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id="toc-Special_cases-sublist" class="vector-toc-list"> <li id="toc-Right_square_pyramid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Right_square_pyramid"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Right square pyramid</span> </div> </a> <ul id="toc-Right_square_pyramid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equilateral_square_pyramid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equilateral_square_pyramid"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Equilateral square pyramid</span> </div> </a> <ul id="toc-Equilateral_square_pyramid-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</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">4</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Works_cited" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Works_cited"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Works cited</span> </div> </a> <ul id="toc-Works_cited-sublist" class="vector-toc-list"> </ul> </li> </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">5</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">Square pyramid</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 25 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-25" 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">25 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/%D9%87%D8%B1%D9%85_%D8%B1%D8%A8%D8%A7%D8%B9%D9%8A" title="هرم رباعي – Arabic" lang="ar" hreflang="ar" data-title="هرم رباعي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B0_%D0%BF%D0%B8%D1%80%D0%B0%D0%BC%D0%B8%D0%B4%D0%B0" title="Квадратна пирамида – Bulgarian" lang="bg" hreflang="bg" data-title="Квадратна пирамида" 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/Pir%C3%A0mide_quadrada" title="Piràmide quadrada – Catalan" lang="ca" hreflang="ca" data-title="Piràmide quadrada" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Pyramid_sgw%C3%A2r" title="Pyramid sgwâr – Welsh" lang="cy" hreflang="cy" data-title="Pyramid sgwâr" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Pir%C3%A1mide_cuadrada" title="Pirámide cuadrada – Spanish" lang="es" hreflang="es" data-title="Pirámide cuadrada" 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/Kvadrata_piramido" title="Kvadrata piramido – Esperanto" lang="eo" hreflang="eo" data-title="Kvadrata piramido" 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/Piramide_karratu" title="Piramide karratu – Basque" lang="eu" hreflang="eu" data-title="Piramide karratu" 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%87%D8%B1%D9%85_%D9%85%D8%B1%D8%A8%D8%B9%E2%80%8C%D8%A7%D9%84%D9%82%D8%A7%D8%B9%D8%AF%D9%87" title="هرم مربع‌القاعده – Persian" lang="fa" hreflang="fa" data-title="هرم مربع‌القاعده" 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/Pyramide_%C3%A0_base_carr%C3%A9e" title="Pyramide à base carrée – French" lang="fr" hreflang="fr" data-title="Pyramide à base carrée" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%AC%EA%B0%81%EB%BF%94" title="사각뿔 – Korean" lang="ko" hreflang="ko" data-title="사각뿔" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Limas_persegi" title="Limas persegi – Indonesian" lang="id" hreflang="id" data-title="Limas persegi" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Piramide_quadrata" title="Piramide quadrata – Italian" lang="it" hreflang="it" data-title="Piramide quadrata" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B0_%D0%BF%D0%B8%D1%80%D0%B0%D0%BC%D0%B8%D0%B4%D0%B0" title="Квадратна пирамида – Macedonian" lang="mk" hreflang="mk" data-title="Квадратна пирамида" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vierkante_piramide" title="Vierkante piramide – Dutch" lang="nl" hreflang="nl" data-title="Vierkante piramide" 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%9B%9B%E8%A7%92%E9%8C%90" title="四角錐 – Japanese" lang="ja" hreflang="ja" data-title="四角錐" 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/Kvadratisk_pyramide" title="Kvadratisk pyramide – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvadratisk pyramide" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Pir%C3%A2mide_quadrada" title="Pirâmide quadrada – Portuguese" lang="pt" hreflang="pt" data-title="Pirâmide quadrada" 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/Piramid%C4%83_p%C4%83trat%C4%83" title="Piramidă pătrată – Romanian" lang="ro" hreflang="ro" data-title="Piramidă pătrată" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B0%D1%8F_%D0%BF%D0%B8%D1%80%D0%B0%D0%BC%D0%B8%D0%B4%D0%B0" title="Квадратная пирамида – Russian" lang="ru" hreflang="ru" data-title="Квадратная пирамида" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvadratna_piramida" title="Kvadratna piramida – Slovenian" lang="sl" hreflang="sl" data-title="Kvadratna piramida" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%A4%E0%AF%81%E0%AE%B0%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%88%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%82%E0%AE%AE%E0%AF%8D%E0%AE%AA%E0%AF%81" title="சதுரப் பட்டைக்கூம்பு – Tamil" lang="ta" hreflang="ta" data-title="சதுரப் பட்டைக்கூம்பு" 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%B8%9E%E0%B8%B5%E0%B8%A3%E0%B8%B0%E0%B8%A1%E0%B8%B4%E0%B8%94%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%88%E0%B8%B1%E0%B8%95%E0%B8%B8%E0%B8%A3%E0%B8%B1%E0%B8%AA" title="พีระมิดสี่เหลี่ยมจัตุรัส – Thai" lang="th" hreflang="th" data-title="พีระมิดสี่เหลี่ยมจัตุรัส" 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/Kare_piramit" title="Kare piramit – Turkish" lang="tr" hreflang="tr" data-title="Kare piramit" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B0_%D0%BF%D1%96%D1%80%D0%B0%D0%BC%D1%96%D0%B4%D0%B0" title="Квадратна піраміда – Ukrainian" lang="uk" hreflang="uk" data-title="Квадратна піраміда" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9B%9B%E8%A7%92%E9%8C%90" title="四角錐 – Chinese" lang="zh" hreflang="zh" data-title="四角錐" data-language-autonym="中文" data-language-local-name="Chinese" 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Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></div></div> </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">Pyramid with a square base</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">Square pyramid</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Square_pyramid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Square_pyramid.png/220px-Square_pyramid.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Square_pyramid.png/330px-Square_pyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/53/Square_pyramid.png 2x" data-file-width="420" data-file-height="259" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a>,<br /><a href="/wiki/Johnson_solid" title="Johnson solid">Johnson</a><br /><span class="texhtml"><a href="/wiki/Triangular_hebesphenorotunda" title="Triangular hebesphenorotunda"><i>J</i><sub>92</sub></a> – <b><i>J</i><sub>1</sub></b> – <a href="/wiki/Pentagonal_pyramid" title="Pentagonal pyramid"><i>J</i><sub>2</sub></a></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Face_(geometry)" title="Face (geometry)">Faces</a></th><td class="infobox-data">4 <a href="/wiki/Triangle" title="Triangle">triangles</a><br />1 <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a></th><td class="infobox-data">8</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></th><td class="infobox-data">5</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Vertex_configuration" title="Vertex configuration">Vertex configuration</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\times (3^{2}\times 4)+1\times (3^{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>×</mo> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times (3^{2}\times 4)+1\times (3^{4})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a413e03ac4eec7753d1900309aa023a202c4c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.901ex; height:3.176ex;" alt="{\displaystyle 4\times (3^{2}\times 4)+1\times (3^{4})}"></span><sup id="cite_ref-FOOTNOTEJohnson1966_1-0" class="reference"><a href="#cite_note-FOOTNOTEJohnson1966-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">Symmetry group</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{4\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4\mathrm {v} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222fdec1d220cb23bdee7f21b3b1fd0f2eb7d930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.584ex; height:2.509ex;" alt="{\displaystyle C_{4\mathrm {v} }}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Volume" title="Volume">Volume</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}l^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}l^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13279d29fccfb5fbfb073dbfa488405738cebc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.085ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}l^{2}h}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dihedral_angle" title="Dihedral angle">Dihedral angle</a> (<a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>)</th><td class="infobox-data">Equilateral square pyramid:<sup id="cite_ref-FOOTNOTEJohnson1966_1-1" class="reference"><a href="#cite_note-FOOTNOTEJohnson1966-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><br /><div><ul><li>triangle-to-triangle: 109.47°</li><li>square-to-triangle: 54.74°</li></ul></div></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dual_polyhedron" title="Dual polyhedron">Dual polyhedron</a></th><td class="infobox-data"><a href="/wiki/Self-dual_polyhedron" class="mw-redirect" title="Self-dual polyhedron">self-dual</a></td></tr><tr><th scope="row" class="infobox-label">Properties</th><td class="infobox-data"><a href="/wiki/Convex_set" title="Convex set">convex</a>,<br /><a href="/wiki/Elementary_polyhedron" class="mw-redirect" title="Elementary polyhedron">elementary</a> (equilateral square pyramid)</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e7dcc3"><a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">Net</a></th></tr><tr><td colspan="2" class="infobox-full-data"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Square_pyramid_net.