Solid angle - Wikipedia

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href="#Practical_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Practical applications</span> </div> </a> <ul id="toc-Practical_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solid_angles_for_common_objects" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solid_angles_for_common_objects"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Solid angles for common objects</span> </div> </a> <button aria-controls="toc-Solid_angles_for_common_objects-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 Solid angles for common objects subsection</span> </button> <ul id="toc-Solid_angles_for_common_objects-sublist" class="vector-toc-list"> <li id="toc-Cone,_spherical_cap,_hemisphere" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cone,_spherical_cap,_hemisphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Cone, spherical cap, hemisphere</span> </div> </a> <ul id="toc-Cone,_spherical_cap,_hemisphere-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tetrahedron" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tetrahedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Tetrahedron</span> </div> </a> <ul id="toc-Tetrahedron-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pyramid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pyramid"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Pyramid</span> </div> </a> <ul id="toc-Pyramid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Latitude-longitude_rectangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Latitude-longitude_rectangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Latitude-longitude rectangle</span> </div> </a> <ul id="toc-Latitude-longitude_rectangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Celestial_objects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Celestial_objects"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Celestial objects</span> </div> </a> <ul id="toc-Celestial_objects-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solid_angles_in_arbitrary_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solid_angles_in_arbitrary_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Solid angles in arbitrary dimensions</span> </div> </a> <ul id="toc-Solid_angles_in_arbitrary_dimensions-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">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span 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Available in 48 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-48" 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">48 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%D9%8A%D8%A9_%D9%85%D8%AC%D8%B3%D9%85%D8%A9" title="زاوية مجسمة – Arabic" lang="ar" hreflang="ar" data-title="زاوية مجسمة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81ngulu_s%C3%B3lidu" title="Ángulu sólidu – Asturian" lang="ast" hreflang="ast" data-title="Ángulu sólidu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A6%D1%8F%D0%BB%D0%B5%D1%81%D0%BD%D1%8B_%D0%B2%D1%83%D0%B3%D0%B0%D0%BB" title="Цялесны вугал – Belarusian" lang="be" hreflang="be" data-title="Цялесны вугал" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%B5%D0%BD_%D1%8A%D0%B3%D1%8A%D0%BB" title="Пространствен ъгъл – Bulgarian" lang="bg" hreflang="bg" data-title="Пространствен ъгъл" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Raumwingl" title="Raumwingl – Bavarian" lang="bar" hreflang="bar" data-title="Raumwingl" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Angle_s%C3%B2lid" title="Angle sòlid – Catalan" lang="ca" hreflang="ca" data-title="Angle sòlid" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C4%94%D1%81%D0%BA%D0%B5%D1%80%D0%BB%D0%B5_%D0%BA%C4%95%D1%82%D0%B5%D1%81" title="Ĕскерле кĕтес – Chuvash" lang="cv" hreflang="cv" data-title="Ĕскерле кĕтес" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Prostorov%C3%BD_%C3%BAhel" title="Prostorový úhel – Czech" lang="cs" hreflang="cs" data-title="Prostorový úhel" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Raumwinkel" title="Raumwinkel – German" lang="de" hreflang="de" data-title="Raumwinkel" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ruuminurk" title="Ruuminurk – Estonian" lang="et" hreflang="et" data-title="Ruuminurk" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%84%CE%B5%CF%81%CE%B5%CE%AC_%CE%B3%CF%89%CE%BD%CE%AF%CE%B1" title="Στερεά γωνία – Greek" lang="el" hreflang="el" data-title="Στερεά γωνία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81ngulo_s%C3%B3lido" title="Ángulo sólido – Spanish" lang="es" hreflang="es" data-title="Ángulo sólido" 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/Spaca_angulo" title="Spaca angulo – Esperanto" lang="eo" hreflang="eo" data-title="Spaca angulo" 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/Angelu_solido" title="Angelu solido – Basque" lang="eu" hreflang="eu" data-title="Angelu solido" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%DB%8C%D9%87_%D9%81%D8%B6%D8%A7%DB%8C%DB%8C" title="زاویه فضایی – Persian" lang="fa" hreflang="fa" data-title="زاویه فضایی" 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/Angle_solide" title="Angle solide – French" lang="fr" hreflang="fr" data-title="Angle solide" 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%9E%85%EC%B2%B4%EA%B0%81" title="입체각 – Korean" lang="ko" hreflang="ko" data-title="입체각" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D6%80%D5%B4%D5%B6%D5%A1%D5%B5%D5%AB%D5%B6_%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6" title="Մարմնային անկյուն – Armenian" lang="hy" hreflang="hy" data-title="Մարմնային անկյուն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%98%E0%A4%A8_%E0%A4%95%E0%A5%8B%E0%A4%A3" title="घन कोण – Hindi" lang="hi" hreflang="hi" data-title="घन कोण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Prostorni_kut" title="Prostorni kut – Croatian" lang="hr" hreflang="hr" data-title="Prostorni kut" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Angolo_solido" title="Angolo solido – Italian" lang="it" hreflang="it" data-title="Angolo solido" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%96%D7%95%D7%95%D7%99%D7%AA_%D7%9E%D7%A8%D7%97%D7%91%D7%99%D7%AA" title="זווית מרחבית – Hebrew" lang="he" hreflang="he" data-title="זווית מרחבית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%98%E0%B2%A8%E0%B2%95%E0%B3%8B%E0%B2%A8" title="ಘನಕೋನ – Kannada" lang="kn" hreflang="kn" data-title="ಘನಕೋನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%95%E1%83%A0%E1%83%AA%E1%83%98%E1%83%97%E1%83%98_%E1%83%99%E1%83%A3%E1%83%97%E1%83%AE%E1%83%94" title="სივრცითი კუთხე – Georgian" lang="ka" hreflang="ka" data-title="სივრცითი კუთხე" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BD%D0%B5%D0%BB%D1%96%D0%BA_%D0%B1%D2%B1%D1%80%D1%8B%D1%88" title="Денелік бұрыш – Kazakh" lang="kk" hreflang="kk" data-title="Денелік бұрыш" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Telpas_le%C5%86%C4%B7is" title="Telpas leņķis – Latvian" lang="lv" hreflang="lv" data-title="Telpas leņķis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%A9rsz%C3%B6g" title="Térszög – Hungarian" lang="hu" hreflang="hu" data-title="Térszög" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sudut_padu" title="Sudut padu – Malay" lang="ms" hreflang="ms" data-title="Sudut padu" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9E%D0%B3%D1%82%D0%BE%D1%80%D0%B3%D1%83%D0%B9%D0%BD_%D3%A9%D0%BD%D1%86%D3%A9%D0%B3" title="Огторгуйн өнцөг – Mongolian" lang="mn" hreflang="mn" data-title="Огторгуйн өнцөг" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ruimtehoek" title="Ruimtehoek – Dutch" lang="nl" hreflang="nl" data-title="Ruimtehoek" 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/%E7%AB%8B%E4%BD%93%E8%A7%92" title="立体角 – Japanese" lang="ja" hreflang="ja" data-title="立体角" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/R%C3%BCmwinkel" title="Rümwinkel – Northern Frisian" lang="frr" hreflang="frr" data-title="Rümwinkel" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Romvinkel" title="Romvinkel – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Romvinkel" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Romvinkel" title="Romvinkel – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Romvinkel" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/%C3%80ngol_s%C3%B2lid" title="Àngol sòlid – Piedmontese" lang="pms" hreflang="pms" data-title="Àngol sòlid" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/K%C4%85t_bry%C5%82owy" title="Kąt bryłowy – Polish" lang="pl" hreflang="pl" data-title="Kąt bryłowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%82ngulo_s%C3%B3lido" title="Ângulo sólido – Portuguese" lang="pt" hreflang="pt" data-title="Ângulo sólido" 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/Unghi_solid" title="Unghi solid – Romanian" lang="ro" hreflang="ro" data-title="Unghi solid" 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%A2%D0%B5%D0%BB%D0%B5%D1%81%D0%BD%D1%8B%D0%B9_%D1%83%D0%B3%D0%BE%D0%BB" title="Телесный угол – Russian" lang="ru" hreflang="ru" data-title="Телесный угол" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/K%C3%ABndi_i_ngurt%C3%AB" title="Këndi i ngurtë – Albanian" lang="sq" hreflang="sq" data-title="Këndi i ngurtë" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Priestorov%C3%BD_uhol" title="Priestorový uhol – Slovak" lang="sk" hreflang="sk" data-title="Priestorový uhol" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Prostorski_kot" title="Prostorski kot – Slovenian" lang="sl" hreflang="sl" data-title="Prostorski kot" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Avaruuskulma" title="Avaruuskulma – Finnish" lang="fi" hreflang="fi" data-title="Avaruuskulma" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rymdvinkel" title="Rymdvinkel – Swedish" lang="sv" hreflang="sv" data-title="Rymdvinkel" data-language-autonym="Svenska" data-language-local-name="Swedish" 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.hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Spherical_angle" class="mw-redirect" title="Spherical angle">spherical angle</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Solid angle</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Angle_solide_coordonnees.