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class="vector-toc-list"> </ul> </li> <li id="toc-Dimensional_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimensional_analysis"> <div class="vector-toc-text"> 1.2 Dimensional analysis </div> </a> <ul id="toc-Dimensional_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conversions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conversions"> <div class="vector-toc-text"> 2 Conversions </div> </a> <button aria-controls="toc-Conversions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Conversions subsection </button> <ul id="toc-Conversions-sublist" class="vector-toc-list"> <li id="toc-Between_degrees" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Between_degrees"> <div class="vector-toc-text"> 2.1 Between degrees </div> </a> <ul id="toc-Between_degrees-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Between_gradians" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Between_gradians"> <div class="vector-toc-text"> 2.2 Between gradians </div> </a> <ul id="toc-Between_gradians-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Usage" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Usage"> <div class="vector-toc-text"> 3 Usage </div> </a> <button aria-controls="toc-Usage-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Usage subsection </button> <ul id="toc-Usage-sublist" class="vector-toc-list"> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> 3.1 Mathematics </div> </a> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics"> <div class="vector-toc-text"> 3.2 Physics </div> </a> <ul id="toc-Physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prefixes_and_variants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prefixes_and_variants"> <div class="vector-toc-text"> 3.3 Prefixes and variants </div> </a> <ul id="toc-Prefixes_and_variants-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 4 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Pre-20th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pre-20th_century"> <div class="vector-toc-text"> 4.1 Pre-20th century </div> </a> <ul id="toc-Pre-20th_century-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_an_SI_unit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_an_SI_unit"> <div class="vector-toc-text"> 4.2 As an SI unit </div> </a> <ul id="toc-As_an_SI_unit-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 5 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 6 Notes </div> </a> <ul id="toc-Notes-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"> 7 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 8 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Radian</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 84 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-84" aria-hidden="true" > 84 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Radiaal" title="Radiaal – Afrikaans" lang="af" hreflang="af" data-title="Radiaal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%A7%D8%AF%D9%8A%D8%A7%D9%86" title="راديان – Arabic" lang="ar" hreflang="ar" data-title="راديان" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Radi%C3%A1n" title="Radián – Asturian" lang="ast" hreflang="ast" data-title="Radián" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Radian" title="Radian – Azerbaijani" lang="az" hreflang="az" data-title="Radian" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B0%E0%A7%87%E0%A6%A1%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8" title="রেডিয়ান – Bangla" lang="bn" hreflang="bn" data-title="রেডিয়ান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D1%8B%D1%8F%D0%BD" title="Радыян – Belarusian" lang="be" hreflang="be" data-title="Радыян" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D1%8B%D1%8F%D0%BD" title="Радыян – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Радыян" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%B0%D0%BD" title="Радиан – Bulgarian" lang="bg" hreflang="bg" data-title="Радиан" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%82%E0%BD%9E%E0%BD%B4%E0%BC%8B%E0%BD%9A%E0%BD%91%E0%BC%8D" title="གཞུ་ཚད། – Tibetan" lang="bo" hreflang="bo" data-title="གཞུ་ཚད།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target">བོད་ཡིག</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Radijan" title="Radijan – Bosnian" lang="bs" hreflang="bs" data-title="Radijan" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Radian" title="Radian – Catalan" lang="ca" hreflang="ca" data-title="Radian" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%B0%D0%BD" title="Радиан – Chuvash" lang="cv" hreflang="cv" data-title="Радиан" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Radi%C3%A1n" title="Radián – Czech" lang="cs" hreflang="cs" data-title="Radián" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Radian" title="Radian – Welsh" lang="cy" hreflang="cy" data-title="Radian" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Radian" title="Radian – Danish" lang="da" hreflang="da" data-title="Radian" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Radiant_(Einheit)" title="Radiant (Einheit) – German" lang="de" hreflang="de" data-title="Radiant (Einheit)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Radiaan" title="Radiaan – Estonian" lang="et" hreflang="et" data-title="Radiaan" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BA%CF%84%CE%AF%CE%BD%CE%B9%CE%BF_(%CE%BC%CE%BF%CE%BD%CE%AC%CE%B4%CE%B1_%CE%BC%CE%AD%CF%84%CF%81%CE%B7%CF%83%CE%B7%CF%82)" title="Ακτίνιο (μονάδα μέτρησης) – Greek" lang="el" hreflang="el" data-title="Ακτίνιο (μονάδα μέτρησης)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Radi%C3%A1n" title="Radián – Spanish" lang="es" hreflang="es" data-title="Radián" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Radiano" title="Radiano – Esperanto" lang="eo" hreflang="eo" data-title="Radiano" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Radian" title="Radian – Basque" lang="eu" hreflang="eu" data-title="Radian" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D8%A7%D8%AF%DB%8C%D8%A7%D9%86" title="رادیان – Persian" lang="fa" hreflang="fa" data-title="رادیان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Radian" title="Radian – French" lang="fr" hreflang="fr" data-title="Radian" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Raidian" title="Raidian – Irish" lang="ga" hreflang="ga" data-title="Raidian" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Radi%C3%A1n" title="Radián – Galician" lang="gl" hreflang="gl" data-title="Radián" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%BC%A7%E5%BA%A6" title="弧度 – Gan" lang="gan" hreflang="gan" data-title="弧度" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%9D%BC%EB%94%94%EC%95%88" title="라디안 – Korean" lang="ko" hreflang="ko" data-title="라디안" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Radiya" title="Radiya – Hausa" lang="ha" hreflang="ha" data-title="Radiya" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target">Hausa</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8C%D5%A1%D5%A4%D5%AB%D5%A1%D5%B6" title="Ռադիան – Armenian" lang="hy" hreflang="hy" data-title="Ռադիան" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B0%E0%A5%87%E0%A4%A1%E0%A4%BF%E0%A4%AF%E0%A4%A8" title="रेडियन – Hindi" lang="hi" hreflang="hi" data-title="रेडियन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Radijan" title="Radijan – Croatian" lang="hr" hreflang="hr" data-title="Radijan" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Radian" title="Radian – Indonesian" lang="id" hreflang="id" data-title="Radian" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Radiano" title="Radiano – Interlingua" lang="ia" hreflang="ia" data-title="Radiano" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Rad%C3%ADanar" title="Radíanar – Icelandic" lang="is" hreflang="is" data-title="Radíanar" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Radiante" title="Radiante – Italian" lang="it" hreflang="it" data-title="Radiante" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A8%D7%93%D7%99%D7%90%D7%9F" title="רדיאן – Hebrew" lang="he" hreflang="he" data-title="רדיאן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%90%E1%83%93%E1%83%98%E1%83%90%E1%83%9C%E1%83%98" title="რადიანი – Georgian" lang="ka" hreflang="ka" data-title="რადიანი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%B0%D0%BD" title="Радиан – Kazakh" lang="kk" hreflang="kk" data-title="Радиан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Radians_(unitas)" title="Radians (unitas) – Latin" lang="la" hreflang="la" data-title="Radians (unitas)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Radi%C4%81ns" title="Radiāns – Latvian" lang="lv" hreflang="lv" data-title="Radiāns" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Radiant_(Eenheet)" title="Radiant (Eenheet) – Luxembourgish" lang="lb" hreflang="lb" data-title="Radiant (Eenheet)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Radianas" title="Radianas – Lithuanian" lang="lt" hreflang="lt" data-title="Radianas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Radi%C3%A1n" title="Radián – Hungarian" lang="hu" hreflang="hu" data-title="Radián" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D1%98%D0%B0%D0%BD" title="Радијан – Macedonian" lang="mk" hreflang="mk" data-title="Радијан" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A5%80" title="त्रिज्यी – Marathi" lang="mr" hreflang="mr" data-title="त्रिज्यी" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Radian" title="Radian – Malay" lang="ms" hreflang="ms" data-title="Radian" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%B0%D0%BD" title="Радиан – Mongolian" lang="mn" hreflang="mn" data-title="Радиан" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Radiaal_(wiskunde)" title="Radiaal (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Radiaal (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%82%B8%E3%82%A2%E3%83%B3" title="ラジアン – Japanese" lang="ja" hreflang="ja" data-title="ラジアン" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Radian" title="Radian – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Radian" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Radian" title="Radian – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Radian" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Radian" title="Radian – Occitan" lang="oc" hreflang="oc" data-title="Radian" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Radian" title="Radian – Uzbek" lang="uz" hreflang="uz" data-title="Radian" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B0%E0%A9%87%E0%A8%A1%E0%A9%80%E0%A8%85%E0%A8%A8" title="ਰੇਡੀਅਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਰੇਡੀਅਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Riedian" title="Riedian – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Riedian" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Rajant" title="Rajant – Piedmontese" lang="pms" hreflang="pms" data-title="Rajant" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Radian" title="Radian – Polish" lang="pl" hreflang="pl" data-title="Radian" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Radiano" title="Radiano – Portuguese" lang="pt" hreflang="pt" data-title="Radiano" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Radian" title="Radian – Romanian" lang="ro" hreflang="ro" data-title="Radian" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%B0%D0%BD" title="Радиан – Russian" lang="ru" hreflang="ru" data-title="Радиан" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Radiani" title="Radiani – Albanian" lang="sq" hreflang="sq" data-title="Radiani" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Radianti" title="Radianti – Sicilian" lang="scn" hreflang="scn" data-title="Radianti" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BB%E0%B7%9A%E0%B6%A9%E0%B7%92%E0%B6%BA%E0%B6%B1%E0%B6%BA" title="රේඩියනය – Sinhala" lang="si" hreflang="si" data-title="රේඩියනය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Radian" title="Radian – Simple English" lang="en-simple" hreflang="en-simple" data-title="Radian" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Radi%C3%A1n" title="Radián – Slovak" lang="sk" hreflang="sk" data-title="Radián" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Radian" title="Radian – Slovenian" lang="sl" hreflang="sl" data-title="Radian" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D8%A7%D8%AF%DB%8C%D8%A7%D9%86" title="ڕادیان – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕادیان" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D1%98%D0%B0%D0%BD" title="Радијан – Serbian" lang="sr" hreflang="sr" data-title="Радијан" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Radijan" title="Radijan – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Radijan" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Radiaani" title="Radiaani – Finnish" lang="fi" hreflang="fi" data-title="Radiaani" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Radian" title="Radian – Swedish" lang="sv" hreflang="sv" data-title="Radian" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%86%E0%AE%B0%E0%AF%88%E0%AE%AF%E0%AE%AE%E0%AF%8D" title="ஆரையம் – Tamil" lang="ta" hreflang="ta" data-title="ஆரையம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B0%E0%B1%87%E0%B0%A1%E0%B0%BF%E0%B0%AF%E0%B0%A8%E0%B1%8D" title="రేడియన్ – Telugu" lang="te" hreflang="te" data-title="రేడియన్" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A3%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2%E0%B8%99" title="เรเดียน – Thai" lang="th" hreflang="th" data-title="เรเดียน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%B0%D0%BD" title="Радиан – Tajik" lang="tg" hreflang="tg" data-title="Радиан" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Radyan" title="Radyan – Turkish" lang="tr" hreflang="tr" data-title="Radyan" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D1%96%D0%B0%D0%BD" title="Радіан – Ukrainian" lang="uk" hreflang="uk" data-title="Радіан" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%DA%88%DB%8C%D9%86" title="ریڈین – Urdu" lang="ur" hreflang="ur" data-title="ریڈین" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Radian" title="Radian – Vietnamese" lang="vi" hreflang="vi" data-title="Radian" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a 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/></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">SI derived unit of angle</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"㎭" redirects here. Not to be confused with <a href="/wiki/Rad_(radiation_unit)" title="Rad (radiation unit)">Rad (radiation unit)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Radian_(disambiguation)" class="mw-disambig" title="Radian (disambiguation)">Radian (disambiguation)</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 ib-unit"><tbody><tr><th colspan="2" class="infobox-above">Radian</th></tr><tr><td colspan="2" class="infobox-image notheme" style="background-color: #f8f9fa;"><a href="/wiki/File:Circle_radians.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Circle_radians.gif/220px-Circle_radians.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Circle_radians.gif/330px-Circle_radians.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Circle_radians.gif/440px-Circle_radians.gif 2x" data-file-width="450" data-file-height="450" /></a><div class="infobox-caption">An arc of a <a href="/wiki/Circle" title="Circle">circle</a> with the same length as the <a href="/wiki/Radius" title="Radius">radius</a> of that circle subtends an angle of 1 radian. The circumference subtends an angle of 2<a href="/wiki/Pi" title="Pi">π</a> radians.