.176ex;" alt="{\displaystyle {\tfrac {n(n-3)}{2}}}" /></span> <i>total</i> diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or <span class="texhtml"><i>n</i> − 3</span> diagonals, and each diagonal is shared by two vertices. </p><p>In general, a regular <i>n</i>-sided polygon has <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 \left\lfloor {\frac {n-2}{2}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lfloor {\frac {n-2}{2}}\right\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea68d80e4c9a20cf1cea12bfeba0bc5a0e7a448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.944ex; height:6.176ex;" alt="{\displaystyle \left\lfloor {\frac {n-2}{2}}\right\rfloor }" /></span> <i>distinct</i> diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square. </p> <table rules="none" border="0" cellspacing="4" cellpadding="0" style="background:transparent;text-align:right"> <tbody><tr valign="top"> <td> <table class="wikitable"> <tbody><tr> <th>Sides</th> <th>Diagonals </th></tr> <tr> <td>3</td> <td>0 </td></tr> <tr> <td>4</td> <td>2 </td></tr> <tr> <td>5</td> <td>5 </td></tr> <tr> <td>6</td> <td>9 </td></tr> <tr> <td>7</td> <td>14 </td></tr> <tr> <td>8</td> <td>20 </td></tr> <tr> <td>9</td> <td>27 </td></tr> <tr> <td>10</td> <td>35 </td></tr></tbody></table> </td> <td> <table class="wikitable"> <tbody><tr> <th>Sides</th> <th>Diagonals </th></tr> <tr> <td>11</td> <td>44 </td></tr> <tr> <td>12</td> <td>54 </td></tr> <tr> <td>13</td> <td>65 </td></tr> <tr> <td>14</td> <td>77 </td></tr> <tr> <td>15</td> <td>90 </td></tr> <tr> <td>16</td> <td>104 </td></tr> <tr> <td>17</td> <td>119 </td></tr> <tr> <td>18</td> <td>135 </td></tr></tbody></table> </td> <td> <table class="wikitable"> <tbody><tr> <th>Sides</th> <th>Diagonals </th></tr> <tr> <td>19</td> <td>152 </td></tr> <tr> <td>20</td> <td>170 </td></tr> <tr> <td>21</td> <td>189 </td></tr> <tr> <td>22</td> <td>209 </td></tr> <tr> <td>23</td> <td>230 </td></tr> <tr> <td>24</td> <td>252 </td></tr> <tr> <td>25</td> <td>275 </td></tr> <tr> <td>26</td> <td>299 </td></tr></tbody></table> </td> <td> <table class="wikitable"> <tbody><tr> <th>Sides</th> <th>Diagonals </th></tr> <tr> <td>27</td> <td>324 </td></tr> <tr> <td>28</td> <td>350 </td></tr> <tr> <td>29</td> <td>377 </td></tr> <tr> <td>30</td> <td>405 </td></tr> <tr> <td>31</td> <td>434 </td></tr> <tr> <td>32</td> <td>464 </td></tr> <tr> <td>33</td> <td>495 </td></tr> <tr> <td>34</td> <td>527 </td></tr></tbody></table> </td> <td> <table class="wikitable"> <tbody><tr> <th>Sides</th> <th>Diagonals </th></tr> <tr> <td>35</td> <td>560 </td></tr> <tr> <td>36</td> <td>594 </td></tr> <tr> <td>37</td> <td>629 </td></tr> <tr> <td>38</td> <td>665 </td></tr> <tr> <td>39</td> <td>702 </td></tr> <tr> <td>40</td> <td>740 </td></tr> <tr> <td>41</td> <td>779 </td></tr> <tr> <td>42</td> <td>819 </td></tr></tbody></table> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Regions_formed_by_diagonals">Regions formed by diagonals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=2" title="Edit section: Regions formed by diagonals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Convex_polygon" title="Convex polygon">convex polygon</a>, if no three diagonals are <a href="/wiki/Concurrent_lines" title="Concurrent lines">concurrent</a> at a single point in the interior, the number of regions that the diagonals divide the interior into is given by<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> </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 {\binom {n}{4}}+{\binom {n-1}{2}}={\frac {(n-1)(n-2)(n^{2}-3n+12)}{24}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mrow> <mn>24</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{4}}+{\binom {n-1}{2}}={\frac {(n-1)(n-2)(n^{2}-3n+12)}{24}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be30ea536f67ce33c5b189bf3655448638ae1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.291ex; height:6.509ex;" alt="{\displaystyle {\binom {n}{4}}+{\binom {n-1}{2}}={\frac {(n-1)(n-2)(n^{2}-3n+12)}{24}}.}" /></span></dd></dl> <p>For <i>n</i>-gons with <i>n</i>=3, 4, ... the number of regions is </p> <dl><dd>1, 4, 11, 25, 50, 91, 154, 246...</dd></dl> <p>This is <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> sequence A006522.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Intersections_of_diagonals">Intersections of diagonals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=3" title="Edit section: Intersections of diagonals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If no three diagonals of a convex polygon are concurrent at a point in the interior, the number of interior intersections of diagonals is given by <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 \textstyle {\binom {n}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {n}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e099ea438a57aa4aaa043feb6604733e1fe87b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle \textstyle {\binom {n}{4}}}" /></span>.