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Square_pyramid_net.svg/280px-Square_pyramid_net.svg.png" decoding="async" width="280" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Square_pyramid_net.svg/420px-Square_pyramid_net.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Square_pyramid_net.svg/560px-Square_pyramid_net.svg.png 2x" data-file-width="480" data-file-height="480" /></a></span></td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>square pyramid</b> is a <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a> with a square base, having a total of five faces. If the <a href="/wiki/Apex_(geometry)" title="Apex (geometry)">apex</a> of the pyramid is directly above the center of the square, it is a <i>right square pyramid</i> with four <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangles</a>; otherwise, it is an <i>oblique square pyramid</i>. When all of the pyramid's edges are equal in length, its triangles are all <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral</a>. It is called an <i>equilateral square pyramid</i>, an example of a <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solid</a>. </p><p>Square pyramids have appeared throughout the history of architecture, with examples being <a href="/wiki/Egyptian_pyramid" class="mw-redirect" title="Egyptian pyramid">Egyptian pyramids</a> and many other similar buildings. They also occur in chemistry in <a href="/wiki/Square_pyramidal_molecular_geometry" title="Square pyramidal molecular geometry">square pyramidal molecular structures</a>. Square pyramids are often used in the construction of other <a href="/wiki/Polyhedra" class="mw-redirect" title="Polyhedra">polyhedra</a>. Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches. </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=Square_pyramid&action=edit&section=1" title="Edit section: Special cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Right_square_pyramid">Right square pyramid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_pyramid&action=edit&section=2" title="Edit section: Right square pyramid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A square pyramid has five vertices, eight edges, and five faces. One face, called the <i>base</i> of the pyramid, is a <a href="/wiki/Square" title="Square">square</a>; the four other faces are <a href="/wiki/Triangle" title="Triangle">triangles</a>.<sup id="cite_ref-FOOTNOTEClissold2020[httpsbooksgooglecombooksidXgW5DwAAQBAJpgPA180_180]_2-0" class="reference"><a href="#cite_note-FOOTNOTEClissold2020[httpsbooksgooglecombooksidXgW5DwAAQBAJpgPA180_180]-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Four of the edges make up the square by connecting its four vertices. The other four edges are known as the <a href="/wiki/Lateral_edge" class="mw-redirect" title="Lateral edge">lateral edges</a> of the pyramid; they meet at the fifth vertex, called the <a href="/wiki/Apex_(geometry)" title="Apex (geometry)">apex</a>.<sup id="cite_ref-FOOTNOTEO'KeeffeHyde2020[httpsbooksgooglecombooksid_MjPDwAAQBAJpgPA141_141]Smith2000[httpsbooksgooglecombooksidB0khWEZmOlwCpgPA98_98]_3-0" class="reference"><a href="#cite_note-FOOTNOTEO'KeeffeHyde2020[httpsbooksgooglecombooksid_MjPDwAAQBAJpgPA141_141]Smith2000[httpsbooksgooglecombooksidB0khWEZmOlwCpgPA98_98]-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> If the pyramid's apex lies on a line erected perpendicularly from the center of the square, it is called a <i>right square pyramid</i>, and the four triangular faces are <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangles</a>. Otherwise, the pyramid has two or more non-isosceles triangular faces and is called an <i>oblique square pyramid</i>.<sup id="cite_ref-FOOTNOTEFreitag2014[httpsbooksgooglecombooksidGYsWAAAAQBAJpgPA598_598]_4-0" class="reference"><a href="#cite_note-FOOTNOTEFreitag2014[httpsbooksgooglecombooksidGYsWAAAAQBAJpgPA598_598]-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The <i>slant height</i> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> of a right square pyramid is defined as the height of one of its isosceles triangles. It can be obtained via the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\sqrt {b^{2}-{\frac {l^{2}}{4}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\sqrt {b^{2}-{\frac {l^{2}}{4}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3404d16cf426ed0df735c0f5c4625813d8faf23e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.635ex; height:6.176ex;" alt="{\displaystyle s={\sqrt {b^{2}-{\frac {l^{2}}{4}}}},}"></span> 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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> is the length of the triangle's base, also one of the square's edges, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is the length of the triangle's legs, which are lateral edges of the pyramid.<sup id="cite_ref-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA177_177]PerryPerry1981[httpsbooksgooglecombooksidDi2uCwAAQBAJpgPA145_145–146]_5-0" class="reference"><a href="#cite_note-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA177_177]PerryPerry1981[httpsbooksgooglecombooksidDi2uCwAAQBAJpgPA145_145–146]-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The height <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> of a right square pyramid can be similarly obtained, with a substitution of the slant height formula giving:<sup id="cite_ref-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA177_177]_6-0" class="reference"><a href="#cite_note-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA177_177]-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\sqrt {s^{2}-{\frac {l^{2}}{4}}}}={\sqrt {b^{2}-{\frac {l^{2}}{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\sqrt {s^{2}-{\frac {l^{2}}{4}}}}={\sqrt {b^{2}-{\frac {l^{2}}{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e6c3e35e9135506b93558b63d72c95c405d554" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.875ex; height:6.176ex;" alt="{\displaystyle h={\sqrt {s^{2}-{\frac {l^{2}}{4}}}}={\sqrt {b^{2}-{\frac {l^{2}}{2}}}}.}"></span> A <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a>'s <a href="/wiki/Surface_area" title="Surface area">surface area</a> is the sum of the areas of its faces. The surface area <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> of a right square pyramid can be expressed as <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=4T+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>T</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4T+S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f95847a111ede2b9a4a15196dda083a281cfbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.98ex; height:2.343ex;" alt="{\displaystyle A=4T+S}"></span>, 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> are the areas of one of its triangles and its base, respectively. The area of a triangle is half of the product of its base and side, with the area of a square being the length of the side squared. This gives the expression:<sup id="cite_ref-FOOTNOTEFreitag2014[httpsbooksgooglecombooksidGYsWAAAAQBAJpgPA798_798]_7-0" class="reference"><a href="#cite_note-FOOTNOTEFreitag2014[httpsbooksgooglecombooksidGYsWAAAAQBAJpgPA798_798]-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=4\left({\frac {1}{2}}ls\right)+l^{2}=2ls+l^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>l</mi> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>l</mi> <mi>s</mi> <mo>+</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\left({\frac {1}{2}}ls\right)+l^{2}=2ls+l^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70aa226353e6ccc810358559ce1a4285b8d62274" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.462ex; height:6.176ex;" alt="{\displaystyle A=4\left({\frac {1}{2}}ls\right)+l^{2}=2ls+l^{2}.}"></span> In general, the volume <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> of a pyramid is equal to one-third of the area of its base multiplied by its height.<sup id="cite_ref-FOOTNOTEAlexanderKoeberlin2014[httpsbooksgooglecombooksidEN_KAgAAQBAJpgPA403_403]_8-0" class="reference"><a href="#cite_note-FOOTNOTEAlexanderKoeberlin2014[httpsbooksgooglecombooksidEN_KAgAAQBAJpgPA403_403]-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Expressed in a formula for a square pyramid, this is:<sup id="cite_ref-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA178_178]_9-0" class="reference"><a href="#cite_note-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA178_178]-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {1}{3}}l^{2}h.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{3}}l^{2}h.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf58854f5bfcf24294e4ab2365bbe048ad5bba1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.618ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{3}}l^{2}h.}"></span> </p><p>Many mathematicians have discovered the formula for calculating the volume of a square pyramid in ancient times. In the <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a>, Egyptian mathematicians demonstrated knowledge of the formula for calculating the volume of a <a href="/wiki/Frustum" title="Frustum">truncated square pyramid</a>, suggesting that they were also acquainted with the volume of a square pyramid, but it is unknown how the formula was derived. Beyond the discovery of the volume of a square pyramid, the problem of finding the slope and height of a square pyramid can be found in the <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a>.<sup id="cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage20mode2upviewtheater_20–22]_10-0" class="reference"><a href="#cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage20mode2upviewtheater_20–22]-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The Babylonian mathematicians also considered the volume of a frustum, but gave an incorrect formula for it.