</div></td></tr><tr><th colspan="2" class="infobox-header">General information</th></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/System_of_units_of_measurement" title="System of units of measurement">Unit system</a></th><td class="infobox-data"><a href="/wiki/SI" class="mw-redirect" title="SI">SI</a></td></tr><tr><th scope="row" class="infobox-label">Unit of</th><td class="infobox-data"><a href="/wiki/Angle" title="Angle">angle</a></td></tr><tr><th scope="row" class="infobox-label">Symbol</th><td class="infobox-data">rad, R<a href="#cite_note-Hall_1909-1">[1]</a></td></tr><tr><th colspan="2" class="infobox-header">Conversions </th></tr><tr class="nowrap"><td>1 rad in ...</td><td>... is equal to ...</td></tr><tr style="display:none"><th colspan="2"> </th></tr><tr><th scope="row" class="infobox-label">   <a href="/wiki/Milliradian" title="Milliradian">milliradians</a></th><td class="infobox-data">   1000 mrad</td></tr><tr><th scope="row" class="infobox-label">   <a href="/wiki/Turn_(unit)" class="mw-redirect" title="Turn (unit)">turns</a></th><td class="infobox-data">   <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>⁠1/2π⁠ turn ≈ 0.159154 turn</td></tr><tr><th scope="row" class="infobox-label">   <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a></th><td class="infobox-data">   <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠180/π⁠° ≈ 57.295779513°</td></tr><tr><th scope="row" class="infobox-label">   <a href="/wiki/Gradian" title="Gradian">gradians</a></th><td class="infobox-data">   <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠200/π⁠ grad ≈ 63.661977g</td></tr></tbody></table><style data-mw-deduplicate="TemplateStyles:r1236303919">@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-has-images-with-white-backgrounds img{background:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-has-images-with-white-backgrounds img{background:white}}</style> The radian, denoted by the symbol rad, is the unit of <a href="/wiki/Angle" title="Angle">angle</a> in the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI) and is the standard unit of angular measure used in many areas of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius.<a href="#cite_note-DDUnits-2">[2]</a> The unit was formerly an <a href="/wiki/SI_supplementary_unit" class="mw-redirect" title="SI supplementary unit">SI supplementary unit</a> and is currently a <a href="/wiki/Dimensionless_unit" class="mw-redirect" title="Dimensionless unit">dimensionless</a> <a href="/wiki/SI_derived_unit" title="SI derived unit">SI derived unit</a>,<a href="#cite_note-DDUnits-2">[2]</a> defined in the SI as 1 rad = 1<a href="#cite_note-3">[3]</a> and expressed in terms of the <a href="/wiki/SI_base_unit" title="SI base unit">SI base unit</a> <a href="/wiki/Metre" title="Metre">metre</a> (m) as rad = m/m.<a href="#cite_note-FOOTNOTEInternational_Bureau_of_Weights_and_Measures2019137-4">[4]</a> Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.<a href="#cite_note-5">[5]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div> One radian is defined as the angle at the center of a circle in a plane that <a href="https://en.wiktionary.org/wiki/subtend" class="extiw" title="wikt:subtend">subtends</a> an arc whose length equals the radius of the circle.<a href="#cite_note-6">[6]</a> More generally, the <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {s}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {s}{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7d68caa29ae24caf4c436f270327373a36367d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.116ex; height:4.676ex;" alt="{\displaystyle \theta ={\frac {s}{r}}}">, where θ is the magnitude in radians of the subtended angle, s is arc length, and r is radius. A <a href="/wiki/Right_angle" title="Right angle">right angle</a> is exactly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}"> radians.<a href="#cite_note-FOOTNOTEInternational_Bureau_of_Weights_and_Measures2019151-7">[7]</a> One <a href="/wiki/Complete_revolution" class="mw-redirect" title="Complete revolution">complete revolution</a>, expressed as an angle in radians, is the length of the circumference divided by the radius, which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi r}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi r}{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d28b7d772d434f6c720e3a34d0c535c2374266d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.379ex; height:5.176ex;" alt="{\displaystyle {\frac {2\pi r}{r}}}">, or 2π. Thus, 2π radians is equal to 360 degrees. The relation 2π rad = 360° can be derived using the formula for <a href="/wiki/Arc_length" title="Arc length">arc length</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>arc</mtext> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>θ</mi> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72e4b265d38d20fabfb63a41e01ab69df09aa268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.336ex; height:4.843ex;" alt="{\textstyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}">. Since radian is the measure of an angle that is subtended by an arc of a length equal to the radius of the circle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> </mrow> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959e703bed7bcda672810995f78c1ccb0a49fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.537ex; height:4.843ex;" alt="{\textstyle 1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)}">. This can be further simplified to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> </mrow> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ab6e40cba37dbb4dbb924f329acf43088d45f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.822ex; height:3.843ex;" alt="{\textstyle 1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}">. Multiplying both sides by 360° gives 360° = 2π rad. <div class="mw-heading mw-heading3"><h3 id="Unit_symbol">Unit symbol</h3></div> The <a href="/wiki/International_Bureau_of_Weights_and_Measures" title="International Bureau of Weights and Measures">International Bureau of Weights and Measures</a><a href="#cite_note-FOOTNOTEInternational_Bureau_of_Weights_and_Measures2019151-7">[7]</a> and <a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">International Organization for Standardization</a><a href="#cite_note-8">[8]</a> specify rad as the symbol for the radian. Alternative symbols that were in use in 1909 are c (the superscript letter c, for "circular measure"), the letter r, or a superscript R,<a href="#cite_note-Hall_1909-1">[1]</a> but these variants are infrequently used, as they may be mistaken for a <a href="/wiki/Degree_symbol" title="Degree symbol">degree symbol</a> (°) or a radius (r). Hence an angle of 1.2 radians would be written today as 1.2 rad; archaic notations include 1.2 r, 1.2rad, 1.2c, or 1.2R. In mathematical writing, the symbol "rad" is often omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the <a href="/wiki/Degree_sign" class="mw-redirect" title="Degree sign">degree sign</a> ° is used. <div class="mw-heading mw-heading3"><h3 id="Dimensional_analysis">Dimensional analysis</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="#As_an_SI_unit">§ As an SI unit</a></div> Plane angle may be defined as <a href="/wiki/%CE%98" class="mw-redirect" title="Θ">θ</a> = s/r, where θ is the magnitude in radians of the subtended angle, s is circular arc length, and r is radius. One radian corresponds to the angle for which s = r, hence 1 radian = 1 m/m = 1.