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Angle_solide_coordonnees.svg/220px-Angle_solide_coordonnees.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Angle_solide_coordonnees.svg/330px-Angle_solide_coordonnees.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Angle_solide_coordonnees.svg/440px-Angle_solide_coordonnees.svg.png 2x" data-file-width="500" data-file-height="500" /></a></span><div class="infobox-caption">Visual representation of a solid angle</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data">Ω</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Steradian" title="Steradian">steradian</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data"><a href="/wiki/Square_degree" title="Square degree">Square degree</a>, <a href="/wiki/Spat_(angular_unit)" title="Spat (angular unit)">spat (angular unit)</a></td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data">m<sup>2</sup>/m<sup>2</sup></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Conserved_quantity" title="Conserved quantity">Conserved</a>?</th><td class="infobox-data">No</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from<br />other quantities</div></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 \Omega =A/r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =A/r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f16c681d42b5a98489eb13bccf7df459eddb77fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.785ex; height:3.176ex;" alt="{\displaystyle \Omega =A/r^{2}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>solid angle</b> (symbol: <span class="texhtml">Ω</span>) is a measure of the amount of the <a href="/wiki/Field_of_view" title="Field of view">field of view</a> from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the <i>apex</i> of the solid angle, and the object is said to <i><a href="/wiki/Subtended_angle" title="Subtended angle">subtend</a></i> its solid angle at that point. </p><p>In the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI), a solid angle is expressed in a <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit</a> called a <i><a href="/wiki/Steradian" title="Steradian">steradian</a></i> (symbol: sr), which is equal to one square radian, sr = rad<sup>2</sup>. One steradian corresponds to one unit of area (of any shape) on the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a> surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total <a href="/wiki/Surface_area" title="Surface area">surface area</a> of the unit sphere, <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\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/057444bf35a0c22b19bcae1ef06e06ecdf8abe56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 4\pi }"></span>. Solid angles can also be measured in squares of angular measures such as <a href="/wiki/Square_degree" title="Square degree">degrees</a>, minutes, and seconds. </p><p>A small object nearby may subtend the same solid angle as a larger object farther away. For example, although the <a href="/wiki/Moon" title="Moon">Moon</a> is much smaller than the <a href="/wiki/Sun" title="Sun">Sun</a>, it is also much closer to <a href="/wiki/Earth" title="Earth">Earth</a>. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle (and therefore apparent size). This is evident during a <a href="/wiki/Solar_eclipse" title="Solar eclipse">solar eclipse</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_properties"><span class="anchor" id="Square_minute"></span><span class="anchor" id="Square_second"></span>Definition and properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=1" title="Edit section: Definition and properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Spherical_polygon_area" class="mw-redirect" title="Spherical polygon area">Spherical polygon area</a></div> <p>The magnitude of an object's solid angle in <a href="/wiki/Steradian" title="Steradian">steradians</a> is equal to the <a href="/wiki/Area_(geometry)" class="mw-redirect" title="Area (geometry)">area</a> of the segment of a <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a>, centered at the apex, that the object covers. Giving the area of a segment of a unit sphere in steradians is analogous to giving the length of an arc of a <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> in radians. Just as the magnitude of a plane angle in radians at the vertex of a circular sector is the ratio of the length of its arc to its radius, the magnitude of a solid angle in steradians is the ratio of the area covered on a sphere by an object to the square of the radius of the sphere. The formula for the magnitude of the solid angle in steradians is </p><p><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 \Omega ={\frac {A}{r^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ={\frac {A}{r^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf6149c6dcd69ff789a370d91f138a4ff1e0fda2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.362ex; height:5.676ex;" alt="{\displaystyle \Omega ={\frac {A}{r^{2}}},}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> is the area (of any shape) on the surface of the sphere 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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is the radius of the sphere. </p><p>Solid angles are often used in <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <a href="/wiki/Physics" title="Physics">physics</a>, and in particular <a href="/wiki/Astrophysics" title="Astrophysics">astrophysics</a>. The solid angle of an object that is very far away is roughly proportional to the ratio of area to squared distance. Here "area" means the area of the object when projected along the viewing direction. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Solid_Angle,_1_Steradian.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Solid_Angle%2C_1_Steradian.svg/220px-Solid_Angle%2C_1_Steradian.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Solid_Angle%2C_1_Steradian.svg/330px-Solid_Angle%2C_1_Steradian.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Solid_Angle%2C_1_Steradian.svg/440px-Solid_Angle%2C_1_Steradian.svg.png 2x" data-file-width="874" data-file-height="874" /></a><figcaption>Any area on a sphere which is equal in area to the square of its radius, when observed from its center, subtends precisely one <a href="/wiki/Steradian" title="Steradian">steradian</a>.</figcaption></figure> <p>The solid angle of a sphere measured from any point in its interior is 4<a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> sr. The solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2<span class="texhtml mvar" style="font-style:italic;">π</span>/3  sr. <span class="anchor" id="Octant"></span>The solid angle subtended at the corner of a cube (an <a href="/wiki/Octant_(geometry)" class="mw-redirect" title="Octant (geometry)">octant</a>) or spanned by a <a href="/wiki/Spherical_octant" class="mw-redirect" title="Spherical octant">spherical octant</a> is <span class="texhtml mvar" style="font-style:italic;">π</span>/2  sr, one-eight of the solid angle of a sphere.<sup id="cite_ref-u421_1-0" class="reference"><a href="#cite_note-u421-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Solid angles can also be measured in <a href="/wiki/Square_degree" title="Square degree">square degrees</a> (1 sr = (<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">180/<span class="texhtml mvar" style="font-style:italic;">π</span></span>)<sup>2</sup> square degrees), in square <a href="/wiki/Arc-minutes" class="mw-redirect" title="Arc-minutes">arc-minutes</a> and square <a href="/wiki/Arc-seconds" class="mw-redirect" title="Arc-seconds">arc-seconds</a>, or in fractions of the sphere (1 sr = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4<span class="texhtml mvar" style="font-style:italic;">π</span></span></span>⁠</span> fractional area), also known as <a href="/wiki/Spat_(angular_unit)" title="Spat (angular unit)">spat</a> (1 sp = 4<span class="texhtml mvar" style="font-style:italic;">π</span> sr). </p><p>In <a href="/wiki/Spherical_coordinates#Integration_and_differentiation_in_spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a> there is a formula for the <a href="/wiki/Differential_of_a_function" title="Differential of a function">differential</a>, </p><p><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 d\Omega =\sin \theta \,d\theta \,d\varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\Omega =\sin \theta \,d\theta \,d\varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95edc40d7613aabd033e626131afdec63461db94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.789ex; height:2.676ex;" alt="{\displaystyle d\Omega =\sin \theta \,d\theta \,d\varphi ,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">θ</span> is the <a href="/wiki/Colatitude" title="Colatitude">colatitude</a> (angle from the North Pole) and <span class="texhtml mvar" style="font-style:italic;">φ</span> is the longitude. </p><p>The solid angle for an arbitrary <a href="/wiki/Oriented_surface" class="mw-redirect" title="Oriented surface">oriented surface</a> <span class="texhtml mvar" style="font-style:italic;">S</span> subtended at a point <span class="texhtml mvar" style="font-style:italic;">P</span> is equal to the solid angle of the projection of the surface <span class="texhtml mvar" style="font-style:italic;">S</span> to the unit sphere with center <span class="texhtml mvar" style="font-style:italic;">P</span>, which can be calculated as the <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a>: </p><p><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 \Omega =\iint _{S}{\frac {{\hat {r}}\cdot {\hat {n}}}{r^{2}}}\,dS\ =\iint _{S}\sin \theta \,d\theta \,d\varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> <mtext> </mtext> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =\iint _{S}{\frac {{\hat {r}}\cdot {\hat {n}}}{r^{2}}}\,dS\ =\iint _{S}\sin \theta \,d\theta \,d\varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/813a8c78d2cde1bd83ae64245c0840975dc83e88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.953ex; height:5.843ex;" alt="{\displaystyle \Omega =\iint _{S}{\frac {{\hat {r}}\cdot {\hat {n}}}{r^{2}}}\,dS\ =\iint _{S}\sin \theta \,d\theta \,d\varphi ,}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {r}}={\vec {r}}/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {r}}={\vec {r}}/r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256efb7ce45460857353617c7a5863cf2168d1dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.824ex; height:2.843ex;" alt="{\displaystyle {\hat {r}}={\vec {r}}/r}"></span> is the <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> corresponding 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 {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span>, the <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position vector</a> of an infinitesimal area of surface <span class="texhtml"><i>dS</i></span> with respect to point <span class="texhtml mvar" style="font-style:italic;">P</span>, and 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 {\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d125dccc556f5c8b0bf98a4f3847590b3f353bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}"></span> represents the unit <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal vector</a> to <span class="texhtml"><i>dS</i></span>. Even if the projection on the unit sphere to the surface <span class="texhtml mvar" style="font-style:italic;">S</span> is not <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a>, the multiple folds are correctly considered according to the surface orientation described by the sign of the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</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 {\hat {r}}\cdot {\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {r}}\cdot {\hat {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753adeed649105d37314c368b0b635b0175b6917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.365ex; height:2.176ex;" alt="{\displaystyle {\hat {r}}\cdot {\hat {n}}}"></span>. </p><p>Thus one can approximate the solid angle subtended by a small <a href="/wiki/Facet" title="Facet">facet</a> having flat surface area <span class="texhtml"><i>dS</i></span>, orientation <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 {\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d125dccc556f5c8b0bf98a4f3847590b3f353bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}"></span>, and distance <span class="texhtml"><i>r</i></span> from the viewer as: </p><p><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 d\Omega =4\pi \left({\frac {dS}{A}}\right)\,({\hat {r}}\cdot {\hat {n}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\Omega =4\pi \left({\frac {dS}{A}}\right)\,({\hat {r}}\cdot {\hat {n}}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40afe9d9c270c0f4ceaa0cb234d3a73d5000c796" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.442ex; height:6.176ex;" alt="{\displaystyle d\Omega =4\pi \left({\frac {dS}{A}}\right)\,({\hat {r}}\cdot {\hat {n}}),}"></span> </p><p>where the <a href="/wiki/Surface_area_of_a_sphere" class="mw-redirect" title="Surface area of a sphere">surface area of a sphere</a> is <span class="texhtml"><i>A</i> = 4<span class="texhtml mvar" style="font-style:italic;">π</span><i>r</i><sup>2</sup></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Practical_applications">Practical applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=2" title="Edit section: Practical applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Defining <a href="/wiki/Luminous_intensity" title="Luminous intensity">luminous intensity</a> and <a href="/wiki/Luminance" title="Luminance">luminance</a>, and the correspondent radiometric quantities <a href="/wiki/Radiant_intensity" title="Radiant intensity">radiant intensity</a> and <a href="/wiki/Radiance" title="Radiance">radiance</a></li> <li>Calculating <a href="/wiki/Spherical_trigonometry#Area_and_spherical_excess" title="Spherical trigonometry">spherical excess</a> <span class="texhtml"><i>E</i></span> of a <a href="/wiki/Spherical_triangle" class="mw-redirect" title="Spherical triangle">spherical triangle</a></li> <li>The calculation of potentials by using the <a href="/wiki/Boundary_element_method" title="Boundary element method">boundary element method</a> (BEM)</li> <li>Evaluating the size of <a href="/wiki/Ligand" title="Ligand">ligands</a> in metal complexes, see <a href="/wiki/Ligand_cone_angle" title="Ligand cone angle">ligand cone angle</a></li> <li>Calculating the <a href="/wiki/Electric_field" title="Electric field">electric field</a> and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> strength around charge distributions</li> <li>Deriving <a href="/wiki/Gauss%27s_Law" class="mw-redirect" title="Gauss's Law">Gauss's Law</a></li> <li>Calculating emissive power and irradiation in heat transfer</li> <li>Calculating cross sections in <a href="/wiki/Rutherford_scattering" class="mw-redirect" title="Rutherford scattering">Rutherford scattering</a></li> <li>Calculating cross sections in <a href="/wiki/Raman_scattering" title="Raman scattering">Raman scattering</a></li> <li>The solid angle of the <a href="/wiki/Acceptance_cone" class="mw-redirect" title="Acceptance cone">acceptance cone</a> of the <a href="/wiki/Optical_fiber" title="Optical fiber">optical fiber</a></li> <li>The computation of nodal densities in meshes.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Solid_angles_for_common_objects">Solid angles for common objects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=3" title="Edit section: Solid angles for common objects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Cone,_spherical_cap,_hemisphere"><span id="Cone.2C_spherical_cap.2C_hemisphere"></span>Cone, spherical cap, hemisphere</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=4" title="Edit section: Cone, spherical cap, hemisphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Steradian_cone_and_cap.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Steradian_cone_and_cap.svg/250px-Steradian_cone_and_cap.svg.png" decoding="async" width="250" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Steradian_cone_and_cap.svg/375px-Steradian_cone_and_cap.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Steradian_cone_and_cap.svg/500px-Steradian_cone_and_cap.svg.png 2x" data-file-width="400" data-file-height="350" /></a><figcaption>Diagram showing a section through the centre of a cone (1) subtending a solid angle of 1 steradian in a sphere of radius r, along with the spherical "cap" (2). The external surface area A of the cap equals r2 only if solid angle of the cone is exactly 1 steradian. Hence, in this figure <span class="texhtml"><i>θ</i> = <i>A</i>/2</span> and <span class="texhtml"><i>r</i> = 1</span>.</figcaption></figure> <p>The solid angle of a <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cone</a> with its apex at the apex of the solid angle, and with <a href="/wiki/Apex_(geometry)" title="Apex (geometry)">apex</a> angle 2<span class="texhtml"><i>θ</i></span>, is the area of a <a href="/wiki/Spherical_cap" title="Spherical cap">spherical cap</a> on a <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a> </p><p><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 \Omega =2\pi \left(1-\cos \theta \right)\ =4\pi \sin ^{2}{\frac {\theta }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =2\pi \left(1-\cos \theta \right)\ =4\pi \sin ^{2}{\frac {\theta }{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f481f6c05e1785cfdedb6700ce01abfc69e359" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.949ex; height:5.343ex;" alt="{\displaystyle \Omega =2\pi \left(1-\cos \theta \right)\ =4\pi \sin ^{2}{\frac {\theta }{2}}.}"></span> </p><p>For small <span class="texhtml"><i>θ</i></span> such that <span class="texhtml">cos <i>θ</i> ≈ 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>θ</i><sup>2</sup></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> this reduces to <span class="texhtml">π<i>θ</i><sup>2</sup></span>, the area of a circle. </p><p>The above is found by computing the following <a href="/wiki/Double_integral" class="mw-redirect" title="Double integral">double integral</a> using the unit <a href="/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates" title="Spherical coordinate system">surface element in spherical coordinates</a>: </p><p><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}\int _{0}^{2\pi }\int _{0}^{\theta }\sin \theta '\,d\theta '\,d\phi &=\int _{0}^{2\pi }d\phi \int _{0}^{\theta }\sin \theta '\,d\theta '\\&=2\pi \int _{0}^{\theta }\sin \theta '\,d\theta '\\&=2\pi \left[-\cos \theta '\right]_{0}^{\theta }\\&=2\pi \left(1-\cos \theta \right).\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> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ϕ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>d</mi> <mi>ϕ</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>θ</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>θ</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{0}^{2\pi }\int _{0}^{\theta }\sin \theta '\,d\theta '\,d\phi &=\int _{0}^{2\pi }d\phi \int _{0}^{\theta }\sin \theta '\,d\theta '\\&=2\pi \int _{0}^{\theta }\sin \theta '\,d\theta '\\&=2\pi \left[-\cos \theta '\right]_{0}^{\theta }\\&=2\pi \left(1-\cos \theta \right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/553bfb7326b3102cfc2e03f81cf7b34da431296b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:44.023ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}\int _{0}^{2\pi }\int _{0}^{\theta }\sin \theta '\,d\theta '\,d\phi &=\int _{0}^{2\pi }d\phi \int _{0}^{\theta }\sin \theta '\,d\theta '\\&=2\pi \int _{0}^{\theta }\sin \theta '\,d\theta '\\&=2\pi \left[-\cos \theta '\right]_{0}^{\theta }\\&=2\pi \left(1-\cos \theta \right).\end{aligned}}}"></span> </p><p>This formula can also be derived without the use of <a href="/wiki/Calculus" title="Calculus">calculus</a>. </p><p> Over 2200 years ago <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> proved that the surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Archimedes-spherical-cap.