<sup id="cite_ref-FOOTNOTEEves1997[httpsbooksgooglecombooksidJ9QcmFHj8EwCpgPA2_2]_11-0" class="reference"><a href="#cite_note-FOOTNOTEEves1997[httpsbooksgooglecombooksidJ9QcmFHj8EwCpgPA2_2]-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> One Chinese mathematician <a href="/wiki/Liu_Hui" title="Liu Hui">Liu Hui</a> also discovered the volume by the method of dissecting a rectangular solid into pieces.<sup id="cite_ref-FOOTNOTEWagner1979_12-0" class="reference"><a href="#cite_note-FOOTNOTEWagner1979-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Equilateral_square_pyramid">Equilateral square pyramid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_pyramid&action=edit&section=3" title="Edit section: Equilateral square pyramid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><span class="mw-3d-wrapper" data-label="3D"><a href="/wiki/File:J1_square_pyramid.stl" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/J1_square_pyramid.stl/220px-J1_square_pyramid.stl.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/J1_square_pyramid.stl/330px-J1_square_pyramid.stl.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/J1_square_pyramid.stl/440px-J1_square_pyramid.stl.png 2x" data-file-width="5120" data-file-height="2880" /></a></span><figcaption>3D model of an equilateral square pyramid</figcaption></figure> <p><span class="anchor" id="Equilateral_square_pyramid"></span>If all triangular edges are of equal length, the four triangles are <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral</a>, and the pyramid's faces are all <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a>, it is an <i>equilateral square pyramid.</i><sup id="cite_ref-FOOTNOTEHocevar1903[httpsbooksgooglecombooksid0OAXAAAAYAAJpgPA44_44]_13-0" class="reference"><a href="#cite_note-FOOTNOTEHocevar1903[httpsbooksgooglecombooksid0OAXAAAAYAAJpgPA44_44]-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angles</a> between adjacent triangular faces 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="{\textstyle \arccos \left(-1/3\right)\approx 109.47^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <msup> <mn>109.47</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \arccos \left(-1/3\right)\approx 109.47^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a36f84b936257e22b35ce99190d3a3920ee45a64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.934ex; height:2.843ex;" alt="{\textstyle \arccos \left(-1/3\right)\approx 109.47^{\circ }}"></span>, and that between the base and each triangular face being half of that, <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="{\textstyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <msup> <mn>54.74</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/033a01423b57ef3378420e6c1a267bc500774a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.144ex; height:3.343ex;" alt="{\textstyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}"></span>.<sup id="cite_ref-FOOTNOTEJohnson1966_1-2" class="reference"><a href="#cite_note-FOOTNOTEJohnson1966-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A <a href="/wiki/Convex_set" title="Convex set">convex</a> polyhedron in which all of the faces are <a href="/wiki/Regular_polygons" class="mw-redirect" title="Regular polygons">regular polygons</a> is called a <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solid</a>. The equilateral square pyramid is among them, enumerated as the first Johnson solid <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 J_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260ffe7da7c858cf114ad89a6c794944ea4e760f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{1}}"></span>.<sup id="cite_ref-FOOTNOTEUehara2020[httpsbooksgooglecombooksid51juDwAAQBAJpgPA62_62]_14-0" class="reference"><a href="#cite_note-FOOTNOTEUehara2020[httpsbooksgooglecombooksid51juDwAAQBAJpgPA62_62]-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Because its edges are all equal in length (that 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 b=l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24332969b83e377a59de965866851796737c9d1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.789ex; height:2.176ex;" alt="{\displaystyle b=l}"></span>), its slant, height, surface area, and volume can be derived by substituting the formulas of a right square pyramid:<sup id="cite_ref-FOOTNOTESimonson2011[httpsbooksgooglecombooksidWs6-DwAAQBAJpgPA123_123]Berman1971see_table_IV,_line_21_15-0" class="reference"><a href="#cite_note-FOOTNOTESimonson2011[httpsbooksgooglecombooksidWs6-DwAAQBAJpgPA123_123]Berman1971see_table_IV,_line_21-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s={\frac {\sqrt {3}}{2}}l\approx 0.866l,&\qquad h={\frac {1}{\sqrt {2}}}l\approx 0.707l,\\A=(1+{\sqrt {3}})l^{2}\approx 2.732l^{2},&\qquad V={\frac {\sqrt {2}}{6}}l^{3}\approx 0.236l^{3}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mi>l</mi> <mo>≈</mo> <mn>0.866</mn> <mi>l</mi> <mo>,</mo> </mtd> <mtd> <mspace width="2em" /> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mi>l</mi> <mo>≈</mo> <mn>0.707</mn> <mi>l</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mn>2.732</mn> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mspace width="2em" /> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>6</mn> </mfrac> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>≈</mo> <mn>0.236</mn> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s={\frac {\sqrt {3}}{2}}l\approx 0.866l,&\qquad h={\frac {1}{\sqrt {2}}}l\approx 0.707l,\\A=(1+{\sqrt {3}})l^{2}\approx 2.732l^{2},&\qquad V={\frac {\sqrt {2}}{6}}l^{3}\approx 0.236l^{3}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312bb79cd287dbb01e5559b754492d8a43303336" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:53.043ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}s={\frac {\sqrt {3}}{2}}l\approx 0.866l,&\qquad h={\frac {1}{\sqrt {2}}}l\approx 0.707l,\\A=(1+{\sqrt {3}})l^{2}\approx 2.732l^{2},&\qquad V={\frac {\sqrt {2}}{6}}l^{3}\approx 0.236l^{3}.\end{aligned}}}"></span> </p><p>Like other right pyramids with a regular polygon as a base, a right square pyramid has <a href="/wiki/Pyramidal_symmetry" class="mw-redirect" title="Pyramidal symmetry">pyramidal symmetry</a>. For the square pyramid, this is the symmetry of <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</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 C_{4\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4\mathrm {v} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222fdec1d220cb23bdee7f21b3b1fd0f2eb7d930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.584ex; height:2.509ex;" alt="{\displaystyle C_{4\mathrm {v} }}"></span>: the pyramid is left invariant by rotations of one-, two-, and three-quarters of a full turn around its <a href="/wiki/Axis_of_symmetry" class="mw-redirect" title="Axis of symmetry">axis of symmetry</a>, the line connecting the apex to the center of the base; and is also <a href="/wiki/Mirror_symmetric" class="mw-redirect" title="Mirror symmetric">mirror symmetric</a> relative to any perpendicular plane passing through a bisector of the base.<sup id="cite_ref-FOOTNOTEJohnson1966_1-3" class="reference"><a href="#cite_note-FOOTNOTEJohnson1966-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It can be represented as the <a href="/wiki/Wheel_graph" title="Wheel graph">wheel graph</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 W_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a976c59bf91696c4af8023b529364d4d9e2dc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{4}}"></span>, meaning its <a href="/wiki/Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">skeleton</a> can be interpreted as a square in which its four vertices connects a vertex in the center called the <a href="/wiki/Universal_vertex" title="Universal vertex">universal vertex</a>.<sup id="cite_ref-FOOTNOTEPisanskiServatius2013[httpsbooksgooglecombooksid3vnEcMCx0HkCpgPA21_21]_16-0" class="reference"><a href="#cite_note-FOOTNOTEPisanskiServatius2013[httpsbooksgooglecombooksid3vnEcMCx0HkCpgPA21_21]-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> It is <a href="/wiki/Self-dual_polyhedron" class="mw-redirect" title="Self-dual polyhedron">self-dual</a>, meaning its <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> is the square pyramid itself.<sup id="cite_ref-FOOTNOTEWohlleben2019[httpsbooksgooglecombooksidrEpjDwAAQBAJpgPA485_485–486]_17-0" class="reference"><a href="#cite_note-FOOTNOTEWohlleben2019[httpsbooksgooglecombooksidrEpjDwAAQBAJpgPA485_485–486]-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>An equilateral square pyramid is an <a href="/wiki/Elementary_polyhedron" class="mw-redirect" title="Elementary polyhedron">elementary polyhedron</a>. This means it cannot be separated by a plane to create two small convex polyhedrons with regular faces.<sup id="cite_ref-FOOTNOTEHartshorne2000[httpsbooksgooglecombooksidEJCSL9S6la0CpgPA464_464]Johnson1966_18-0" class="reference"><a href="#cite_note-FOOTNOTEHartshorne2000[httpsbooksgooglecombooksidEJCSL9S6la0CpgPA464_464]Johnson1966-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_pyramid&action=edit&section=4" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:206px;max-width:206px"><div class="thumbimage" style="height:135px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:All_Gizah_Pyramids.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/All_Gizah_Pyramids.jpg/204px-All_Gizah_Pyramids.jpg" decoding="async" width="204" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/All_Gizah_Pyramids.jpg/306px-All_Gizah_Pyramids.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/All_Gizah_Pyramids.jpg/408px-All_Gizah_Pyramids.jpg 2x" data-file-width="4372" data-file-height="2906" /></a></span></div><div class="thumbcaption">The <a href="/wiki/Egyptian_pyramids" title="Egyptian pyramids">Egyptian pyramids</a> are examples of square pyramidal buildings in architecture.</div></div><div class="tsingle" style="width:182px;max-width:182px"><div class="thumbimage" style="height:135px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Piramide_Chichen-Itza_-_panoramio_(2).