<a href="#cite_note-9">[9]</a> However, rad is only to be used to express angles, not to express ratios of lengths in general.<a href="#cite_note-FOOTNOTEInternational_Bureau_of_Weights_and_Measures2019151-7">[7]</a> A similar calculation using <a href="/wiki/Circular_sector#Area" title="Circular sector">the area of a circular sector</a> θ = 2A/r2 gives 1 radian as 1 m2/m2 = 1.<a href="#cite_note-10">[10]</a> The key fact is that the radian is a <a href="/wiki/Dimensionless_unit" class="mw-redirect" title="Dimensionless unit">dimensionless unit</a> equal to <a href="/wiki/1" title="1">1</a>. In SI 2019, the SI radian is defined accordingly as 1 rad = 1.<a href="#cite_note-11">[11]</a> It is a long-established practice in mathematics and across all areas of science to make use of rad = 1.<a href="#cite_note-FOOTNOTEInternational_Bureau_of_Weights_and_Measures2019137-4">[4]</a><a href="#cite_note-12">[12]</a> Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations".<a href="#cite_note-13">[13]</a> For example, an object hanging by a string from a pulley will rise or drop by y = rθ centimetres, where r is the magnitude of the radius of the pulley in centimetres and θ is the magnitude of the angle through which the pulley turns in radians. When multiplying r by θ, the unit radian does not appear in the product, nor does the unit centimetre—because both factors are magnitudes (numbers). Similarly in the formula for the <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of a rolling wheel, ω = v/r, radians appear in the units of ω but not on the right hand side.<a href="#cite_note-14">[14]</a> Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".<a href="#cite_note-15">[15]</a> Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is "pedagogically unsatisfying".<a href="#cite_note-16">[16]</a> In 1993 the <a href="/wiki/American_Association_of_Physics_Teachers" title="American Association of Physics Teachers">American Association of Physics Teachers</a> Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of <a href="/wiki/Angle_measure" class="mw-redirect" title="Angle measure">angle measure</a> (rad), <a href="/wiki/Angular_speed" class="mw-redirect" title="Angular speed">angular speed</a> (rad/s), <a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a> (rad/s2), and <a href="/wiki/Torsion_constant#Torsional_Rigidity_(GJ)_and_Stiffness_(GJ/L)" title="Torsion constant">torsional stiffness</a> (N⋅m/rad), and not in the quantities of <a href="/wiki/Torque" title="Torque">torque</a> (N⋅m) and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> (kg⋅m2/s).<a href="#cite_note-17">[17]</a> At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a <a href="/wiki/Base_unit_of_measurement" title="Base unit of measurement">base unit of measurement</a> for a <a href="/wiki/Base_quantity" class="mw-redirect" title="Base quantity">base quantity</a> (and dimension) of "plane angle".<a href="#cite_note-18">[18]</a><a href="#cite_note-FOOTNOTEMohrPhillips2015-19">[19]</a><a href="#cite_note-Quincey-20">[20]</a> Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for the <a href="/wiki/Area_of_a_circle" title="Area of a circle">area of a circle</a>, πr2. The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations".<a href="#cite_note-FOOTNOTEQuincey2016-21">[21]</a> A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use.<a href="#cite_note-Quincey-20">[20]</a> In particular, Quincey identifies Torrens' proposal to introduce a constant <a href="/wiki/%CE%97" class="mw-redirect" title="Η">η</a> equal to 1 inverse radian (1 rad−1) in a fashion similar to the <a href="/wiki/Vacuum_permittivity#Historical_origin_of_the_parameter_ε0" title="Vacuum permittivity">introduction of the constant ε0</a>.<a href="#cite_note-FOOTNOTEQuincey2016-21">[21]</a><a href="#cite_note-22">[a]</a> With this change the formula for the angle subtended at the center of a circle, s = rθ, is modified to become s = ηrθ, and the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> for the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> of an angle θ becomes:<a href="#cite_note-Quincey-20">[20]</a><a href="#cite_note-FOOTNOTETorrens1986-23">[22]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mi>η</mi> <mi>θ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>η</mi> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>η</mi> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>η</mi> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7df4a16dcdaff73203b89de36f938b4155ccf136" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:79.49ex; height:6.009ex;" alt="{\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\eta \theta =\theta /{\text{rad}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>η</mi> <mi>θ</mi> <mo>=</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rad</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\eta \theta =\theta /{\text{rad}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276b9dd82af828185ef24250fd0a78dc2edb1057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.406ex; height:2.843ex;" alt="{\displaystyle x=\eta \theta =\theta /{\text{rad}}}"> is the angle in radians. The capitalized function Sin is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed,<a href="#cite_note-FOOTNOTETorrens1986-23">[22]</a> while sin is the traditional function on <a href="/wiki/Pure_number" class="mw-redirect" title="Pure number">pure numbers</a> which assumes its argument is a dimensionless number in radians.<a href="#cite_note-FOOTNOTEMohrShirleyPhillipsTrott20226-24">[23]</a> The capitalised symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sin} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sin} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f133e3f320f11daba5f13277480b2ec04cb3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.232ex; height:2.176ex;" alt="{\displaystyle \operatorname {Sin} }"> can be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }"> if it is clear that the complete form is meant.<a href="#cite_note-Quincey-20">[20]</a><a href="#cite_note-FOOTNOTEMohrShirleyPhillipsTrott20228–9-25">[24]</a> Current SI can be considered relative to this framework as a <a href="/wiki/Natural_unit" class="mw-redirect" title="Natural unit">natural unit</a> system where the equation η = 1 is assumed to hold, or similarly, 1 rad = 1. This radian convention allows the omission of η in mathematical formulas.<a href="#cite_note-FOOTNOTEQuincey2021-26">[25]</a> Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal.<a href="#cite_note-27">[26]</a> For example, the <a href="/wiki/Boost_(C%2B%2B_libraries)" title="Boost (C++ libraries)">Boost</a> units library defines angle units with a <code>plane_angle</code> dimension,<a href="#cite_note-28">[27]</a> and <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>'s unit system similarly considers angles to have an angle dimension.