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Archimedes-spherical-cap.png/220px-Archimedes-spherical-cap.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Archimedes-spherical-cap.png/330px-Archimedes-spherical-cap.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c5/Archimedes-spherical-cap.png 2x" data-file-width="355" data-file-height="284" /></a><figcaption>Archimedes' theorem that surface area of the region of sphere below horizontal plane H in given diagram is equal to area of a circle of radius t.</figcaption></figure><p>In the above coloured diagram this radius is given as </p><p><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 2r\sin {\frac {\theta }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2r\sin {\frac {\theta }{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45476a3732d427d758c4f3f86215caf6c8eb6537" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.486ex; height:5.343ex;" alt="{\displaystyle 2r\sin {\frac {\theta }{2}}.}"></span> In the adjacent black & white diagram this radius is given as "t". </p><p>Hence for a unit sphere the solid angle of the spherical cap is given as </p><p><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 \Omega =4\pi \sin ^{2}{\frac {\theta }{2}}=2\pi \left(1-\cos \theta \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =4\pi \sin ^{2}{\frac {\theta }{2}}=2\pi \left(1-\cos \theta \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74412aa3558d931cce1b1178e41e8613b8e23f80" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.368ex; height:5.343ex;" alt="{\displaystyle \Omega =4\pi \sin ^{2}{\frac {\theta }{2}}=2\pi \left(1-\cos \theta \right).}"></span> </p><p><span class="anchor" id="hemisphere"></span> When <span class="texhtml"><i>θ</i></span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, the spherical cap becomes a <a href="/wiki/Sphere" title="Sphere">hemisphere</a> having a solid angle 2<span class="texhtml mvar" style="font-style:italic;">π</span>. </p><p>The solid angle of the complement of the cone is </p><p><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 4\pi -\Omega =2\pi \left(1+\cos \theta \right)=4\pi \cos ^{2}{\frac {\theta }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi -\Omega =2\pi \left(1+\cos \theta \right)=4\pi \cos ^{2}{\frac {\theta }{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee979e8fb9d6fb6b93c64f576fa8a33be8723ad7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.572ex; height:5.343ex;" alt="{\displaystyle 4\pi -\Omega =2\pi \left(1+\cos \theta \right)=4\pi \cos ^{2}{\frac {\theta }{2}}.}"></span> </p><p>This is also the solid angle of the part of the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> that an astronomical observer positioned at latitude <span class="texhtml"><i>θ</i></span> can see as the Earth rotates. At the equator all of the celestial sphere is visible; at either pole, only one half. </p><p>The solid angle subtended by a segment of a spherical cap cut by a plane at angle <span class="texhtml mvar" style="font-style:italic;"><i>γ</i></span> from the cone's axis and passing through the cone's apex can be calculated by the formula<sup id="cite_ref-Mazonka_4-0" class="reference"><a href="#cite_note-Mazonka-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><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 \Omega =2\left[\arccos \left({\frac {\sin \gamma }{\sin \theta }}\right)-\cos \theta \arccos \left({\frac {\tan \gamma }{\tan \theta }}\right)\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>γ</mi> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =2\left[\arccos \left({\frac {\sin \gamma }{\sin \theta }}\right)-\cos \theta \arccos \left({\frac {\tan \gamma }{\tan \theta }}\right)\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6801ed323a0f13ac7a1460c6549fa0b15002269d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.095ex; height:6.176ex;" alt="{\displaystyle \Omega =2\left[\arccos \left({\frac {\sin \gamma }{\sin \theta }}\right)-\cos \theta \arccos \left({\frac {\tan \gamma }{\tan \theta }}\right)\right].}"></span> </p><p>For example, if <span class="texhtml"><i>γ</i> = −<i>θ</i></span>, then the formula reduces to the spherical cap formula above: the first term becomes <span class="texhtml mvar" style="font-style:italic;">π</span>, and the second <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">π</span> cos <i>θ</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Tetrahedron">Tetrahedron</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=5" title="Edit section: Tetrahedron"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let OABC be the vertices of a <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> with an origin at O subtended by the triangular face ABC 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 {\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f41e23bc49994564f046f8b266f685ab97ea8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.55ex; height:3.176ex;" alt="{\displaystyle {\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}}"></span> are the vector positions of the vertices A, B and C. Define the <a href="/wiki/Vertex_angle" class="mw-redirect" title="Vertex angle">vertex angle</a> <span class="texhtml mvar" style="font-style:italic;">θ<sub>a</sub></span> to be the angle BOC and define <span class="texhtml mvar" style="font-style:italic;">θ<sub>b</sub></span>, <span class="texhtml mvar" style="font-style:italic;">θ<sub>c</sub></span> correspondingly. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{ab}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{ab}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09a2b61c2920ae19fc79de9222a69b433fd0bac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.193ex; height:2.509ex;" alt="{\displaystyle \phi _{ab}}"></span> be the <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> between the planes that contain the tetrahedral faces OAC and OBC and define <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 \phi _{ac}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{ac}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34262635d927e7451a6ac8fcf4eb20f92447190b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.199ex; height:2.509ex;" alt="{\displaystyle \phi _{ac}}"></span>, <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 \phi _{bc}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{bc}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60812249c965560a0383a9a0b1e433045b947753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.035ex; height:2.509ex;" alt="{\displaystyle \phi _{bc}}"></span> correspondingly. The solid angle <span class="texhtml">Ω</span> subtended by the triangular surface ABC is given by </p><p><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 \Omega =\left(\phi _{ab}+\phi _{bc}+\phi _{ac}\right)\ -\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>−</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =\left(\phi _{ab}+\phi _{bc}+\phi _{ac}\right)\ -\pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eeec55a4175c611931921054e0997e15117d1a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.481ex; height:2.843ex;" alt="{\displaystyle \Omega =\left(\phi _{ab}+\phi _{bc}+\phi _{ac}\right)\ -\pi .}"></span> </p><p>This follows from the theory of <a href="/wiki/Spherical_excess" class="mw-redirect" title="Spherical excess">spherical excess</a> and it leads to the fact that there is an analogous theorem to the theorem that <i>"The sum of internal angles of a planar triangle is equal to <span class="texhtml mvar" style="font-style:italic;">π</span>"</i>, for the sum of the four internal solid angles of a tetrahedron as follows: </p><p><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 \sum _{i=1}^{4}\Omega _{i}=2\sum _{i=1}^{6}\phi _{i}\ -4\pi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </munderover> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </munderover> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mo>−</mo> <mn>4</mn> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{4}\Omega _{i}=2\sum _{i=1}^{6}\phi _{i}\ -4\pi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0647b4a789a0ebfeabae27581bbb91db49e2c7a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.357ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{4}\Omega _{i}=2\sum _{i=1}^{6}\phi _{i}\ -4\pi ,}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0182dbf29b54844c92fd9b0311778a02a38398ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.185ex; height:2.509ex;" alt="{\displaystyle \phi _{i}}"></span> ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>A useful formula for calculating the solid angle of the tetrahedron at the origin O that is purely a function of the vertex angles <span class="texhtml mvar" style="font-style:italic;">θ<sub>a</sub></span>, <span class="texhtml mvar" style="font-style:italic;">θ<sub>b</sub></span>, <span class="texhtml mvar" style="font-style:italic;">θ<sub>c</sub></span> is given by <a href="/wiki/Simon_Antoine_Jean_L%27Huilier" title="Simon Antoine Jean L'Huilier">L'Huilier</a>'s theorem<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> as </p><p><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 \tan \left({\frac {1}{4}}\Omega \right)={\sqrt {\tan \left({\frac {\theta _{s}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{a}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{b}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{c}}{2}}\right)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi mathvariant="normal">Ω</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \left({\frac {1}{4}}\Omega \right)={\sqrt {\tan \left({\frac {\theta _{s}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{a}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{b}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{c}}{2}}\right)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee071c96fb7fcf9a40391ade64f0cf1d3544a5b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:71.307ex; height:7.509ex;" alt="{\displaystyle \tan \left({\frac {1}{4}}\Omega \right)={\sqrt {\tan \left({\frac {\theta _{s}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{a}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{b}}{2}}\right)\tan \left({\frac {\theta _{s}-\theta _{c}}{2}}\right)}},}"></span> </p><p>where <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 \theta _{s}={\frac {\theta _{a}+\theta _{b}+\theta _{c}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{s}={\frac {\theta _{a}+\theta _{b}+\theta _{c}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb9b92701a270b613790400284b0d7089d56523" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.611ex; height:5.343ex;" alt="{\displaystyle \theta _{s}={\frac {\theta _{a}+\theta _{b}+\theta _{c}}{2}}.