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Piramide_Chichen-Itza_-_panoramio_%282%29.jpg/180px-Piramide_Chichen-Itza_-_panoramio_%282%29.jpg" decoding="async" width="180" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Piramide_Chichen-Itza_-_panoramio_%282%29.jpg/270px-Piramide_Chichen-Itza_-_panoramio_%282%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Piramide_Chichen-Itza_-_panoramio_%282%29.jpg/360px-Piramide_Chichen-Itza_-_panoramio_%282%29.jpg 2x" data-file-width="4608" data-file-height="3456" /></a></span></div><div class="thumbcaption">One of the <a href="/wiki/Mesoamerican_pyramids" title="Mesoamerican pyramids">Mesoamerican pyramids</a>, a similar building to the Egyptian, has flat tops and stairs at the faces</div></div></div></div></div> <p>In architecture, the <a href="/wiki/Egyptian_pyramids" title="Egyptian pyramids">pyramids built in ancient Egypt</a> are examples of buildings shaped like square pyramids.<sup id="cite_ref-FOOTNOTEKinseyMoorePrassidis2011[httpsbooksgooglecombooksidfFpuDwAAQBAJpgRA1-PA371_371]_19-0" class="reference"><a href="#cite_note-FOOTNOTEKinseyMoorePrassidis2011[httpsbooksgooglecombooksidfFpuDwAAQBAJpgRA1-PA371_371]-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Pyramidology" title="Pyramidology">Pyramidologists</a> have put forward various suggestions for the design of the <a href="/wiki/Great_Pyramid_of_Giza" title="Great Pyramid of Giza">Great Pyramid of Giza</a>, including a theory based on the <a href="/wiki/Kepler_triangle" title="Kepler triangle">Kepler triangle</a> and the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>. However, modern scholars favor descriptions using integer ratios, as being more consistent with the knowledge of Egyptian mathematics and proportion.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Mesoamerican_pyramids" title="Mesoamerican pyramids">Mesoamerican pyramids</a> are also ancient pyramidal buildings similar to the Egyptian; they differ in having flat tops and stairs ascending their faces.<sup id="cite_ref-FOOTNOTEFeder2010[httpsbooksgooglecombooksidRlRz2symkAsCpgPA34_34]TakacsCline2015[httpsbooksgooglecombooksidSPcvCgAAQBAJpgPA16_16]_21-0" class="reference"><a href="#cite_note-FOOTNOTEFeder2010[httpsbooksgooglecombooksidRlRz2symkAsCpgPA34_34]TakacsCline2015[httpsbooksgooglecombooksidSPcvCgAAQBAJpgPA16_16]-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Modern buildings whose designs imitate the Egyptian pyramids include the <a href="/wiki/Louvre_Pyramid" title="Louvre Pyramid">Louvre Pyramid</a> and the casino hotel <a href="/wiki/Luxor_Las_Vegas" title="Luxor Las Vegas">Luxor Las Vegas</a>.<sup id="cite_ref-FOOTNOTEJarvisNaested2012[httpsbooksgooglecombooksidNWzsz8vioZwCpgPA172_172]Simonson2011[httpsbooksgooglecombooksidWs6-DwAAQBAJpgPA122_122]_22-0" class="reference"><a href="#cite_note-FOOTNOTEJarvisNaested2012[httpsbooksgooglecombooksidNWzsz8vioZwCpgPA172_172]Simonson2011[httpsbooksgooglecombooksidWs6-DwAAQBAJpgPA122_122]-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Stereochemistry" title="Stereochemistry">stereochemistry</a>, an <a href="/wiki/Atom_cluster" class="mw-redirect" title="Atom cluster">atom cluster</a> can have a <a href="/wiki/Square_pyramidal_molecular_geometry" title="Square pyramidal molecular geometry">square pyramidal geometry</a>. A square pyramidal molecule has a <a href="/wiki/Main-group_element" title="Main-group element">main-group element</a> with one active <a href="/wiki/Lone_pair" title="Lone pair">lone pair</a>, which can be described by a model that predicts the geometry of molecules known as <a href="/wiki/VSEPR_theory" title="VSEPR theory">VSEPR theory</a>.<sup id="cite_ref-FOOTNOTEPetrucciHarwoodHerring2002[httpsbooksgooglecombooksidEZEoAAAAYAAJpgPA414_414]_23-0" class="reference"><a href="#cite_note-FOOTNOTEPetrucciHarwoodHerring2002[httpsbooksgooglecombooksidEZEoAAAAYAAJpgPA414_414]-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Examples of molecules with this structure include <a href="/wiki/Chlorine_pentafluoride" title="Chlorine pentafluoride">chlorine pentafluoride</a>, <a href="/wiki/Bromine_pentafluoride" title="Bromine pentafluoride">bromine pentafluoride</a>, and <a href="/wiki/Iodine_pentafluoride" title="Iodine pentafluoride">iodine pentafluoride</a>.<sup id="cite_ref-FOOTNOTEEmeléus1969[httpsbooksgooglecombooksid9SkSBQAAQBAJpgPA13_13]_24-0" class="reference"><a href="#cite_note-FOOTNOTEEmeléus1969[httpsbooksgooglecombooksid9SkSBQAAQBAJpgPA13_13]-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Tetrakishexahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/150px-Tetrakishexahedron.jpg" decoding="async" width="150" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/225px-Tetrakishexahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/300px-Tetrakishexahedron.jpg 2x" data-file-width="767" data-file-height="737" /></a><figcaption><a href="/wiki/Tetrakis_hexahedra" class="mw-redirect" title="Tetrakis hexahedra">Tetrakis hexahedra</a>, a construction of polyhedra by augmentation involving square pyramids</figcaption></figure> <p>The base of a square pyramid can be attached to a square face of another polyhedron to construct new polyhedra, an example of <a href="/wiki/Augmentation_(geometry)" class="mw-redirect" title="Augmentation (geometry)">augmentation</a>. For example, a <a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a> can be constructed by attaching the base of an equilateral square pyramid onto each face of a cube.<sup id="cite_ref-FOOTNOTEDemeySmessaert2017_25-0" class="reference"><a href="#cite_note-FOOTNOTEDemeySmessaert2017-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Attaching <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a> or <a href="/wiki/Antiprisms" class="mw-redirect" title="Antiprisms">antiprisms</a> to pyramids is known as <a href="/wiki/Elongation_(geometry)" class="mw-redirect" title="Elongation (geometry)">elongation</a> or <a href="/wiki/Gyroelongation" class="mw-redirect" title="Gyroelongation">gyroelongation</a>, respectively.<sup id="cite_ref-FOOTNOTESlobodanObradovićÐukanović2015_26-0" class="reference"><a href="#cite_note-FOOTNOTESlobodanObradovićÐukanović2015-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Some of the other Johnson solids can be constructed by either augmenting square pyramids or augmenting other shapes with square pyramids: <a href="/wiki/Elongated_square_pyramid" title="Elongated square pyramid">elongated square pyramid</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 J_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb88fe79a7d9957cc72f045df791174dc8ff750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{8}}"></span>, <a href="/wiki/Gyroelongated_square_pyramid" title="Gyroelongated square pyramid">gyroelongated square pyramid</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 J_{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/017c2f3e393e343e0ad65bf0628e83d3dbc4a3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{10}}"></span>, <a href="/wiki/Elongated_square_bipyramid" title="Elongated square bipyramid">elongated square bipyramid</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 J_{15}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{15}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1ea8bf006fbdf1b74b14bf00c1c3df24460f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{15}}"></span>, <a href="/wiki/Gyroelongated_square_bipyramid" title="Gyroelongated square bipyramid">gyroelongated square bipyramid</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 J_{17}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{17}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1624a94a0f0901572cf3efe9e4011864a6cf459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{17}}"></span>, <a href="/wiki/Augmented_triangular_prism" title="Augmented triangular prism">augmented triangular prism</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 J_{49}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>49</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{49}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d605933a376de528e8626b530664f253fade9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{49}}"></span>, <a href="/wiki/Biaugmented_triangular_prism" title="Biaugmented triangular prism">biaugmented triangular prism</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 J_{50}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{50}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b0dcc5ad0a20d3893012d6a7da77e5cfc48386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{50}}"></span>, <a href="/wiki/Triaugmented_triangular_prism" title="Triaugmented triangular prism">triaugmented triangular prism</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 J_{51}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>51</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{51}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08681f4eb2c120b643cea750a6b9f0acb0250855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{51}}"></span>, <a href="/wiki/Augmented_pentagonal_prism" title="Augmented pentagonal prism">augmented pentagonal prism</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 J_{52}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>52</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{52}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79d45f3a381ee0a3af3e0b9b53d47749ddb212e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{52}}"></span>, <a href="/wiki/Biaugmented_pentagonal_prism" title="Biaugmented pentagonal prism">biaugmented pentagonal prism</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 J_{53}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>53</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{53}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea47d8c0b1e0e63aef3f8d834b292f738b70dc39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{53}}"></span>, <a href="/wiki/Augmented_hexagonal_prism" title="Augmented hexagonal prism">augmented hexagonal prism</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 