<a href="#cite_note-FOOTNOTEMohrShirleyPhillipsTrott20223-29">[28]</a><a href="#cite_note-30">[29]</a> <div class="mw-heading mw-heading2"><h2 id="Conversions">Conversions</h2></div> <table class="wikitable" style="text-align:center;float:right;clear:right;margin-left:1em;"> <caption>Conversion of common angles </caption> <tbody><tr> <th><a href="/wiki/Turn_(angle)" title="Turn (angle)">Turns</a> </th> <th><a class="mw-selflink selflink">Radians</a> </th> <th><a href="/wiki/Degree_(angle)" title="Degree (angle)">Degrees</a> </th> <th><a href="/wiki/Gradian" title="Gradian">Gradians</a> </th></tr> <tr> <td>0 turn </td> <td>0 rad </td> <td>0° </td> <td>0g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/72⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/36⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/72⁠ rad </td> <td>5° </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5+5/9⁠g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/24⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/12⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/24⁠ rad </td> <td>15° </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠16+2/3⁠g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/16⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/8⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/16⁠ rad </td> <td>22.5° </td> <td>25g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/12⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/6⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/12⁠ rad </td> <td>30° </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠33+1/3⁠g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/10⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/5⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/10⁠ rad </td> <td>36° </td> <td>40g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/4⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/8⁠ rad </td> <td>45° </td> <td>50g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2π or 𝜏⁠ turn </td> <td>1 rad </td> <td><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style>approx. 57.3° </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319">approx. 63.7g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/3⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/6⁠ rad </td> <td>60° </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠66+2/3⁠g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/5⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2π or 𝜏/5⁠ rad </td> <td>72° </td> <td>80g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/4⁠ rad </td> <td>90° </td> <td>100g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2π or 𝜏/3⁠ rad </td> <td>120° </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠133+1/3⁠g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/5⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4π or 2𝜏 or α/5⁠ rad </td> <td>144° </td> <td>160g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ turn </td> <td>π or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝜏/2⁠ rad </td> <td>180° </td> <td>200g </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠ turn </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3π or ρ/2⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3𝜏/4⁠ rad </td> <td>270° </td> <td>300g </td></tr> <tr> <td>1 turn </td> <td>𝜏 or 2π rad </td> <td>360° </td> <td>400g </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Between_degrees">Between degrees</h3></div> As stated, one radian is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {180^{\circ }}/{\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {180^{\circ }}/{\pi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c30e963e934eeedc7a8db2b1c6271e3ab6c910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.036ex; height:2.843ex;" alt="{\displaystyle {180^{\circ }}/{\pi }}">. Thus, to convert from radians to degrees, multiply by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {180^{\circ }}/{\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {180^{\circ }}/{\pi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c30e963e934eeedc7a8db2b1c6271e3ab6c910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.036ex; height:2.843ex;" alt="{\displaystyle {180^{\circ }}/{\pi }}">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{angle in degrees}}={\text{angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>angle in degrees</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>angle in radians</mtext> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{angle in degrees}}={\text{angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d0aecff9df80c8451bd90170aee6d9c9d3ca3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.715ex; height:5.343ex;" alt="{\displaystyle {\text{angle in degrees}}={\text{angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}"></dd></dl> For example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mi>π</mi> </mfrac> </mrow> <mo>≈</mo> <msup> <mn>57.2958</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2942db06bf85253a68de787c5325a313e6ac4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.202ex; height:5.343ex;" alt="{\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> <mo>=</mo> <mn>2.5</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mi>π</mi> </mfrac> </mrow> <mo>≈</mo> <msup> <mn>143.2394</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3bd0b667a34c7f0b6fa53d84e59f556cacb8915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.983ex; height:5.343ex;" alt="{\displaystyle 2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mi>π</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c99c65c21ada1bc1baca5000c3a634b482d07b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.916ex; height:5.343ex;" alt="{\displaystyle {\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}"></dd></dl> Conversely, to convert from degrees to radians, multiply by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\pi }/{180}{\text{ rad}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>180</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\pi }/{180}{\text{ rad}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f10c1de2043d1efa612037bfd917ed2a38ecb047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.929ex; height:2.843ex;" alt="{\displaystyle {\pi }/{180}{\text{ rad}}}">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{angle in radians}}={\text{angle in degrees}}\cdot {\frac {\pi }{180}}{\text{ rad}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>angle in radians</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>angle in degrees</mtext> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{angle in radians}}={\text{angle in degrees}}\cdot {\frac {\pi }{180}}{\text{ rad}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0b689c2522a282bd5675883572730fd3529741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.608ex; height:4.676ex;" alt="{\displaystyle {\text{angle in radians}}={\text{angle in degrees}}\cdot {\frac {\pi }{180}}{\text{ rad}}}"></dd></dl> For example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{\circ }=1\cdot {\frac {\pi }{180}}{\text{ rad}}\approx 0.0175{\text{ rad}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> <mo>≈</mo> <mn>0.0175</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{\circ }=1\cdot {\frac {\pi }{180}}{\text{ rad}}\approx 0.0175{\text{ rad}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cadb5f3cc75e2cf51186ef0dfe9d7ca876502621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.932ex; height:4.676ex;" alt="{\displaystyle 1^{\circ }=1\cdot {\frac {\pi }{180}}{\text{ rad}}\approx 0.0175{\text{ rad}}}"></dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 23^{\circ }=23\cdot {\frac {\pi }{180}}{\text{ rad}}\approx 0.4014{\text{ rad}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>23</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mn>23</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> <mo>≈</mo> <mn>0.4014</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 23^{\circ }=23\cdot {\frac {\pi }{180}}{\text{ rad}}\approx 0.4014{\text{ rad}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd399e1f1e1ac68e34382036f274b4d5b164268b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.257ex; height:4.676ex;" alt="{\displaystyle 23^{\circ }=23\cdot {\frac {\pi }{180}}{\text{ rad}}\approx 0.4014{\text{ rad}}}"> Radians can be converted to <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turns</a> (one turn is the angle corresponding to a revolution) by