}"></span> </p><p>Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f41e23bc49994564f046f8b266f685ab97ea8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.55ex; height:3.176ex;" alt="{\displaystyle {\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}}"></span> be the vector positions of the vertices A, B and C, and let <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span> be the magnitude of each vector (the origin-point distance). The solid angle <span class="texhtml">Ω</span> subtended by the triangular surface ABC is:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p><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 \tan \left({\frac {1}{2}}\Omega \right)={\frac {\left|{\vec {a}}\ {\vec {b}}\ {\vec {c}}\right|}{abc+\left({\vec {a}}\cdot {\vec {b}}\right)c+\left({\vec {a}}\cdot {\vec {c}}\right)b+\left({\vec {b}}\cdot {\vec {c}}\right)a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi mathvariant="normal">Ω</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mi>c</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mi>b</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \left({\frac {1}{2}}\Omega \right)={\frac {\left|{\vec {a}}\ {\vec {b}}\ {\vec {c}}\right|}{abc+\left({\vec {a}}\cdot {\vec {b}}\right)c+\left({\vec {a}}\cdot {\vec {c}}\right)b+\left({\vec {b}}\cdot {\vec {c}}\right)a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1c5eeb500549deecf720c7ab47d513225e3e39" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:50.681ex; height:9.509ex;" alt="{\displaystyle \tan \left({\frac {1}{2}}\Omega \right)={\frac {\left|{\vec {a}}\ {\vec {b}}\ {\vec {c}}\right|}{abc+\left({\vec {a}}\cdot {\vec {b}}\right)c+\left({\vec {a}}\cdot {\vec {c}}\right)b+\left({\vec {b}}\cdot {\vec {c}}\right)a}},}"></span> </p><p>where <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 \left|{\vec {a}}\ {\vec {b}}\ {\vec {c}}\right|={\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\vec {a}}\ {\vec {b}}\ {\vec {c}}\right|={\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60256bc77b0620a46ba98e799d4388aa0a56289c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:18.976ex; height:4.009ex;" alt="{\displaystyle \left|{\vec {a}}\ {\vec {b}}\ {\vec {c}}\right|={\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}"></span> </p><p>denotes the <a href="/wiki/Triple_product" title="Triple product">scalar triple product</a> of the three vectors 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 {\vec {a}}\cdot {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\cdot {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00c5274227b517684c55880c88f45414fe44936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.003ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}\cdot {\vec {b}}}"></span> denotes the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a>. </p><p>Care must be taken here to avoid negative or incorrect solid angles. One source of potential errors is that the scalar triple product can be negative if <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">c</span> have the wrong <a href="/wiki/Determinant" title="Determinant">winding</a>. Computing the absolute value is a sufficient solution since no other portion of the equation depends on the winding. The other pitfall arises when the scalar triple product is positive but the divisor is negative. In this case returns a negative value that must be increased by <span class="texhtml mvar" style="font-style:italic;">π</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Pyramid">Pyramid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=6" title="Edit section: Pyramid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The solid angle of a four-sided right rectangular <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a> with <a href="/wiki/Apex_(geometry)" title="Apex (geometry)">apex</a> angles <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> (<a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angles</a> measured to the opposite side faces of the pyramid) is <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 \Omega =4\arcsin \left(\sin \left({a \over 2}\right)\sin \left({b \over 2}\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>4</mn> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =4\arcsin \left(\sin \left({a \over 2}\right)\sin \left({b \over 2}\right)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd454cd299f5297908bda7934ad93b4a22b8d3c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.716ex; height:6.176ex;" alt="{\displaystyle \Omega =4\arcsin \left(\sin \left({a \over 2}\right)\sin \left({b \over 2}\right)\right).}"></span> </p><p>If both the side lengths (<span class="texhtml"><i>α</i></span> and <span class="texhtml"><i>β</i></span>) of the base of the pyramid and the distance (<span class="texhtml"><i>d</i></span>) from the center of the base rectangle to the apex of the pyramid (the center of the sphere) are known, then the above equation can be manipulated to give </p><p><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 \Omega =4\arctan {\frac {\alpha \beta }{2d{\sqrt {4d^{2}+\alpha ^{2}+\beta ^{2}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>4</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mrow> <mn>2</mn> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =4\arctan {\frac {\alpha \beta }{2d{\sqrt {4d^{2}+\alpha ^{2}+\beta ^{2}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc55e4cd9c2b3d77d58d3365a6c17bb83a37963" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.413ex; height:6.676ex;" alt="{\displaystyle \Omega =4\arctan {\frac {\alpha \beta }{2d{\sqrt {4d^{2}+\alpha ^{2}+\beta ^{2}}}}}.}"></span> </p><p>The solid angle of a right <span class="texhtml mvar" style="font-style:italic;">n</span>-gonal pyramid, where the pyramid base is a regular <span class="texhtml mvar" style="font-style:italic;">n</span>-sided polygon of circumradius <span class="texhtml mvar" style="font-style:italic;">r</span>, with a pyramid height <span class="texhtml mvar" style="font-style:italic;">h</span> is </p><p><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 \Omega =2\pi -2n\arctan \left({\frac {\tan \left({\pi \over n}\right)}{\sqrt {1+{r^{2} \over h^{2}}}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =2\pi -2n\arctan \left({\frac {\tan \left({\pi \over n}\right)}{\sqrt {1+{r^{2} \over h^{2}}}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaafabcadd7ae949813a0f3c1d016588b25a7f7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:34.013ex; height:13.176ex;" alt="{\displaystyle \Omega =2\pi -2n\arctan \left({\frac {\tan \left({\pi \over n}\right)}{\sqrt {1+{r^{2} \over h^{2}}}}}\right).}"></span> </p><p>The solid angle of an arbitrary pyramid with an <span class="texhtml"><i>n</i></span>-sided base defined by the sequence of unit vectors representing edges <span class="texhtml">{<i>s</i><sub>1</sub>, <i>s</i><sub>2</sub>}, ... <i>s</i><sub><i>n</i></sub></span> can be efficiently computed by:<sup id="cite_ref-Mazonka_4-1" class="reference"><a href="#cite_note-Mazonka-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><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 \Omega =2\pi -\arg \prod _{j=1}^{n}\left(\left(s_{j-1}s_{j}\right)\left(s_{j}s_{j+1}\right)-\left(s_{j-1}s_{j+1}\right)+i\left[s_{j-1}s_{j}s_{j+1}\right]\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>arg</mi> <mo>⁡</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mrow> <mo>[</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =2\pi -\arg \prod _{j=1}^{n}\left(\left(s_{j-1}s_{j}\right)\left(s_{j}s_{j+1}\right)-\left(s_{j-1}s_{j+1}\right)+i\left[s_{j-1}s_{j}s_{j+1}\right]\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0474c9860232795ae0d57fb0ee32f2979b7c8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:64.518ex; height:7.176ex;" alt="{\displaystyle \Omega =2\pi -\arg \prod _{j=1}^{n}\left(\left(s_{j-1}s_{j}\right)\left(s_{j}s_{j+1}\right)-\left(s_{j-1}s_{j+1}\right)+i\left[s_{j-1}s_{j}s_{j+1}\right]\right).}"></span> </p><p>where parentheses (* *) is a <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> and square brackets [* * *] is a <a href="/wiki/Scalar_triple_product" class="mw-redirect" title="Scalar triple product">scalar triple product</a>, and <span class="texhtml mvar" style="font-style:italic;">i</span> is an <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. Indices are cycled: <span class="texhtml"><i>s</i><sub>0</sub> = <i>s</i><sub><i>n</i></sub></span> and <span class="texhtml"><i>s</i><sub>1</sub> = <i>s</i><sub><i>n</i> + 1</sub></span>. The complex products add the phase associated with each vertex angle of the polygon. However, a multiple of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span> is lost in the branch cut of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arg</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec03a9c123925f400a40064ca491d268f9312956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.237ex; height:2.009ex;" alt="{\displaystyle \arg }"></span> and must be kept track of separately. Also, the running product of complex phases must scaled occasionally to avoid underflow in the limit of nearly parallel segments. </p> <div class="mw-heading mw-heading3"><h3 id="Latitude-longitude_rectangle">Latitude-longitude rectangle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=7" title="Edit section: Latitude-longitude rectangle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The solid angle of a latitude-longitude rectangle on a <a href="/wiki/Globe" title="Globe">globe</a> is <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 \left(\sin \phi _{\mathrm {N} }-\sin \phi _{\mathrm {S} }\right)\left(\theta _{\mathrm {E} }-\theta _{\mathrm {W} }\,\!