J_{54}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>54</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{54}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/216ef107eb10997ee46f794c4a3e48872552473a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{54}}"></span>, <a href="/wiki/Parabiaugmented_hexagonal_prism" title="Parabiaugmented hexagonal prism">parabiaugmented hexagonal prism</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 J_{55}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{55}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c61bf76783ccb801808945aa65fb29ddedeac1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{55}}"></span>, <a href="/wiki/Metabiaugmented_hexagonal_prism" title="Metabiaugmented hexagonal prism">metabiaugmented hexagonal prism</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 J_{56}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{56}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7cd68af48e04627e1cdeab27ce665ce88e573f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{56}}"></span>, <a href="/wiki/Triaugmented_hexagonal_prism" title="Triaugmented hexagonal prism">triaugmented hexagonal prism</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 J_{57}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>57</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{57}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0006730a285589d055e642ce093c4abc1542a51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{57}}"></span>, and <a href="/wiki/Augmented_sphenocorona" title="Augmented sphenocorona">augmented sphenocorona</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 J_{87}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>87</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{87}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff84fbf46da4f5b03fd603797431c2186ff6d7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{87}}"></span>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </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=Square_pyramid&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal number</a>, a natural number that counts the number of stacked spheres in a square pyramid.</li></ul> <div style="clear:both;" class=""></div> <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=Square_pyramid&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_pyramid&action=edit&section=7" 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 reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-FOOTNOTEJohnson1966-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEJohnson1966_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEJohnson1966_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEJohnson1966_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEJohnson1966_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFJohnson1966">Johnson (1966)</a>.</span> </li> <li id="cite_note-FOOTNOTEClissold2020[httpsbooksgooglecombooksidXgW5DwAAQBAJpgPA180_180]-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEClissold2020[httpsbooksgooglecombooksidXgW5DwAAQBAJpgPA180_180]_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFClissold2020">Clissold (2020)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XgW5DwAAQBAJ&pg=PA180">180</a>.</span> </li> <li id="cite_note-FOOTNOTEO'KeeffeHyde2020[httpsbooksgooglecombooksid_MjPDwAAQBAJpgPA141_141]Smith2000[httpsbooksgooglecombooksidB0khWEZmOlwCpgPA98_98]-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEO'KeeffeHyde2020[httpsbooksgooglecombooksid_MjPDwAAQBAJpgPA141_141]Smith2000[httpsbooksgooglecombooksidB0khWEZmOlwCpgPA98_98]_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFO'KeeffeHyde2020">O'Keeffe & Hyde (2020)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA141">141</a>; <a href="#CITEREFSmith2000">Smith (2000)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B0khWEZmOlwC&pg=PA98">98</a>.</span> </li> <li id="cite_note-FOOTNOTEFreitag2014[httpsbooksgooglecombooksidGYsWAAAAQBAJpgPA598_598]-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFreitag2014[httpsbooksgooglecombooksidGYsWAAAAQBAJpgPA598_598]_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFreitag2014">Freitag (2014)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GYsWAAAAQBAJ&pg=PA598">598</a>.</span> </li> <li id="cite_note-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA177_177]PerryPerry1981[httpsbooksgooglecombooksidDi2uCwAAQBAJpgPA145_145–146]-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA177_177]PerryPerry1981[httpsbooksgooglecombooksidDi2uCwAAQBAJpgPA145_145–146]_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLarcombe1929">Larcombe (1929)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SAE9AAAAIAAJ&pg=PA177">177</a>; <a href="#CITEREFPerryPerry1981">Perry & Perry (1981)</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Di2uCwAAQBAJ&pg=PA145">145–146</a>.</span> </li> <li id="cite_note-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA177_177]-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA177_177]_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLarcombe1929">Larcombe (1929)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SAE9AAAAIAAJ&pg=PA177">177</a>.</span> </li> <li id="cite_note-FOOTNOTEFreitag2014[httpsbooksgooglecombooksidGYsWAAAAQBAJpgPA798_798]-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFreitag2014[httpsbooksgooglecombooksidGYsWAAAAQBAJpgPA798_798]_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFreitag2014">Freitag (2014)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GYsWAAAAQBAJ&pg=PA798">798</a>.</span> </li> <li id="cite_note-FOOTNOTEAlexanderKoeberlin2014[httpsbooksgooglecombooksidEN_KAgAAQBAJpgPA403_403]-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAlexanderKoeberlin2014[httpsbooksgooglecombooksidEN_KAgAAQBAJpgPA403_403]_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlexanderKoeberlin2014">Alexander & Koeberlin (2014)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EN_KAgAAQBAJ&pg=PA403">403</a>.</span> </li> <li id="cite_note-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA178_178]-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELarcombe1929[httpsbooksgooglecombooksidSAE9AAAAIAAJpgPA178_178]_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLarcombe1929">Larcombe (1929)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SAE9AAAAIAAJ&pg=PA178">178</a>.</span> </li> <li id="cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage20mode2upviewtheater_20–22]-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage20mode2upviewtheater_20–22]_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCromwell1997">Cromwell (1997)</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom/page/20/mode/2up?view=theater">20–22</a>.</span> </li> <li id="cite_note-FOOTNOTEEves1997[httpsbooksgooglecombooksidJ9QcmFHj8EwCpgPA2_2]-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEves1997[httpsbooksgooglecombooksidJ9QcmFHj8EwCpgPA2_2]_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEves1997">Eves (1997)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J9QcmFHj8EwC&pg=PA2">2</a>.</span> </li> <li id="cite_note-FOOTNOTEWagner1979-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWagner1979_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWagner1979">Wagner (1979)</a>.</span> </li> <li id="cite_note-FOOTNOTEHocevar1903[httpsbooksgooglecombooksid0OAXAAAAYAAJpgPA44_44]-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHocevar1903[httpsbooksgooglecombooksid0OAXAAAAYAAJpgPA44_44]_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHocevar1903">Hocevar (1903)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0OAXAAAAYAAJ&pg=PA44">44</a>.</span> </li> <li id="cite_note-FOOTNOTEUehara2020[httpsbooksgooglecombooksid51juDwAAQBAJpgPA62_62]-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEUehara2020[httpsbooksgooglecombooksid51juDwAAQBAJpgPA62_62]_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFUehara2020">Uehara (2020)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62">62</a>.</span> </li> <li id="cite_note-FOOTNOTESimonson2011[httpsbooksgooglecombooksidWs6-DwAAQBAJpgPA123_123]Berman1971see_table_IV,_line_21-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESimonson2011[httpsbooksgooglecombooksidWs6-DwAAQBAJpgPA123_123]Berman1971see_table_IV,_line_21_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSimonson2011">Simonson (2011)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ws6-DwAAQBAJ&pg=PA123">123</a>; <a href="#CITEREFBerman1971">Berman (1971)</a>, see table IV, line 21.</span> </li> <li id="cite_note-FOOTNOTEPisanskiServatius2013[httpsbooksgooglecombooksid3vnEcMCx0HkCpgPA21_21]-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPisanskiServatius2013[httpsbooksgooglecombooksid3vnEcMCx0HkCpgPA21_21]_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPisanskiServatius2013">Pisanski & Servatius (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21">21</a>.</span> </li> <li id="cite_note-FOOTNOTEWohlleben2019[httpsbooksgooglecombooksidrEpjDwAAQBAJpgPA485_485–486]-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWohlleben2019[httpsbooksgooglecombooksidrEpjDwAAQBAJpgPA485_485–486]_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWohlleben2019">Wohlleben (2019)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rEpjDwAAQBAJ&pg=PA485">485–486</a>.</span> </li> <li id="cite_note-FOOTNOTEHartshorne2000[httpsbooksgooglecombooksidEJCSL9S6la0CpgPA464_464]Johnson1966-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHartshorne2000[httpsbooksgooglecombooksidEJCSL9S6la0CpgPA464_464]Johnson1966_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHartshorne2000">Hartshorne (2000)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EJCSL9S6la0C&pg=PA464">464</a>; <a href="#CITEREFJohnson1966">Johnson (1966)</a>.</span> </li> <li id="cite_note-FOOTNOTEKinseyMoorePrassidis2011[httpsbooksgooglecombooksidfFpuDwAAQBAJpgRA1-PA371_371]-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKinseyMoorePrassidis2011[httpsbooksgooglecombooksidfFpuDwAAQBAJpgRA1-PA371_371]_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKinseyMoorePrassidis2011">Kinsey, Moore & Prassidis (2011)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fFpuDwAAQBAJ&pg=RA1-PA371">371</a>.