dividing the number of radians by 2π. <div class="mw-heading mw-heading3"><h3 id="Between_gradians">Between gradians</h3></div> One revolution is <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><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 }"> radians, which equals one <a href="/wiki/Turn_(unit)" class="mw-redirect" title="Turn (unit)">turn</a>, which is by definition 400 <a href="/wiki/Gradian" title="Gradian">gradians</a> (400 <a href="/wiki/Gon_(unit)" class="mw-redirect" title="Gon (unit)">gons</a> or 400g). To convert from radians to gradians multiply by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 200^{\text{g}}/\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>200</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>g</mtext> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 200^{\text{g}}/\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1d0ea57ccd08deffef7bd0dffec07d0dc32d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.036ex; height:2.843ex;" alt="{\displaystyle 200^{\text{g}}/\pi }">, and to convert from gradians to radians multiply by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /200{\text{ rad}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>200</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /200{\text{ rad}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079aef1503512ec03135eb36bf00ac751ab827b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.929ex; height:2.843ex;" alt="{\displaystyle \pi /200{\text{ rad}}}">. For example, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.2{\text{ rad}}=1.2\cdot {\frac {200^{\text{g}}}{\pi }}\approx 76.3944^{\text{g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> <mo>=</mo> <mn>1.2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>200</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>g</mtext> </mrow> </msup> <mi>π</mi> </mfrac> </mrow> <mo>≈</mo> <msup> <mn>76.3944</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>g</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.2{\text{ rad}}=1.2\cdot {\frac {200^{\text{g}}}{\pi }}\approx 76.3944^{\text{g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f561b0ab4763c0812325fbcb1ea56aca4438b5a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.82ex; height:5.343ex;" alt="{\displaystyle 1.2{\text{ rad}}=1.2\cdot {\frac {200^{\text{g}}}{\pi }}\approx 76.3944^{\text{g}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 50^{\text{g}}=50\cdot {\frac {\pi }{200}}{\text{ rad}}\approx 0.7854{\text{ rad}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>50</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>g</mtext> </mrow> </msup> <mo>=</mo> <mn>50</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>200</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> <mo>≈</mo> <mn>0.7854</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> rad</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 50^{\text{g}}=50\cdot {\frac {\pi }{200}}{\text{ rad}}\approx 0.7854{\text{ rad}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ed60f7ddefc6c5cc50abb31b4169b404d2a43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.257ex; height:4.676ex;" alt="{\displaystyle 50^{\text{g}}=50\cdot {\frac {\pi }{200}}{\text{ rad}}\approx 0.7854{\text{ rad}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2></div> <div class="mw-heading mw-heading3"><h3 id="Mathematics">Mathematics</h3></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Radian-common.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Radian-common.svg/357px-Radian-common.svg.png" decoding="async" width="357" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Radian-common.svg/536px-Radian-common.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Radian-common.svg/714px-Radian-common.svg.png 2x" data-file-width="690" data-file-height="445" /></a><figcaption>Some common angles, measured in radians. All the large polygons in this diagram are <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a>.</figcaption></figure> In <a href="/wiki/Calculus" title="Calculus">calculus</a> and most other branches of mathematics beyond practical <a href="/wiki/Geometry" title="Geometry">geometry</a>, angles are measured in radians. This is because radians have a mathematical naturalness that leads to a more elegant formulation of some important results. Results in <a href="/wiki/Analysis_(mathematics)" class="mw-redirect" title="Analysis (mathematics)">analysis</a> involving <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> can be elegantly stated when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a> formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{h\rightarrow 0}{\frac {\sin h}{h}}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>h</mi> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{h\rightarrow 0}{\frac {\sin h}{h}}=1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26bb2c7c053f12ce143283d40434c161841b938c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.125ex; height:5.509ex;" alt="{\displaystyle \lim _{h\rightarrow 0}{\frac {\sin h}{h}}=1,}"></dd></dl> which is the basis of many other identities in mathematics, including <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\sin x=\cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\sin x=\cos x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0239876162b11c79d29c30db912b151793e03c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.268ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\sin x=\cos x}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cbddd7a23b3b22be9501d8314ba9b93357404d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.908ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.}"></dd></dl> Because of these and other properties, the trigonometric functions appear in solutions to mathematical problems that are not obviously related to the functions' geometrical meanings (for example, the solutions to the <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {d^{2}y}{dx^{2}}}=-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {d^{2}y}{dx^{2}}}=-y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494e04d09142eac983a501a8fdf6308612679273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.53ex; height:4.676ex;" alt="{\displaystyle {\tfrac {d^{2}y}{dx^{2}}}=-y}">, the evaluation of the integral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int {\frac {dx}{1+x^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int {\frac {dx}{1+x^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddfe7c9360bc927a2ee753b62a6e89e6d59a1ce9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:7.16ex; height:4.176ex;" alt="{\displaystyle \textstyle \int {\frac {dx}{1+x^{2}}},}"> and so on). In all such cases, it is appropriate that the arguments of the functions are treated as (dimensionless) numbers—without any reference to angles. The trigonometric functions of angles also have simple and elegant series expansions when radians are used. For example, when x is the angle expressed in radians, the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> for sin x becomes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d39cc4bdec2487b57e0c58e878ccebc21de6d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.779ex; height:5.843ex;" alt="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}"></dd></dl> If y were the angle x but expressed in degrees, i.e. y = πx / 180, then the series would contain messy factors involving powers of π/180: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin y={\frac {\pi }{180}}x-\left({\frac {\pi }{180}}\right)^{3}\ {\frac {x^{3}}{3!}}+\left({\frac {\pi }{180}}\right)^{5}\ {\frac {x^{5}}{5!}}-\left({\frac {\pi }{180}}\right)^{7}\ {\frac {x^{7}}{7!