\right)\;\mathrm {sr} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">r</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sin \phi _{\mathrm {N} }-\sin \phi _{\mathrm {S} }\right)\left(\theta _{\mathrm {E} }-\theta _{\mathrm {W} }\,\!\right)\;\mathrm {sr} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf840153d418e1e33277c845f9d81a0af3fc349f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.515ex; height:2.843ex;" alt="{\displaystyle \left(\sin \phi _{\mathrm {N} }-\sin \phi _{\mathrm {S} }\right)\left(\theta _{\mathrm {E} }-\theta _{\mathrm {W} }\,\!\right)\;\mathrm {sr} ,}"></span> where <span class="texhtml"><i>φ</i><sub>N</sub></span> and <span class="texhtml"><i>φ</i><sub>S</sub></span> are north and south lines of <a href="/wiki/Latitude" title="Latitude">latitude</a> (measured from the <a href="/wiki/Equator" title="Equator">equator</a> in <a href="/wiki/Radian" title="Radian">radians</a> with angle increasing northward), and <span class="texhtml"><i>θ</i><sub>E</sub></span> and <span class="texhtml"><i>θ</i><sub>W</sub></span> are east and west lines of <a href="/wiki/Longitude" title="Longitude">longitude</a> (where the angle in radians increases eastward).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Mathematically, this represents an arc of angle <span class="texhtml"><i>ϕ</i><sub>N</sub> − <i>ϕ</i><sub>S</sub></span> swept around a sphere by <span class="texhtml"><i>θ</i><sub>E</sub> − <i>θ</i><sub>W</sub></span> radians. When longitude spans 2<span class="texhtml mvar" style="font-style:italic;">π</span> radians and latitude spans <span class="texhtml mvar" style="font-style:italic;">π</span> radians, the solid angle is that of a sphere. </p><p>A latitude-longitude rectangle should not be confused with the solid angle of a rectangular pyramid. All four sides of a rectangular pyramid intersect the sphere's surface in <a href="/wiki/Great_circle" title="Great circle">great circle</a> arcs. With a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not. </p> <div class="mw-heading mw-heading3"><h3 id="Celestial_objects">Celestial objects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=8" title="Edit section: Celestial objects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By using the definition of <a href="/wiki/Angular_diameter" title="Angular diameter">angular diameter</a>, the formula for the solid angle of a celestial object can be defined in terms of the radius of the object, <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197e66194eb64577670e2a100026bff6fb15d236" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\textstyle R}"></span>, and the distance from the observer to the object, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>: </p><p><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 \Omega =2\pi \left(1-{\frac {\sqrt {d^{2}-R^{2}}}{d}}\right):d\geq R.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>d</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <mi>d</mi> <mo>≥</mo> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =2\pi \left(1-{\frac {\sqrt {d^{2}-R^{2}}}{d}}\right):d\geq R.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782cd672742ad3a2e964c1d470b741e2b89136f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.095ex; height:7.509ex;" alt="{\displaystyle \Omega =2\pi \left(1-{\frac {\sqrt {d^{2}-R^{2}}}{d}}\right):d\geq R.}"></span> </p><p>By inputting the appropriate average values for the <a href="/wiki/Sun" title="Sun">Sun</a> and the <a href="/wiki/Moon" title="Moon">Moon</a> (in relation to Earth), the average solid angle of the Sun is <span class="nowrap"><span data-sort-value="6995679400000000000♠"></span>6.794<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−5</sup></span> steradians and the average solid angle of the <a href="/wiki/Moon" title="Moon">Moon</a> is <span class="nowrap"><span data-sort-value="6995641800000000000♠"></span>6.418<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−5</sup></span> steradians. In terms of the total celestial sphere, the <a href="/wiki/Sun" title="Sun">Sun</a> and the <a href="/wiki/Moon" title="Moon">Moon</a> subtend average <i>fractional areas</i> of <span class="nowrap"><span data-sort-value="6996540600000000000♠"></span>0.000<span style="margin-left:.25em;">5406</span></span>% (<span class="nowrap"><span data-sort-value="7000540600000000000♠"></span>5.406 <a href="/wiki/Part_per_million" class="mw-redirect" title="Part per million">ppm</a></span>) and <span class="nowrap"><span data-sort-value="6996510700000000000♠"></span>0.000<span style="margin-left:.25em;">5107</span></span>% (<span class="nowrap"><span data-sort-value="6994510699999999999♠"></span>5.107 ppm</span>), respectively. As these solid angles are about the same size, the Moon can cause both total and annular solar <a href="/wiki/Solar_eclipse" title="Solar eclipse">eclipses</a> depending on the distance between the Earth and the Moon during the eclipse. </p> <div class="mw-heading mw-heading2"><h2 id="Solid_angles_in_arbitrary_dimensions">Solid angles in arbitrary dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=9" title="Edit section: Solid angles in arbitrary dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The solid angle subtended by the complete (<span class="texhtml mvar" style="font-style:italic;">d − 1</span>)-dimensional spherical surface of the unit sphere in <a href="/wiki/Euclidean_space" title="Euclidean space"><span class="texhtml"><i>d</i></span>-dimensional Euclidean space</a> can be defined in any number of dimensions <span class="texhtml"><i>d</i></span>. One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula <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 \Omega _{d}={\frac {2\pi ^{\frac {d}{2}}}{\Gamma \left({\frac {d}{2}}\right)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{d}={\frac {2\pi ^{\frac {d}{2}}}{\Gamma \left({\frac {d}{2}}\right)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/016b623cebc313b2747225f50571a4046ceaed69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:13.663ex; height:9.343ex;" alt="{\displaystyle \Omega _{d}={\frac {2\pi ^{\frac {d}{2}}}{\Gamma \left({\frac {d}{2}}\right)}},}"></span> where <span class="texhtml">Γ</span> is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>. When <span class="texhtml"><i>d</i></span> is an integer, the gamma function can be computed explicitly.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> It follows that <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 \Omega _{d}={\begin{cases}{\frac {1}{\left({\frac {d}{2}}-1\right)!}}2\pi ^{\frac {d}{2}}\ &d{\text{ even}}\\{\frac {\left({\frac {1}{2}}\left(d-1\right)\right)!}{(d-1)!}}2^{d}\pi ^{{\frac {1}{2}}(d-1)}\ &d{\text{ odd}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> </mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mtext> </mtext> </mtd> <mtd> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mtext> </mtext> </mtd> <mtd> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> odd</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{d}={\begin{cases}{\frac {1}{\left({\frac {d}{2}}-1\right)!}}2\pi ^{\frac {d}{2}}\ &d{\text{ even}}\\{\frac {\left({\frac {1}{2}}\left(d-1\right)\right)!}{(d-1)!}}2^{d}\pi ^{{\frac {1}{2}}(d-1)}\ &d{\text{ odd}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbaf6b586f4b6d5cc3b1ab81e9aa071863e1eb1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:36.534ex; height:12.176ex;" alt="{\displaystyle \Omega _{d}={\begin{cases}{\frac {1}{\left({\frac {d}{2}}-1\right)!}}2\pi ^{\frac {d}{2}}\ &d{\text{ even}}\\{\frac {\left({\frac {1}{2}}\left(d-1\right)\right)!}{(d-1)!}}2^{d}\pi ^{{\frac {1}{2}}(d-1)}\ &d{\text{ odd}}.\end{cases}}}"></span> </p><p>This gives the expected results of 4<span class="texhtml mvar" style="font-style:italic;">π</span> steradians for the 3D sphere bounded by a surface of area <span class="texhtml">4π<i>r</i><sup>2</sup></span> and 2<span class="texhtml mvar" style="font-style:italic;">π</span> radians for the 2D circle bounded by a circumference of length <span class="texhtml">2π<i>r</i></span>. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval <span class="texhtml">[−<i>r</i>, <i>r</i>]</span> and this is bounded by two limiting points. </p><p>The counterpart to the vector formula in arbitrary dimension was derived by Aomoto<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and independently by Ribando.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> It expresses them as an infinite multivariate <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</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 \Omega =\Omega _{d}{\frac {\left|\det(V)\right|}{(4\pi )^{d/2}}}\sum _{{\vec {a}}\in \mathbb {N} _{0}^{\binom {d}{2}}}\left[{\frac {(-2)^{\sum _{i<j}a_{ij}}}{\prod _{i<j}a_{ij}!}}\prod _{i}\Gamma \left({\frac {1+\sum _{m\neq i}a_{im}}{2}}\right)\right]{\vec {\alpha }}^{\vec {a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>d</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </msubsup> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>≠</mo> <mi>i</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =\Omega _{d}{\frac {\left|\det(V)\right|}{(4\pi )^{d/2}}}\sum _{{\vec {a}}\in \mathbb {N} _{0}^{\binom {d}{2}}}\left[{\frac {(-2)^{\sum _{i<j}a_{ij}}}{\prod _{i<j}a_{ij}!}}\prod _{i}\Gamma \left({\frac {1+\sum _{m\neq i}a_{im}}{2}}\right)\right]{\vec {\alpha }}^{\vec {a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc134d3a32a327f1989c92b95bb5fc82e4c9f1e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:65.755ex; height:10.676ex;" alt="{\displaystyle \Omega =\Omega _{d}{\frac {\left|\det(V)\right|}{(4\pi )^{d/2}}}\sum _{{\vec {a}}\in \mathbb {N} _{0}^{\binom {d}{2}}}\left[{\frac {(-2)^{\sum _{i<j}a_{ij}}}{\prod _{i<j}a_{ij}!}}\prod _{i}\Gamma \left({\frac {1+\sum _{m\neq i}a_{im}}{2}}\right)\right]{\vec {\alpha }}^{\vec {a}}.