<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><sup id="cite_ref-youtube_8-0" class="reference"><a href="#cite_note-youtube-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This holds, for example, for any <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a> with an odd number of sides. The formula follows from the fact that each intersection is uniquely determined by the four endpoints of the two intersecting diagonals: the number of intersections is thus the number of combinations of the <i>n</i> vertices four at a time. </p> <div class="mw-heading mw-heading3"><h3 id="Regular_polygons">Regular polygons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=4" title="Edit section: Regular polygons"><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">See also: <a href="/wiki/Quadrilateral#Diagonals" title="Quadrilateral">Quadrilateral § Diagonals</a>, <a href="/wiki/Hexagon#Convex_equilateral_hexagon" title="Hexagon">Hexagon § Convex equilateral hexagon</a>, and <a href="/wiki/Heptagon#Diagonals_and_heptagonal_triangle" title="Heptagon">Heptagon § Diagonals and heptagonal triangle</a></div> <p>Although the number of distinct diagonals in a polygon increases as its number of sides increases, the length of any diagonal can be calculated. </p><p>In a regular n-gon with side length <i>a</i>, the length of the <i>xth</i> shortest distinct diagonal 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 \sin \left({\frac {\pi (x+1)}{n}}\right)\csc \left({\frac {\pi }{n}}\right)\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>csc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left({\frac {\pi (x+1)}{n}}\right)\csc \left({\frac {\pi }{n}}\right)\cdot a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191645838bf7de34623670553066d17dc5ec26c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.87ex; height:6.343ex;" alt="{\displaystyle \sin \left({\frac {\pi (x+1)}{n}}\right)\csc \left({\frac {\pi }{n}}\right)\cdot a}" /></span></dd></dl> <p>This formula shows that as the number of sides approaches infinity, the <i>xth</i> shortest diagonal approaches the length <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+1)a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+1)a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/732b8523812193ddafa26990cecd8d2daa34c42a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.372ex; height:2.843ex;" alt="{\displaystyle (x+1)a}" /></span>⁠</span>. Additionally, the formula for the shortest diagonal simplifies in the case of x = 1: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left({\frac {2\pi }{n}}\right)\csc \left({\frac {\pi }{n}}\right)\cdot a=2\cos \left({\frac {\pi }{n}}\right)\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>csc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left({\frac {2\pi }{n}}\right)\csc \left({\frac {\pi }{n}}\right)\cdot a=2\cos \left({\frac {\pi }{n}}\right)\cdot a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0927eea1378e9fea4decf6c140392209a4a8ccbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.565ex; height:6.176ex;" alt="{\displaystyle \sin \left({\frac {2\pi }{n}}\right)\csc \left({\frac {\pi }{n}}\right)\cdot a=2\cos \left({\frac {\pi }{n}}\right)\cdot a}" /></span></dd></dl> <p>If the number of sides is even, the longest diagonal will be equivalent to the diameter of the polygon's circumcircle because the long diagonals all intersect each other at the polygon's center. </p><p>Special cases include: </p><p>A <a href="/wiki/Square" title="Square">square</a> has two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\approx 1.414.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1.414.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx 1.414.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7785ce3771362d45d00313d4f94b3738ba193efe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.14ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\approx 1.414.}" /></span> </p><p>A <a href="/wiki/Regular_pentagon" class="mw-redirect" title="Regular pentagon">regular pentagon</a> has five diagonals all of the same length. The ratio of a diagonal to a side is the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>≈</mo> <mn>1.618.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db100d7ae8b559acce5063f1d3cfe52a9be8ba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.979ex; height:5.843ex;" alt="{\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618.}" /></span> </p><p>A regular <a href="/wiki/Hexagon" title="Hexagon">hexagon</a> has nine diagonals: the six shorter ones are equal to each other in length; the three longer ones are equal to each other in length and intersect each other at the center of the hexagon. The ratio of a long diagonal to a side is 2, and the ratio of a short diagonal to a side is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}" /></span>. </p><p>A regular <a href="/wiki/Heptagon" title="Heptagon">heptagon</a> has 14 diagonals. The seven shorter ones equal each other, and the seven longer ones equal each other. The reciprocal of the side equals the sum of the reciprocals of a short and a long diagonal. </p> <div class="mw-heading mw-heading2"><h2 id="Polyhedrons">Polyhedrons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=5" title="Edit section: Polyhedrons"><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">See also: <a href="/wiki/Face_diagonal" title="Face diagonal">Face diagonal</a> and <a href="/wiki/Space_diagonal" title="Space diagonal">Space diagonal</a></div> <p>A <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> (a <a href="/wiki/Solid_object" class="mw-redirect" title="Solid object">solid object</a> in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>, bounded by <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">two-dimensional</a> <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a>) may have two different types of diagonals: face diagonals