}}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mi>x</mi> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin y={\frac {\pi }{180}}x-\left({\frac {\pi }{180}}\right)^{3}\ {\frac {x^{3}}{3!}}+\left({\frac {\pi }{180}}\right)^{5}\ {\frac {x^{5}}{5!}}-\left({\frac {\pi }{180}}\right)^{7}\ {\frac {x^{7}}{7!}}+\cdots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5cd0b253dd5679de35d2926a8dc4e63feeeed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:64.13ex; height:5.843ex;" alt="{\displaystyle \sin y={\frac {\pi }{180}}x-\left({\frac {\pi }{180}}\right)^{3}\ {\frac {x^{3}}{3!}}+\left({\frac {\pi }{180}}\right)^{5}\ {\frac {x^{5}}{5!}}-\left({\frac {\pi }{180}}\right)^{7}\ {\frac {x^{7}}{7!}}+\cdots .}"></dd></dl> In a similar spirit, if angles are involved, mathematically important relationships between the sine and cosine functions and the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> (see, for example, <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>) can be elegantly stated when the functions' arguments are angles expressed in radians (and messy otherwise). More generally, in complex-number theory, the arguments of these functions are (dimensionless, possibly complex) numbers—without any reference to physical angles at all. <div class="mw-heading mw-heading3"><h3 id="Physics">Physics</h3></div> The radian is widely used in <a href="/wiki/Physics" title="Physics">physics</a> when angular measurements are required. For example, <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> is typically expressed in the unit <a href="/wiki/Radian_per_second" title="Radian per second">radian per second</a> (rad/s). One revolution per second corresponds to 2π radians per second. Similarly, the unit used for <a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a> is often radian per second per second (rad/s2). For the purpose of <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>, the units of angular velocity and angular acceleration are s−1 and s−2 respectively. Likewise, the phase angle difference of two waves can also be expressed using the radian as the unit. For example, if the phase angle difference of two waves is (n⋅2π) radians, where n is an integer, they are considered to be in <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a>, whilst if the phase angle difference of two waves is (n⋅2π + π) radians, with n an integer, they are considered to be in antiphase. A unit of reciprocal radian or inverse radian (rad-1) is involved in derived units such as meter per radian (for <a href="/wiki/Angular_wavelength" class="mw-redirect" title="Angular wavelength">angular wavelength</a>) or newton-metre per radian (for torsional stiffness). <div class="mw-heading mw-heading3"><h3 id="Prefixes_and_variants">Prefixes and variants</h3></div> <a href="/wiki/Metric_prefix" title="Metric prefix">Metric prefixes</a> for submultiples are used with radians. A <a href="/wiki/Milliradian" title="Milliradian">milliradian</a> (mrad) is a thousandth of a radian (0.001 rad), i.e. 1 rad = 103 mrad. There are 2<a href="/wiki/Pi" title="Pi">π</a> × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6283⁠ of the angle subtended by a full circle. This unit of angular measurement of a circle is in common use by <a href="/wiki/Telescopic_sight" title="Telescopic sight">telescopic sight</a> manufacturers using <a href="/wiki/Stadiametric_rangefinding" title="Stadiametric rangefinding">(stadiametric) rangefinding</a> in <a href="/wiki/Reticle" title="Reticle">reticles</a>. The <a href="/wiki/Beam_divergence" title="Beam divergence">divergence</a> of <a href="/wiki/Laser" title="Laser">laser</a> beams is also usually measured in milliradians. The <a href="/wiki/Angular_mil" class="mw-redirect" title="Angular mil">angular mil</a> is an approximation of the milliradian used by <a href="/wiki/NATO" title="NATO">NATO</a> and other military organizations in <a href="/wiki/Gun" title="Gun">gunnery</a> and <a href="/wiki/Sniper#Targeting,_tactics_and_techniques" title="Sniper">targeting</a>. Each angular mil represents <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6400⁠ of a circle and is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠15/8⁠% or 1.875% smaller than the milliradian. For the small angles typically found in targeting work, the convenience of using the number 6400 in calculation outweighs the small mathematical errors it introduces. In the past, other gunnery systems have used different approximations to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2000π⁠; for example Sweden used the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6300⁠ streck and the USSR used <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6000⁠. Being based on the milliradian, the NATO mil subtends roughly 1 m at a range of 1000 m (at such small angles, the curvature is negligible). Prefixes smaller than milli- are useful in measuring extremely small angles. Microradians (μrad, 10−6 rad) and nanoradians (nrad, 10−9 rad) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is the <a href="/wiki/Arc_second" class="mw-redirect" title="Arc second">arc second</a>, which is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/648,000⁠ rad (around 4.8481 microradians). <table class="wikitable" style="text-align:center; margin:0 auto;"> <caption>SI multiples of radian (rad) </caption> <tbody><tr> <th scope="col" colspan="3">Submultiples </th> <th scope="col" colspan="3">Multiples </th></tr> <tr style="background:#d4d4d4;"> <th scope="col">Value </th> <th scope="col">SI symbol </th> <th scope="col">Name </th> <th scope="col">Value </th> <th scope="col">SI symbol </th> <th scope="col">Name </th></tr> <tr> <td>10−1 rad </td> <td>drad </td> <td>deciradian </td> <td>101 rad </td> <td>darad </td> <td>decaradian </td></tr> <tr> <td>10−2 rad </td> <td>crad </td> <td>centiradian </td> <td>102 rad </td> <td>hrad </td> <td>hectoradian </td></tr> <tr> <td>10−3 rad </td> <td>mrad </td> <td><a href="/wiki/Milliradian" title="Milliradian">milliradian</a> </td> <td>103 rad </td> <td>krad </td> <td>kiloradian </td></tr> <tr> <td>10−6 rad </td> <td>μrad </td> <td>microradian </td> <td>106 rad </td> <td>Mrad </td> <td>megaradian </td></tr> <tr> <td>10−9 rad </td> <td>nrad </td> <td>nanoradian </td> <td>109 rad </td> <td>Grad </td> <td>gigaradian </td></tr> <tr> <td>10−12 rad </td> <td>prad </td> <td>picoradian </td> <td>1012 rad </td> <td>Trad </td> <td>teraradian </td></tr> <tr> <td>10−15 rad </td> <td>frad </td> <td>femtoradian </td> <td>1015 rad </td> <td>Prad </td> <td>petaradian </td></tr> <tr> <td>10−18 rad </td> <td>arad </td> <td>attoradian </td> <td>1018 rad </td> <td>Erad </td> <td>exaradian </td></tr> <tr> <td>10−21 rad </td> <td>zrad </td> <td>zeptoradian </td> <td>1021 rad </td> <td>Zrad </td> <td>zettaradian </td></tr> <tr> <td>10−24 rad </td> <td>yrad </td> <td>yoctoradian </td> <td>1024 rad </td> <td>Yrad </td> <td>yottaradian </td></tr> <tr> <td>10−27 rad </td> <td>rrad </td> <td>rontoradian </td> <td>1027 rad </td> <td>Rrad </td> <td>ronnaradian </td></tr> <tr> <td>10−30 rad </td> <td>qrad </td> <td>quectoradian </td> <td>1030 rad </td> <td>Qrad </td> <td>quettaradian </td></tr> </tbody></table> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <div class="mw-heading mw-heading3"><h3 id="Pre-20th_century">Pre-20th century</h3></div> The idea of measuring angles by the length of the arc was in use by mathematicians quite early. For example, <a href="/wiki/Al-Kashi" class="mw-redirect" title="Al-Kashi">al-Kashi</a> (c. 1400) used so-called diameter parts as units, where one diameter part was <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/60⁠ radian. They also used sexagesimal subunits of the diameter part.