}"></span> Given <span class="texhtml mvar" style="font-style:italic;">d</span> unit vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1e99e843fffeae2c302f24e96edb76b2cbf915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.975ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{i}}"></span> defining the angle, let <span class="texhtml mvar" style="font-style:italic;">V</span> denote the matrix formed by combining them so the <span class="texhtml mvar" style="font-style:italic;">i</span>th column 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 {\vec {v}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1e99e843fffeae2c302f24e96edb76b2cbf915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.975ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{i}}"></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 \alpha _{ij}={\vec {v}}_{i}\cdot {\vec {v}}_{j}=\alpha _{ji},\alpha _{ii}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{ij}={\vec {v}}_{i}\cdot {\vec {v}}_{j}=\alpha _{ji},\alpha _{ii}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6064b350e803731788f670a974b58c27ea14fc45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.015ex; height:3.009ex;" alt="{\displaystyle \alpha _{ij}={\vec {v}}_{i}\cdot {\vec {v}}_{j}=\alpha _{ji},\alpha _{ii}=1}"></span>. The variables <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 \alpha _{ij},1\leq i<j\leq d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{ij},1\leq i<j\leq d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cae56896d751558a6bbc2732b87795de7df7610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.433ex; height:2.843ex;" alt="{\displaystyle \alpha _{ij},1\leq i<j\leq d}"></span> form a multivariable <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 {\vec {\alpha }}=(\alpha _{12},\dotsc ,\alpha _{1d},\alpha _{23},\dotsc ,\alpha _{d-1,d})\in \mathbb {R} ^{\binom {d}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>d</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\alpha }}=(\alpha _{12},\dotsc ,\alpha _{1d},\alpha _{23},\dotsc ,\alpha _{d-1,d})\in \mathbb {R} ^{\binom {d}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38f81d240d4b1ef53e31c0d607a6be7e4285e06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.324ex; height:4.509ex;" alt="{\displaystyle {\vec {\alpha }}=(\alpha _{12},\dotsc ,\alpha _{1d},\alpha _{23},\dotsc ,\alpha _{d-1,d})\in \mathbb {R} ^{\binom {d}{2}}}"></span>. For a "congruent" integer multiexponent <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 {\vec {a}}=(a_{12},\dotsc ,a_{1d},a_{23},\dotsc ,a_{d-1,d})\in \mathbb {N} _{0}^{\binom {d}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>d</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=(a_{12},\dotsc ,a_{1d},a_{23},\dotsc ,a_{d-1,d})\in \mathbb {N} _{0}^{\binom {d}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b48423679647d36859f501c212335743a37206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.682ex; height:5.009ex;" alt="{\displaystyle {\vec {a}}=(a_{12},\dotsc ,a_{1d},a_{23},\dotsc ,a_{d-1,d})\in \mathbb {N} _{0}^{\binom {d}{2}},}"></span> define <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 {\vec {\alpha }}^{\vec {a}}=\prod \alpha _{ij}^{a_{ij}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </msup> <mo>=</mo> <mo>∏</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\vec {\alpha }}^{\vec {a}}=\prod \alpha _{ij}^{a_{ij}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f668fcdbe9ea4fb5e3cff34203ced75be7b12f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.033ex; height:4.343ex;" alt="{\textstyle {\vec {\alpha }}^{\vec {a}}=\prod \alpha _{ij}^{a_{ij}}}"></span>. Note that here <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 \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab7e98123f0def29a1cd3df96a0b7a58f4202c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} _{0}}"></span> = non-negative integers, or natural numbers beginning with 0. The notation <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 \alpha _{ji}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{ji}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df57a9e94d996a97766949dbcfb2050b5557493a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.965ex; height:2.343ex;" alt="{\displaystyle \alpha _{ji}}"></span> for <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>i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>></mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j>i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb45e6bd25f2486ca8b3052e74e27c11fa0d1761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:4.886ex; height:2.509ex;" alt="{\displaystyle j>i}"></span> means the variable <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 \alpha _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768573314617898450b38de70be02c6341de6235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.965ex; height:2.343ex;" alt="{\displaystyle \alpha _{ij}}"></span>, similarly for the exponents <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_{ji}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ji}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3630693f8dd03d56a80ad9b4fde40e3e14020962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ji}}"></span>. Hence, the term <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 \sum _{m\neq l}a_{lm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>≠</mo> <mi>l</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{m\neq l}a_{lm}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67771520a15636ac4aae3dd9ac9560ad4dd9bdd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.68ex; height:3.343ex;" alt="{\textstyle \sum _{m\neq l}a_{lm}}"></span> means the sum over all terms in <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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> in which l appears as either the first or second index. Where this series converges, it converges to the solid angle defined by the vectors. </p> <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=Solid_angle&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-u421-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-u421_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/octant">"octant"</a>. <i>PlanetMath.org</i>. 2013-03-22<span class="reference-accessdate">. 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(2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00454-006-1253-4">"Measuring Solid Angles Beyond Dimension Three"</a>. <i>Discrete & Computational Geometry</i>. <b>36</b> (3): <span class="nowrap">479–</span>487. <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/10.1007%2Fs00454-006-1253-4">10.1007/s00454-006-1253-4</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+%26+Computational+Geometry&rft.atitle=Measuring+Solid+Angles+Beyond+Dimension+Three&rft.volume=36&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E479-%3C%2Fspan%3E487&rft.date=2006&rft_id=info%3Adoi%2F10.1007%2Fs00454-006-1253-4&rft.aulast=Ribando&rft.aufirst=Jason+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs00454-006-1253-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolid+angle" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaffey1954" class="citation journal cs1">Jaffey, A. H. (1954). "Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables". <i>Rev. Sci. Instrum</i>. <b>25</b> (4): <span class="nowrap">349–</span>354. <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/1954RScI...25..349J">1954RScI...25..349J</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1771061">10.1063/1.1771061</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rev.+Sci.+Instrum.&rft.atitle=Solid+angle+subtended+by+a+circular+aperture+at+point+and+spread+sources%3A+formulas+and+some+tables&rft.volume=25&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E349-%3C%2Fspan%3E354&rft.date=1954&rft_id=info%3Adoi%2F10.1063%2F1.1771061&rft_id=info%3Abibcode%2F1954RScI...25..349J&rft.aulast=Jaffey&rft.aufirst=A.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolid+angle" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMasket1957" class="citation journal cs1">Masket, A. Victor (1957). "Solid angle contour integrals, series, and tables". <i>Rev. Sci. 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M.; Prata, M. J.; Kalla, S. L.; Abbas, M. I.; Oner, F.; Galiano, E. (2007). "Some further analytical results on the solid angle subtended at a point by a circular disk using elliptic integrals". <i>Nucl. Instrum. Methods Phys. Res. A</i>. <b>580</b>: <span class="nowrap">149–</span>152. <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/2007NIMPA.580..149T">2007NIMPA.580..149T</a>. <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%2Fj.nima.2007.05.055">10.1016/j.nima.2007.05.055</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nucl.+Instrum.+Methods+Phys.+Res.+A&rft.atitle=Some+further+analytical+results+on+the+solid+angle+subtended+at+a+point+by+a+circular+disk+using+elliptic+integrals&rft.volume=580&rft.pages=%3Cspan+class%3D%22nowrap%22%3E149-%3C%2Fspan%3E152&rft.date=2007&rft_id=info%3Adoi%2F10.1016%2Fj.nima.2007.05.055&rft_id=info%3Abibcode%2F2007NIMPA.580..149T&rft.aulast=Timus&rft.aufirst=D.+M.&rft.au=Prata%2C+M.+J.&rft.au=Kalla%2C+S.+L.&rft.au=Abbas%2C+M.+I.&rft.au=Oner%2C+F.&rft.au=Galiano%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolid+angle" class="Z3988"></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><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" /></a></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:Solid_angle" class="extiw" title="commons:Category:Solid angle">Solid angle</a></span>.</div></div> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_angle&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Arthur P. Norton, A Star Atlas, Gall and Inglis, Edinburgh, 1969.