on the various faces, connecting non-adjacent vertices on the same face; and space diagonals, entirely in the interior of the polyhedron (except for the endpoints on the vertices). </p> <div class="mw-heading mw-heading2"><h2 id="Higher_dimensions">Higher dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=6" title="Edit section: Higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="N-Cube">N-Cube</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=7" title="Edit section: N-Cube"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The lengths of an n-dimensional <a href="/wiki/Hypercube" title="Hypercube">hypercube</a>'s diagonals can be calculated by <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>. The longest diagonal of an n-cube is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2994734eae382ce30100fb17b9447fd8e99f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}" /></span>. Additionally, there are <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 2^{n-1}{\binom {n}{x+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-1}{\binom {n}{x+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f39a305aa4fc62ccf58d14c564e7138ee90620de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.235ex; height:6.176ex;" alt="{\displaystyle 2^{n-1}{\binom {n}{x+1}}}" /></span> of the <i>xth</i> shortest diagonal. As an example, a 5-cube would have the diagonals: </p> <table rules="none" border="0" cellspacing="4" cellpadding="0" style="background:transparent;text-align:right"> <tbody><tr valign="top"> <td> <table class="wikitable"> <tbody><tr> <th>Diagonal length</th> <th>Number of diagonals </th></tr> <tr> <td><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></td> <td>160 </td></tr> <tr> <td><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span></td> <td>160 </td></tr> <tr> <td>2</td> <td>80 </td></tr> <tr> <td><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span></td> <td>16 </td></tr></tbody></table> </td></tr></tbody></table> <p>Its total number of diagonals is 416. In general, an n-cube has a total of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n-1}(2^{n}-n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-1}(2^{n}-n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed6002f09e0d34d97c27b29336b395d46c93025b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.91ex; height:3.176ex;" alt="{\displaystyle 2^{n-1}(2^{n}-n-1)}" /></span> diagonals. This follows from the more general form of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v(v-1)}{2}}-e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v(v-1)}{2}}-e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fbdbe774f9ed6807a97e24b9fefd4de44f59ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.827ex; height:5.676ex;" alt="{\displaystyle {\frac {v(v-1)}{2}}-e}" /></span> which describes the total number of face and space diagonals in convex polytopes.<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> Here, v represents the number of vertices and e represents the number of edges. </p> <div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=8" title="Edit section: Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By analogy, the <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> <i>X</i>×<i>X</i> of any set <i>X</i> with itself, consisting of all pairs <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f9e25892f6d000349b8bb6578a59567efbdd63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle (x,x)}" /></span>⁠</span>, is called the diagonal, and is the <a href="/wiki/Graph_of_a_relation" class="mw-redirect" title="Graph of a relation">graph</a> of the <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a> <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a> on <i>X</i> or equivalently the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the <a href="/wiki/Identity_function" title="Identity function">identity function</a> from <i>X</i> to <i>X</i>. This plays an important part in geometry; for example, the <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">mapping</a> <i>F</i> from <i>X</i> to itself may be obtained by intersecting the graph of <i>F</i> with the diagonal. </p><p>In geometric studies, the idea of intersecting the diagonal <i>with itself</i> is common, not directly, but by perturbing it within an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a>. This is related at a deep level with the <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> and the zeros of <a href="/wiki/Vector_field" title="Vector field">vector fields</a>. For example, the <a href="/wiki/Circle" title="Circle">circle</a> <i>S</i><sup>1</sup> has <a href="/wiki/Betti_number" title="Betti number">Betti numbers</a> 1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this is to look at the diagonal on the two-<a href="/wiki/Torus" title="Torus">torus</a> <i>S</i><sup>1</sup>×S<sup>1</sup> and observe that it can move <i>off itself</i> by the small motion (<i>θ</i>, <i>θ</i>) to (<i>θ</i>, <i>θ</i> + <i>ε</i>). In general, the intersection number of the graph of a function with the diagonal may be computed using homology via the <a href="/wiki/Lefschetz_fixed-point_theorem" title="Lefschetz fixed-point theorem">Lefschetz fixed-point theorem</a>; the self-intersection of the diagonal is the special case of the identity function. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diagonal&action=edit&section=9" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHarper2018" class="citation web cs1">Harper, Douglas R. (2018). <a rel="nofollow" class="external text" href="https://www.etymonline.com/search?q=diagonal">"diagonal (adj.)"</a>. <i><a href="/wiki/Online_Etymology_Dictionary" class="mw-redirect" title="Online Etymology Dictionary">Online Etymology Dictionary</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Online+Etymology+Dictionary&rft.atitle=diagonal+%28adj.%29&rft.date=2018&rft.aulast=Harper&rft.aufirst=Douglas+R.&rft_id=https%3A%2F%2Fwww.etymonline.com%2Fsearch%3Fq%3Ddiagonal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiagonal" 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">Strabo, Geography 2.1.36–37</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">Euclid, Elements book 11, proposition 28</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">Euclid, Elements book 11, proposition 38</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHonsberger1973" class="citation book cs1">Honsberger (1973). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalgems0001hons/page/99/?q=%22diagonals+of+a+square+divide%22">"A Problem in Combinatorics"</a></span>. <i>Mathematical Gems</i>. Mathematical Association of America. <span class="nowrap">Ch. 9</span>, pp. 99–107. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88385-301-9" title="Special:BookSources/0-88385-301-9"><bdi>0-88385-301-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+Problem+in+Combinatorics&rft.btitle=Mathematical+Gems&rft.pages=%3Cspan+class%3D%22nowrap%22%3ECh.+9%3C%2Fspan%3E%2C+pp.-99-107&rft.pub=Mathematical+Association+of+America&rft.date=1973&rft.isbn=0-88385-301-9&rft.au=Honsberger&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalgems0001hons%2Fpage%2F99%2F%3Fq%3D%2522diagonals%2Bof%2Ba%2Bsquare%2Bdivide%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiagonal" class="Z3988"></span> <div class="paragraphbreak" style="margin-top:0.5em"></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFreeman1976" class="citation journal cs1">Freeman, J. 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"The Number of Regions Determined by a Convex Polygon". <i>Mathematics Magazine</i>. <b>49</b> (1): <span class="nowrap">23–</span>25. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2689875">10.2307/2689875</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2689875">2689875</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=The+Number+of+Regions+Determined+by+a+Convex+Polygon&rft.volume=49&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E23-%3C%2Fspan%3E25&rft.date=1976&rft_id=info%3Adoi%2F10.2307%2F2689875&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2689875%23id-name%3DJSTOR&rft.aulast=Freeman&rft.aufirst=J.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiagonal" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSloane_"A006522"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A006522">"Sequence A006522"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA006522&rft_id=https%3A%2F%2Foeis.org%2FA006522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiagonal" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Poonen, Bjorn; Rubinstein, Michael. "The number of intersection points made by the diagonals of a regular polygon". <i>SIAM J. Discrete Math</i>. 11 (1998), no. 1, 135–156; <a rel="nofollow" class="external text" href="https://math.mit.edu/~poonen/papers/ngon.pdf">link to a version on Poonen's website</a> </span> </li> <li id="cite_note-youtube-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-youtube_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREF3Blue1Brown2015" class="citation audio-visual cs1">3Blue1Brown (2015-05-23). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=K8P8uFahAgc"><i>Circle Division Solution (old version)</i></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-09-01</span></span> – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Circle+Division+Solution+%28old+version%29&rft.date=2015-05-23&rft.au=3Blue1Brown&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DK8P8uFahAgc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiagonal" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_AV_media" title="Template:Cite AV media">cite AV media</a>}}</code>: CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>)</span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.themathdoctors.org/counting-diagonals-of-a-polyhedron/">"Counting Diagonals of a Polyhedron – the Math Doctors"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Counting+Diagonals+of+a+Polyhedron+%E2%80%93+the+Math+Doctors&rft_id=https%3A%2F%2Fwww.themathdoctors.org%2Fcounting-diagonals-of-a-polyhedron%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiagonal" class="Z3988"></span></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=Diagonal&action=edit&section=10" 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"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/diagonal" class="extiw" title="wiktionary:diagonal">diagonal</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/polygondiagonal.html">Diagonals of a polygon</a> with interactive animation</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PolygonDiagonal.html">Polygon diagonal</a> from <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>66 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> 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