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFHerz-Fischler2000">Herz-Fischler (2000)</a> surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle. See <a href="#CITEREFRossi2004">Rossi (2004)</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/architechture-and-mathematics-in-ancient-egypt-corianna-rossi-2003/page/67/">67–68</a>, quoting that "there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence 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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> itself as a number, do not fit with the extant Middle Kingdom mathematical sources"; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56. See also <a href="#CITEREFRossiTout2002">Rossi & Tout (2002)</a> and <a href="#CITEREFMarkowsky1992">Markowsky (1992)</a>.</span> </li> <li id="cite_note-FOOTNOTEFeder2010[httpsbooksgooglecombooksidRlRz2symkAsCpgPA34_34]TakacsCline2015[httpsbooksgooglecombooksidSPcvCgAAQBAJpgPA16_16]-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFeder2010[httpsbooksgooglecombooksidRlRz2symkAsCpgPA34_34]TakacsCline2015[httpsbooksgooglecombooksidSPcvCgAAQBAJpgPA16_16]_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFeder2010">Feder (2010)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RlRz2symkAsC&pg=PA34">34</a>; <a href="#CITEREFTakacsCline2015">Takacs & Cline (2015)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SPcvCgAAQBAJ&pg=PA16">16</a>.</span> </li> <li id="cite_note-FOOTNOTEJarvisNaested2012[httpsbooksgooglecombooksidNWzsz8vioZwCpgPA172_172]Simonson2011[httpsbooksgooglecombooksidWs6-DwAAQBAJpgPA122_122]-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJarvisNaested2012[httpsbooksgooglecombooksidNWzsz8vioZwCpgPA172_172]Simonson2011[httpsbooksgooglecombooksidWs6-DwAAQBAJpgPA122_122]_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJarvisNaested2012">Jarvis & Naested (2012)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NWzsz8vioZwC&pg=PA172">172</a>; <a href="#CITEREFSimonson2011">Simonson (2011)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ws6-DwAAQBAJ&pg=PA122">122</a>.</span> </li> <li id="cite_note-FOOTNOTEPetrucciHarwoodHerring2002[httpsbooksgooglecombooksidEZEoAAAAYAAJpgPA414_414]-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPetrucciHarwoodHerring2002[httpsbooksgooglecombooksidEZEoAAAAYAAJpgPA414_414]_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPetrucciHarwoodHerring2002">Petrucci, Harwood & Herring (2002)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EZEoAAAAYAAJ&pg=PA414">414</a>.</span> </li> <li id="cite_note-FOOTNOTEEmeléus1969[httpsbooksgooglecombooksid9SkSBQAAQBAJpgPA13_13]-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEmeléus1969[httpsbooksgooglecombooksid9SkSBQAAQBAJpgPA13_13]_24-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEmeléus1969">Emeléus (1969)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9SkSBQAAQBAJ&pg=PA13">13</a>.</span> </li> <li id="cite_note-FOOTNOTEDemeySmessaert2017-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDemeySmessaert2017_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDemeySmessaert2017">Demey & Smessaert (2017)</a>.</span> </li> <li id="cite_note-FOOTNOTESlobodanObradovićÐukanović2015-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESlobodanObradovićÐukanović2015_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSlobodanObradovićÐukanović2015">Slobodan, Obradović & Ðukanović (2015)</a>.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><a href="#CITEREFRajwade2001">Rajwade (2001)</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84">84–89</a>. See Table 12.3, 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 P_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5949c8b1de44005a1af3a11188361f2a830842d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.711ex; height:2.509ex;" alt="{\displaystyle P_{n}}"></span> denotes the <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-sided</span> prism and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"></span> denotes the <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-sided</span> antiprism.</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_pyramid&action=edit&section=8" title="Edit section: Works cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><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="CITEREFAlexanderKoeberlin2014" class="citation book cs1">Alexander, Daniel C.; Koeberlin, Geralyn M. (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EN_KAgAAQBAJ"><i>Elementary Geometry for College Students</i></a> (6th ed.). Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-285-19569-8" title="Special:BookSources/978-1-285-19569-8"><bdi>978-1-285-19569-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Geometry+for+College+Students&rft.edition=6th&rft.pub=Cengage+Learning&rft.date=2014&rft.isbn=978-1-285-19569-8&rft.aulast=Alexander&rft.aufirst=Daniel+C.&rft.au=Koeberlin%2C+Geralyn+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEN_KAgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerman1971" class="citation journal cs1">Berman, Martin (1971). "Regular-faced convex polyhedra". <i>Journal of the Franklin Institute</i>. <b>291</b> (5): 329–352. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0016-0032%2871%2990071-8">10.1016/0016-0032(71)90071-8</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0290245">0290245</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Franklin+Institute&rft.atitle=Regular-faced+convex+polyhedra&rft.volume=291&rft.issue=5&rft.pages=329-352&rft.date=1971&rft_id=info%3Adoi%2F10.1016%2F0016-0032%2871%2990071-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D290245%23id-name%3DMR&rft.aulast=Berman&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClissold2020" class="citation book cs1">Clissold, Caroline (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XgW5DwAAQBAJ"><i>Maths 5–11: A Guide for Teachers</i></a>. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-26907-3" title="Special:BookSources/978-0-429-26907-3"><bdi>978-0-429-26907-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Maths+5%E2%80%9311%3A+A+Guide+for+Teachers&rft.pub=Taylor+%26+Francis&rft.date=2020&rft.isbn=978-0-429-26907-3&rft.aulast=Clissold&rft.aufirst=Caroline&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXgW5DwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCromwell1997" class="citation book cs1">Cromwell, Peter R. (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom"><i>Polyhedra</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55432-9" title="Special:BookSources/978-0-521-55432-9"><bdi>978-0-521-55432-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyhedra&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=978-0-521-55432-9&rft.aulast=Cromwell&rft.aufirst=Peter+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpolyhedra0000crom&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDemeySmessaert2017" class="citation journal cs1">Demey, Lorenz; Smessaert, Hans (2017). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsym9100204">"Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation"</a>. <i>Symmetry</i>. <b>9</b> (10): 204. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017Symm....9..204D">2017Symm....9..204D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsym9100204">10.3390/sym9100204</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Symmetry&rft.atitle=Logical+and+Geometrical+Distance+in+Polyhedral+Aristotelian+Diagrams+in+Knowledge+Representation&rft.volume=9&rft.issue=10&rft.pages=204&rft.date=2017&rft_id=info%3Adoi%2F10.3390%2Fsym9100204&rft_id=info%3Abibcode%2F2017Symm....9..204D&rft.aulast=Demey&rft.aufirst=Lorenz&rft.au=Smessaert%2C+Hans&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fsym9100204&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmeléus1969" class="citation book cs1"><a href="/wiki/Harry_Julius_Emel%C3%A9us" title="Harry Julius Emeléus">Emeléus, H. J.</a> (1969). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9SkSBQAAQBAJ"><i>The Chemistry of Fluorine and Its Compounds</i></a>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4832-7304-4" title="Special:BookSources/978-1-4832-7304-4"><bdi>978-1-4832-7304-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Chemistry+of+Fluorine+and+Its+Compounds&rft.pub=Academic+Press&rft.date=1969&rft.isbn=978-1-4832-7304-4&rft.aulast=Emel%C3%A9us&rft.aufirst=H.+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9SkSBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1997" class="citation book cs1"><a href="/wiki/Howard_Eves" title="Howard Eves">Eves, Howard</a> (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J9QcmFHj8EwC"><i>Foundations and Fundamental Concepts of Mathematics</i></a> (3rd ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-69609-6" title="Special:BookSources/978-0-486-69609-6"><bdi>978-0-486-69609-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+and+Fundamental+Concepts+of+Mathematics&rft.edition=3rd&rft.pub=Dover+Publications&rft.date=1997&rft.isbn=978-0-486-69609-6&rft.aulast=Eves&rft.aufirst=Howard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJ9QcmFHj8EwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeder2010" class="citation book cs1">Feder, Kenneth L. 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ABC-CLIO. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-313-37919-2" title="Special:BookSources/978-0-313-37919-2"><bdi>978-0-313-37919-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+Dubious+Archaeology%3A+From+Atlantis+to+the+Walam+Olum%3A+From+Atlantis+to+the+Walam+Olum&rft.pub=ABC-CLIO&rft.date=2010&rft.isbn=978-0-313-37919-2&rft.aulast=Feder&rft.aufirst=Kenneth+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRlRz2symkAsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreitag2014" class="citation book cs1">Freitag, Mark A. 