</li> <li>M. G. Kendall, A Course in the Geometry of N Dimensions, No. 8 of Griffin's Statistical Monographs & Courses, ed. M. G. Kendall, Charles Griffin & Co. Ltd, London, 1961</li> <li><span class="citation mathworld" id="Reference-Mathworld-Solid_Angle"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SolidAngle.html">"Solid Angle"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Solid+Angle&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSolidAngle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolid+angle" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist 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title="Template talk:Classical mechanics SI units"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics_SI_units" title="Special:EditPage/Template:Classical mechanics SI units"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classical_mechanics_SI_units66" style="font-size:114%;margin:0 4em"><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0;"><table class="wikitable" style="text-align:center;line-height:0.9;border-collapse:collapse;margin:auto;border:none;background:none;"> <tbody><tr> <td colspan="4" style="border:none;backgound:none; font-weight:bold;">Linear/translational quantities</td> <td rowspan="12" style="border:none;backgound:none;"></td> <td colspan="4" style="border:none;backgound:none; font-weight:bold;">Angular/rotational quantities</td> </tr> <tr> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;">L</th> <th style="font-weight:normal;">L<sup>2</sup></th> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;"><span class="texhtml"><i>θ</i></span></th> <th style="font-weight:normal;"><span class="texhtml"><i>θ</i></span><sup>2</sup></th> </tr> <tr> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: <span class="texhtml"><i>t</i></span><br /><a href="/wiki/Second" title="Second">s</a></td> <td><a href="/wiki/Absement" title="Absement">absement</a>: <span class="texhtml"><b>A</b></span><br /><a href="/wiki/Meter_second" class="mw-redirect" title="Meter second">m s</a></td> <td></td> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: <span class="texhtml"><i>t</i></span><br /><a href="/wiki/Second" title="Second">s</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Distance" title="Distance">distance</a>: <span class="texhtml"><i>d</i></span>, <span class="nowrap"><a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a>: <span class="texhtml"><b>r</b></span>, <span class="texhtml"><b>s</b></span>, <span class="texhtml"><b>x</b></span></span>, <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a><br /><a href="/wiki/Metre" title="Metre">m</a></td> <td><a href="/wiki/Area" title="Area">area</a>: <span class="texhtml"><i>A</i></span><br /><a href="/wiki/Square_metre" title="Square metre">m<sup>2</sup></a></td> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Angle" title="Angle">angle</a>: <span class="texhtml"><i>θ</i></span>, <a href="/wiki/Angular_displacement" title="Angular displacement">angular displacement</a>: <span class="texhtml"><i><b>θ</b></i></span><br /><a href="/wiki/Radian" title="Radian">rad</a></td> <td><span class="nowrap"><a class="mw-selflink selflink">solid angle</a>: <span class="texhtml">Ω</span><br /><a href="/wiki/Steradian" title="Steradian">rad<sup>2</sup>, sr</a></span></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/Frequency" title="Frequency">frequency</a>: <span class="texhtml"><i>f</i></span></span><br /><a href="/wiki/Inverse_second" title="Inverse second">s<sup>−1</sup></a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Speed" title="Speed">speed</a>: <span class="texhtml"><i>v</i></span>, <a href="/wiki/Velocity" title="Velocity">velocity</a>: <span class="texhtml"><b>v</b></span><br /><a href="/wiki/Metre_per_second" title="Metre per second">m s<sup>−1</sup></a></td> <td><a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a>: <span class="texhtml"><i>ν</i></span>,<br /><a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">specific angular momentum</a>: <span class="texhtml"><b>h</b></span><br />m<sup>2</sup> s<sup>−1</sup></td> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/Frequency" title="Frequency">frequency</a>: <span class="texhtml"><i>f</i></span></span>, <span class="nowrap"><a href="/wiki/Rotational_speed" class="mw-redirect" title="Rotational speed">rotational speed</a>: <span class="texhtml"><i>n</i></span></span>, <span class="nowrap"><a href="/wiki/Rotational_velocity" class="mw-redirect" title="Rotational velocity">rotational velocity</a>: <span class="texhtml"><i><b>n</b></i></span></span><br /><a href="/wiki/Inverse_second" title="Inverse second">s<sup>−1</sup></a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Angular_speed" class="mw-redirect" title="Angular speed">angular speed</a>: <span class="texhtml"><i>ω</i></span>, <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>: <span class="texhtml"><i><b>ω</b></i></span><br /><a href="/wiki/Radian_per_second" title="Radian per second">rad<span style="letter-spacing:0.1em"> </span>s<sup>−1</sup></a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−2</sup></th> <td></td> <td><a href="/wiki/Acceleration" title="Acceleration">acceleration</a>: <span class="texhtml"><b>a</b></span><br /><a href="/wiki/Metre_per_second_squared" title="Metre per second squared">m s<sup>−2</sup></a></td> <td></td> <th style="font-weight:normal;">T<sup>−2</sup></th> <td><span class="nowrap"><a href="/wiki/Rotational_acceleration" class="mw-redirect" title="Rotational acceleration">rotational acceleration</a></span><br /><a href="/wiki/Inverse_square_second" class="mw-redirect" title="Inverse square second">s<sup>−2</sup></a></td> <td><a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a>: <span class="texhtml"><i><b>α</b></i></span><br /><a href="/wiki/Radian_per_second_squared" class="mw-redirect" title="Radian per second squared">rad<span style="letter-spacing:0.1em"> </span>s<sup>−2</sup></a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a href="/wiki/Jerk_(physics)" title="Jerk (physics)">jerk</a>: <span class="texhtml"><b>j</b></span><br />m s<sup>−3</sup></td> <td></td> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a href="/wiki/Jerk_(physics)#Jerk_in_rotation" title="Jerk (physics)">angular jerk</a>: <span class="texhtml"><b>ζ</b></span><br />rad<span style="letter-spacing:0.1em"> </span>s<sup>−3</sup></td> <td></td> </tr> <tr style="border-top: 3px double #a2a9b1;"> <th style="font-weight:normal;">M</th> <td><a href="/wiki/Mass" title="Mass">mass</a>: <span class="texhtml"><i>m</i></span><br /><a href="/wiki/Kilogram" title="Kilogram">kg</a></td> <td><a href="/wiki/Moment_(physics)" title="Moment (physics)">weighted position</a>: <span class="texhtml"><i>M</i> ⟨<i>x</i>⟩ = ∑ <i>m</i> <i>x</i></span> </td> <td><a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>: <span class="texhtml"><i>I</i></span><br /><a href="/wiki/Kilogram_square_metre" class="mw-redirect" title="Kilogram square metre">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup></a></td> <th style="font-weight:normal;">ML</th> <td></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−1</sup></th> <td><a href="/wiki/Mass_flow_rate" title="Mass flow rate">Mass flow rate</a>: <span class="texhtml"><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 {\dot {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad59b9876301e8fb75b9ddbf08de594b87251d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.176ex;" alt="{\displaystyle {\dot {m}}}"></span></span><br /><a href="/wiki/Kilogram_per_second" class="mw-redirect" title="Kilogram per second">kg<span style="letter-spacing:0.1em"> </span>s<sup>−1</sup></a></td> <td><a href="/wiki/Momentum" title="Momentum">momentum</a>: <span class="texhtml"><b>p</b></span>, <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a>: <span class="texhtml"><b>J</b></span><br /><a href="/wiki/Kilogram_metre_per_second" class="mw-redirect" title="Kilogram metre per second">kg<span style="letter-spacing:0.1em"> </span>m s<sup>−1</sup></a>, <a href="/wiki/Newton_second" class="mw-redirect" title="Newton second">N s</a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: <span class="texhtml">𝒮</span>, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: <span class="texhtml">ℵ</span><br /><a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> <th style="font-weight:normal;">MLT<sup>−1</sup></th> <td></td> <td><a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>: <span class="texhtml"><b>L</b></span>, <a href="/wiki/List_of_equations_in_classical_mechanics#Derived_dynamic_quantities" title="List of equations in classical mechanics">angular impulse</a>: <span class="texhtml">Δ<b>L</b></span><br /><a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m rad s<sup>−1</sup></a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−2</sup></th> <td></td> <td><a href="/wiki/Force" title="Force">force</a>: <span class="texhtml"><b>F</b></span>, <a href="/wiki/Weight" title="Weight">weight</a>: <span class="texhtml"><b>F</b><sub>g</sub></span><br /><span style="margin-right:0.1em;">kg </span> m s<sup>−2</sup>, <a href="/wiki/Newton_(unit)" title="Newton (unit)">N</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: <span class="texhtml"><i>E</i></span>, <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>: <span class="texhtml"><i>W</i></span>, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: <span class="texhtml"><i>L</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/Joule" title="Joule">J</a></td> <th style="font-weight:normal;">MLT<sup>−2</sup></th> <td></td> <td><a href="/wiki/Torque" title="Torque">torque</a>: <span class="texhtml"><i><b>τ</b></i></span>, <a href="/wiki/Torque#Terminology" title="Torque">moment</a>: <span class="texhtml"><b>M</b></span><br /><span style="margin-right:0.1em;">kg</span> m rad s<sup>−2</sup>, <a href="/wiki/Newton-metre" title="Newton-metre">N m</a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−3</sup></th> <td></td> <td><a href="/wiki/Yank_(physics)" class="mw-redirect" title="Yank (physics)">yank</a>: <span class="texhtml"><b>Y</b></span><br /><span style="margin-right:0.1em;">kg</span> m s<sup>−3</sup>, N s<sup>−1</sup></td> <td><a href="/wiki/Power_(physics)" title="Power 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Solid angle - Wikipedia