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"Were the Fibonacci series and the Golden Section known in ancient Egypt?". <i><a href="/wiki/Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a></i>. <b>29</b> (2): 101–113. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.2001.2334">10.1006/hmat.2001.2334</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11311%2F997099">11311/997099</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Were+the+Fibonacci+series+and+the+Golden+Section+known+in+ancient+Egypt%3F&rft.volume=29&rft.issue=2&rft.pages=101-113&rft.date=2002&rft_id=info%3Ahdl%2F11311%2F997099&rft_id=info%3Adoi%2F10.1006%2Fhmat.2001.2334&rft.aulast=Rossi&rft.aufirst=Corinna&rft.au=Tout%2C+Christopher+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimonson2011" class="citation book cs1">Simonson, Shai (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ws6-DwAAQBAJ"><i>Rediscovering Mathematics: You Do the Math</i></a>. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-912-4" title="Special:BookSources/978-0-88385-912-4"><bdi>978-0-88385-912-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rediscovering+Mathematics%3A+You+Do+the+Math&rft.pub=Mathematical+Association+of+America&rft.date=2011&rft.isbn=978-0-88385-912-4&rft.aulast=Simonson&rft.aufirst=Shai&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWs6-DwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlobodanObradovićÐukanović2015" class="citation journal cs1">Slobodan, Mišić; Obradović, Marija; Ðukanović, Gordana (2015). <a rel="nofollow" class="external text" href="https://www.heldermann-verlag.de/jgg/jgg19/j19h1misi.pdf">"Composite Concave Cupolae as Geometric and Architectural Forms"</a> <span class="cs1-format">(PDF)</span>. <i>Journal for Geometry and Graphics</i>. <b>19</b> (1): 79–91.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+for+Geometry+and+Graphics&rft.atitle=Composite+Concave+Cupolae+as+Geometric+and+Architectural+Forms&rft.volume=19&rft.issue=1&rft.pages=79-91&rft.date=2015&rft.aulast=Slobodan&rft.aufirst=Mi%C5%A1i%C4%87&rft.au=Obradovi%C4%87%2C+Marija&rft.au=%C3%90ukanovi%C4%87%2C+Gordana&rft_id=https%3A%2F%2Fwww.heldermann-verlag.de%2Fjgg%2Fjgg19%2Fj19h1misi.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith2000" class="citation book cs1">Smith, James T. 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Routledge. p. 16. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-317-45839-5" title="Special:BookSources/978-1-317-45839-5"><bdi>978-1-317-45839-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Ancient+World&rft.pages=16&rft.pub=Routledge&rft.date=2015&rft.isbn=978-1-317-45839-5&rft.aulast=Takacs&rft.aufirst=Sarolta+Anna&rft.au=Cline%2C+Eric+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSPcvCgAAQBAJ%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUehara2020" class="citation book cs1">Uehara, Ryuhei (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=51juDwAAQBAJ"><i>Introduction to Computational Origami: The World of New Computational Geometry</i></a>. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-981-15-4470-5">10.1007/978-981-15-4470-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-15-4470-5" title="Special:BookSources/978-981-15-4470-5"><bdi>978-981-15-4470-5</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220150682">220150682</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Computational+Origami%3A+The+World+of+New+Computational+Geometry&rft.pub=Springer&rft.date=2020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220150682%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-981-15-4470-5&rft.isbn=978-981-15-4470-5&rft.aulast=Uehara&rft.aufirst=Ryuhei&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D51juDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWagner1979" class="citation journal cs1">Wagner, Donald Blackmore (1979). "An early Chinese derivation of the volume of a pyramid: Liu Hui, third century A.D.". <i>Historia Mathematica</i>. <b>6</b> (2): 164–188. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2879%2990076-4">10.1016/0315-0860(79)90076-4</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=An+early+Chinese+derivation+of+the+volume+of+a+pyramid%3A+Liu+Hui%2C+third+century+A.D.&rft.volume=6&rft.issue=2&rft.pages=164-188&rft.date=1979&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2879%2990076-4&rft.aulast=Wagner&rft.aufirst=Donald+Blackmore&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWohlleben2019" class="citation conference cs1">Wohlleben, Eva (2019). "Duality in Non-Polyhedral Bodies Part I: Polyliner". In Cocchiarella, Luigi (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rEpjDwAAQBAJ"><i>ICGG 2018 – Proceedings of the 18th International Conference on Geometry and Graphics: 40th Anniversary – Milan, Italy, August 3–7, 2018</i></a>. International Conference on Geometry and Graphics. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-95588-9">10.1007/978-3-319-95588-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-95588-9" title="Special:BookSources/978-3-319-95588-9"><bdi>978-3-319-95588-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Duality+in+Non-Polyhedral+Bodies+Part+I%3A+Polyliner&rft.btitle=ICGG+2018+%E2%80%93+Proceedings+of+the+18th+International+Conference+on+Geometry+and+Graphics%3A+40th+Anniversary+%E2%80%93+Milan%2C+Italy%2C+August+3%E2%80%937%2C+2018&rft.pub=Springer&rft.date=2019&rft_id=info%3Adoi%2F10.1007%2F978-3-319-95588-9&rft.isbn=978-3-319-95588-9&rft.aulast=Wohlleben&rft.aufirst=Eva&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrEpjDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+pyramid" class="Z3988"></span></li></ul> </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=Square_pyramid&action=edit&section=9" 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:Square_pyramid_(J1)" class="extiw" title="commons:Category:Square pyramid (J1)">Square pyramid (J1)</a></span>.</div></div> </div> <ul><li><cite id="Reference-Mathworld-Square_pyramid"><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/SquarePyramid.html">Square pyramid</a>" ("<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/JohnsonSolid.html">Johnson solid</a>") at <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071008222854/http://polyhedra.org/poly/show/45/square_pyramid">Square Pyramid</a> – Interactive Polyhedron Model</li> <li><a rel="nofollow" class="external text" href="https://www.georgehart.com/virtual-polyhedra/vp.html">Virtual Reality Polyhedra</a> georgehart.com: The Encyclopedia of Polyhedra (<a href="/wiki/VRML" title="VRML">VRML</a> <a rel="nofollow" class="external text" href="https://www.georgehart.com/virtual-polyhedra/vrml/square_pyramid_(J1).wrl">model</a> <a rel="nofollow" class="external text" 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navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output 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style="font-size:114%;margin:0 4em"><a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solids</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramids</a>, <a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupolae</a> and <a href="/wiki/Rotunda_(geometry)" title="Rotunda (geometry)">rotundae</a></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 class="mw-selflink selflink">square pyramid</a></li> <li><a href="/wiki/Pentagonal_pyramid" title="Pentagonal pyramid">pentagonal pyramid</a></li> <li><a href="/wiki/Triangular_cupola" title="Triangular cupola">triangular cupola</a></li> <li><a href="/wiki/Square_cupola" title="Square cupola">square cupola</a></li> <li><a href="/wiki/Pentagonal_cupola" title="Pentagonal cupola">pentagonal cupola</a></li> <li><a href="/wiki/Pentagonal_rotunda" title="Pentagonal rotunda">pentagonal rotunda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modified <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</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/Elongated_triangular_pyramid" title="Elongated triangular pyramid">elongated triangular pyramid</a></li> <li><a href="/wiki/Elongated_square_pyramid" title="Elongated square pyramid">elongated square pyramid</a></li> <li><a href="/wiki/Elongated_pentagonal_pyramid" title="Elongated pentagonal pyramid">elongated pentagonal pyramid</a></li> <li><a href="/wiki/Gyroelongated_square_pyramid" title="Gyroelongated square pyramid">gyroelongated square pyramid</a></li> <li><a href="/wiki/Gyroelongated_pentagonal_pyramid" title="Gyroelongated pentagonal pyramid">gyroelongated pentagonal pyramid</a></li> <li><a href="/wiki/Triangular_bipyramid" title="Triangular bipyramid">triangular bipyramid</a></li> <li><a href="/wiki/Pentagonal_bipyramid" title="Pentagonal bipyramid">pentagonal bipyramid</a></li> <li><a href="/wiki/Elongated_triangular_bipyramid" title="Elongated triangular bipyramid">elongated triangular bipyramid</a></li> <li><a href="/wiki/Elongated_square_bipyramid" title="Elongated square bipyramid">elongated square bipyramid</a></li> <li><a href="/wiki/Elongated_pentagonal_bipyramid" title="Elongated pentagonal bipyramid">elongated pentagonal bipyramid</a></li> <li><a href="/wiki/Gyroelongated_square_bipyramid" title="Gyroelongated square bipyramid">gyroelongated square bipyramid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modified <a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupolae</a> and <a href="/wiki/Rotunda_(geometry)" title="Rotunda (geometry)">rotundae</a></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/Elongated_triangular_cupola" title="Elongated triangular cupola">elongated triangular cupola</a></li> <li><a href="/wiki/Elongated_square_cupola" title="Elongated square cupola">elongated square cupola</a></li> <li><a href="/wiki/Elongated_pentagonal_cupola" title="Elongated pentagonal cupola">elongated pentagonal cupola</a></li> <li><a href="/wiki/Elongated_pentagonal_rotunda" title="Elongated pentagonal rotunda">elongated pentagonal rotunda</a></li> <li><a href="/wiki/Gyroelongated_triangular_cupola" title="Gyroelongated triangular cupola">gyroelongated triangular cupola</a></li> <li><a href="/wiki/Gyroelongated_square_cupola" title="Gyroelongated square cupola">gyroelongated square cupola</a></li> <li><a href="/wiki/Gyroelongated_pentagonal_cupola" title="Gyroelongated pentagonal cupola">gyroelongated pentagonal cupola</a></li> <li><a href="/wiki/Gyroelongated_pentagonal_rotunda" title="Gyroelongated pentagonal rotunda">gyroelongated pentagonal rotunda</a></li> <li><a href="/wiki/Gyrobifastigium" title="Gyrobifastigium">gyrobifastigium</a></li> <li><a href="/wiki/Triangular_orthobicupola" title="Triangular orthobicupola">triangular orthobicupola</a></li> <li><a href="/wiki/Square_orthobicupola" title="Square orthobicupola">square orthobicupola</a></li> <li><a href="/wiki/Square_gyrobicupola" title="Square gyrobicupola">square gyrobicupola</a></li> <li><a href="/wiki/Pentagonal_orthobicupola" title="Pentagonal orthobicupola">pentagonal orthobicupola</a></li> <li><a href="/wiki/Pentagonal_gyrobicupola" title="Pentagonal gyrobicupola">pentagonal gyrobicupola</a></li> <li><a href="/wiki/Pentagonal_orthocupolarotunda" title="Pentagonal orthocupolarotunda">pentagonal orthocupolarotunda</a></li> <li><a href="/wiki/Pentagonal_gyrocupolarotunda" title="Pentagonal gyrocupolarotunda">pentagonal gyrocupolarotunda</a></li> <li><a href="/wiki/Pentagonal_orthobirotunda" title="Pentagonal orthobirotunda">pentagonal orthobirotunda</a></li> <li><a href="/wiki/Elongated_triangular_orthobicupola" title="Elongated triangular orthobicupola">elongated triangular orthobicupola</a></li> <li><a href="/wiki/Elongated_triangular_gyrobicupola" title="Elongated triangular gyrobicupola">elongated triangular gyrobicupola</a></li> <li><a href="/wiki/Elongated_square_gyrobicupola" title="Elongated square gyrobicupola">elongated square gyrobicupola</a></li> <li><a href="/wiki/Elongated_pentagonal_orthobicupola" title="Elongated pentagonal orthobicupola">elongated pentagonal orthobicupola</a></li> <li><a href="/wiki/Elongated_pentagonal_gyrobicupola" title="Elongated pentagonal gyrobicupola">elongated pentagonal gyrobicupola</a></li> <li><a href="/wiki/Elongated_pentagonal_orthocupolarotunda" title="Elongated pentagonal orthocupolarotunda">elongated pentagonal orthocupolarotunda</a></li> <li><a href="/wiki/Elongated_pentagonal_gyrocupolarotunda" title="Elongated pentagonal gyrocupolarotunda">elongated pentagonal gyrocupolarotunda</a></li> <li><a href="/wiki/Elongated_pentagonal_orthobirotunda" title="Elongated pentagonal orthobirotunda">elongated pentagonal orthobirotunda</a></li> <li><a href="/wiki/Elongated_pentagonal_gyrobirotunda" title="Elongated pentagonal gyrobirotunda">elongated pentagonal gyrobirotunda</a></li> <li><a href="/wiki/Gyroelongated_triangular_bicupola" title="Gyroelongated triangular bicupola">gyroelongated triangular bicupola</a></li> <li><a href="/wiki/Gyroelongated_square_bicupola" title="Gyroelongated square bicupola">gyroelongated square bicupola</a></li> <li><a href="/wiki/Gyroelongated_pentagonal_bicupola" title="Gyroelongated pentagonal bicupola">gyroelongated pentagonal bicupola</a></li> <li><a href="/wiki/Gyroelongated_pentagonal_cupolarotunda" title="Gyroelongated pentagonal cupolarotunda">gyroelongated pentagonal cupolarotunda</a></li> <li><a href="/wiki/Gyroelongated_pentagonal_birotunda" title="Gyroelongated pentagonal birotunda">gyroelongated pentagonal birotunda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Augmented <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</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/Augmented_triangular_prism" title="Augmented triangular prism">augmented triangular prism</a></li> <li><a href="/wiki/Biaugmented_triangular_prism" title="Biaugmented triangular prism">biaugmented triangular prism</a></li> <li><a href="/wiki/Triaugmented_triangular_prism" title="Triaugmented triangular prism">triaugmented triangular prism</a></li> <li><a href="/wiki/Augmented_pentagonal_prism" title="Augmented pentagonal prism">augmented pentagonal prism</a></li> <li><a href="/wiki/Biaugmented_pentagonal_prism" title="Biaugmented pentagonal prism">biaugmented pentagonal prism</a></li> <li><a href="/wiki/Augmented_hexagonal_prism" title="Augmented hexagonal prism">augmented hexagonal prism</a></li> <li><a href="/wiki/Parabiaugmented_hexagonal_prism" title="Parabiaugmented hexagonal prism">parabiaugmented hexagonal prism</a></li> <li><a href="/wiki/Metabiaugmented_hexagonal_prism" title="Metabiaugmented hexagonal prism">metabiaugmented hexagonal prism</a></li> <li><a href="/wiki/Triaugmented_hexagonal_prism" title="Triaugmented hexagonal prism">triaugmented hexagonal prism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modified <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a></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/Augmented_dodecahedron" title="Augmented dodecahedron">augmented dodecahedron</a></li> <li><a href="/wiki/Parabiaugmented_dodecahedron" title="Parabiaugmented dodecahedron">parabiaugmented dodecahedron</a></li> <li><a href="/wiki/Metabiaugmented_dodecahedron" title="Metabiaugmented dodecahedron">metabiaugmented dodecahedron</a></li> <li><a href="/wiki/Triaugmented_dodecahedron" title="Triaugmented dodecahedron">triaugmented dodecahedron</a></li> <li><a href="/wiki/Metabidiminished_icosahedron" title="Metabidiminished icosahedron">metabidiminished icosahedron</a></li> <li><a href="/wiki/Tridiminished_icosahedron" title="Tridiminished icosahedron">tridiminished icosahedron</a></li> <li><a href="/wiki/Augmented_tridiminished_icosahedron" title="Augmented tridiminished icosahedron">augmented tridiminished icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modified <a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</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/Augmented_truncated_tetrahedron" title="Augmented truncated tetrahedron">augmented truncated tetrahedron</a></li> <li><a href="/wiki/Augmented_truncated_cube" title="Augmented truncated cube">augmented truncated cube</a></li> <li><a href="/wiki/Biaugmented_truncated_cube" title="Biaugmented truncated cube">biaugmented truncated cube</a></li> <li><a href="/wiki/Augmented_truncated_dodecahedron" title="Augmented truncated dodecahedron">augmented truncated dodecahedron</a></li> <li><a href="/wiki/Parabiaugmented_truncated_dodecahedron" title="Parabiaugmented truncated dodecahedron">parabiaugmented truncated dodecahedron</a></li> <li><a href="/wiki/Metabiaugmented_truncated_dodecahedron" title="Metabiaugmented truncated dodecahedron">metabiaugmented truncated dodecahedron</a></li> <li><a href="/wiki/Triaugmented_truncated_dodecahedron" title="Triaugmented truncated dodecahedron">triaugmented truncated dodecahedron</a></li> <li><a href="/wiki/Gyrate_rhombicosidodecahedron" title="Gyrate rhombicosidodecahedron">gyrate rhombicosidodecahedron</a></li> <li><a href="/wiki/Parabigyrate_rhombicosidodecahedron" title="Parabigyrate rhombicosidodecahedron">parabigyrate rhombicosidodecahedron</a></li> <li><a href="/wiki/Metabigyrate_rhombicosidodecahedron" title="Metabigyrate rhombicosidodecahedron">metabigyrate rhombicosidodecahedron</a></li> <li><a href="/wiki/Trigyrate_rhombicosidodecahedron" title="Trigyrate rhombicosidodecahedron">trigyrate rhombicosidodecahedron</a></li> <li><a href="/wiki/Diminished_rhombicosidodecahedron" title="Diminished rhombicosidodecahedron">diminished rhombicosidodecahedron</a></li> <li><a href="/wiki/Paragyrate_diminished_rhombicosidodecahedron" title="Paragyrate diminished rhombicosidodecahedron">paragyrate diminished rhombicosidodecahedron</a></li> <li><a href="/wiki/Metagyrate_diminished_rhombicosidodecahedron" title="Metagyrate diminished rhombicosidodecahedron">metagyrate diminished rhombicosidodecahedron</a></li> <li><a href="/wiki/Bigyrate_diminished_rhombicosidodecahedron" title="Bigyrate diminished rhombicosidodecahedron">bigyrate diminished rhombicosidodecahedron</a></li> <li><a href="/wiki/Parabidiminished_rhombicosidodecahedron" title="Parabidiminished rhombicosidodecahedron">parabidiminished rhombicosidodecahedron</a></li> <li><a href="/wiki/Metabidiminished_rhombicosidodecahedron" title="Metabidiminished rhombicosidodecahedron">metabidiminished rhombicosidodecahedron</a></li> <li><a href="/wiki/Gyrate_bidiminished_rhombicosidodecahedron" title="Gyrate bidiminished rhombicosidodecahedron">gyrate bidiminished rhombicosidodecahedron</a></li> <li><a href="/wiki/Tridiminished_rhombicosidodecahedron" title="Tridiminished rhombicosidodecahedron">tridiminished rhombicosidodecahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Elementary_polyhedron" class="mw-redirect" title="Elementary polyhedron">elementary solids</a></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/Snub_disphenoid" title="Snub disphenoid">snub disphenoid</a></li> <li><a href="/wiki/Snub_square_antiprism" title="Snub square antiprism">snub square antiprism</a></li> <li><a href="/wiki/Sphenocorona" title="Sphenocorona">sphenocorona</a></li> <li><a href="/wiki/Augmented_sphenocorona" title="Augmented sphenocorona">augmented sphenocorona</a></li> <li><a href="/wiki/Sphenomegacorona" title="Sphenomegacorona">sphenomegacorona</a></li> <li><a href="/wiki/Hebesphenomegacorona" title="Hebesphenomegacorona">hebesphenomegacorona</a></li> <li><a href="/wiki/Disphenocingulum" title="Disphenocingulum">disphenocingulum</a></li> <li><a href="/wiki/Bilunabirotunda" title="Bilunabirotunda">bilunabirotunda</a></li> <li><a href="/wiki/Triangular_hebesphenorotunda" title="Triangular hebesphenorotunda">triangular hebesphenorotunda</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>(See also <a href="/wiki/List_of_Johnson_solids" title="List of Johnson solids">List of Johnson solids</a>, a sortable table)</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=Square_pyramid&oldid=1258206960">https://en.wikipedia.org/w/index.php?title=Square_pyramid&oldid=1258206960</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:Elementary_polyhedron" title="Category:Elementary polyhedron">Elementary polyhedron</a></li><li><a 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