<a href="#cite_note-31">[30]</a> Newton in 1672 spoke of "the angular quantity of a body's circular motion", but used it only as a relative measure to develop an astronomical algorithm.<a href="#cite_note-Roche-32">[31]</a> The concept of the radian measure is normally credited to <a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a>, who died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book, Harmonia mensurarum.<a href="#cite_note-33">[32]</a> In a chapter of editorial comments, Smith gave what is probably the first published calculation of one radian in degrees, citing a note of Cotes that has not survived. Smith described the radian in everything but name – "Now this number is equal to 180 degrees as the radius of a circle to the <a href="/wiki/Semicircumference" class="mw-redirect" title="Semicircumference">semicircumference</a>, this is as 1 to 3.141592653589" –, and recognized its naturalness as a unit of angular measure.<a href="#cite_note-34">[33]</a><a href="#cite_note-35">[34]</a> In 1765, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> implicitly adopted the radian as a unit of angle.<a href="#cite_note-Roche-32">[31]</a> Specifically, Euler defined angular velocity as "The angular speed in rotational motion is the speed of that point, the distance of which from the axis of gyration is expressed by one."<a href="#cite_note-36">[35]</a> Euler was probably the first to adopt this convention, referred to as the radian convention, which gives the simple formula for angular velocity ω = v/r. As discussed in <a href="#Dimensional_analysis">§ Dimensional analysis</a>, the radian convention has been widely adopted, while dimensionally consistent formulations require the insertion of a dimensional constant, for example ω = v/(ηr).<a href="#cite_note-FOOTNOTEQuincey2021-26">[25]</a> Prior to the term radian becoming widespread, the unit was commonly called circular measure of an angle.<a href="#cite_note-37">[36]</a> The term radian first appeared in print on 5 June 1873, in examination questions set by <a href="/wiki/James_Thomson_(engineer)" title="James Thomson (engineer)">James Thomson</a> (brother of <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a>) at <a href="/wiki/Queen%27s_University_Belfast" title="Queen's University Belfast">Queen's College</a>, <a href="/wiki/Belfast" title="Belfast">Belfast</a>. He had used the term as early as 1871, while in 1869, <a href="/wiki/Thomas_Muir_(mathematician)" title="Thomas Muir (mathematician)">Thomas Muir</a>, then of the <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>, vacillated between the terms rad, radial, and radian. In 1874, after a consultation with James Thomson, Muir adopted radian.<a href="#cite_note-38">[37]</a><a href="#cite_note-39">[38]</a><a href="#cite_note-40">[39]</a> The name radian was not universally adopted for some time after this. Longmans' School Trigonometry still called the radian circular measure when published in 1890.<a href="#cite_note-41">[40]</a> In 1893 <a href="/wiki/Alexander_Macfarlane" title="Alexander Macfarlane">Alexander Macfarlane</a> wrote "the true analytical argument for the circular ratios is not the ratio of the arc to the radius, but the ratio of twice the area of a sector to the square on the radius."<a href="#cite_note-42">[41]</a> However, the paper was withdrawn from the published proceedings of mathematical congress held in connection with <a href="/wiki/World%27s_Columbian_Exposition" title="World's Columbian Exposition">World's Columbian Exposition</a> in Chicago (acknowledged at page 167), and privately published in his Papers on Space Analysis (1894). Macfarlane reached this idea or ratios of areas while considering the basis for <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a> which is analogously defined.<a href="#cite_note-43">[42]</a> <div class="mw-heading mw-heading3"><h3 id="As_an_SI_unit">As an SI unit</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="#Dimensional_analysis">§ Dimensional analysis</a></div> As Paul Quincey et al. write, "the status of angles within the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI) has long been a source of controversy and confusion."<a href="#cite_note-44">[43]</a> In 1960, the <a href="/wiki/General_Conference_on_Weights_and_Measures" title="General Conference on Weights and Measures">CGPM</a> established the SI and the radian was classified as a "supplementary unit" along with the <a href="/wiki/Steradian" title="Steradian">steradian</a>. This special class was officially regarded "either as base units or as derived units", as the CGPM could not reach a decision on whether the radian was a base unit or a derived unit.<a href="#cite_note-45">[44]</a> Richard Nelson writes "This ambiguity [in the classification of the supplemental units] prompted a spirited discussion over their proper interpretation."<a href="#cite_note-Nelson-46">[45]</a> In May 1980 the <a href="/wiki/Consultative_Committee_for_Units" class="mw-redirect" title="Consultative Committee for Units">Consultative Committee for Units (CCU)</a> considered a proposal for making radians an SI base unit, using a constant α0 = 1 rad,<a href="#cite_note-47">[46]</a><a href="#cite_note-FOOTNOTEQuincey2021-26">[25]</a> but turned it down to avoid an upheaval to current practice.<a href="#cite_note-FOOTNOTEQuincey2021-26">[25]</a> In October 1980 the CGPM decided that supplementary units were dimensionless derived units for which the CGPM allowed the freedom of using them or not using them in expressions for SI derived units,<a href="#cite_note-Nelson-46">[45]</a> on the basis that "[no formalism] exists which is at the same time coherent and convenient and in which the quantities plane angle and solid angle might be considered as base quantities" and that "[the possibility of treating the radian and steradian as SI base units] compromises the internal coherence of the SI based on only seven base units".<a href="#cite_note-FOOTNOTEInternational_Bureau_of_Weights_and_Measures2019174–175-48">[47]</a> In 1995 the CGPM eliminated the class of supplementary units and defined the radian and the steradian as "dimensionless derived units, the names and symbols of which may, but need not, be used in expressions for other SI derived units, as is convenient".<a href="#cite_note-FOOTNOTEInternational_Bureau_of_Weights_and_Measures2019179-49">[48]</a> Mikhail Kalinin writing in 2019 has criticized the 1980 CGPM decision as "unfounded" and says that the 1995 CGPM decision used inconsistent arguments and introduced "numerous discrepancies, inconsistencies, and contradictions in the wordings of the SI".<a href="#cite_note-50">[49]</a> At the 2013 meeting of the CCU, Peter Mohr gave a presentation on alleged inconsistencies arising from defining the radian as a dimensionless unit rather than a base unit. CCU President Ian M. Mills declared this to be a "formidable problem" and the CCU Working Group on Angles and Dimensionless Quantities in the SI was established.<a href="#cite_note-51">[50]</a> The CCU met in 2021, but did not reach a consensus. A small number of members argued strongly that the radian should be a base unit, but the majority felt the status quo was acceptable or that the change would cause more problems than it would solve. A task group was established to "review the historical use of SI supplementary units and consider whether reintroduction would be of benefit", among other activities.<a href="#cite_note-52">[51]</a><a href="#cite_note-53">[52]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Angular_frequency" title="Angular frequency">Angular frequency</a></li> <li><a href="/wiki/Minute_and_second_of_arc" title="Minute and second of arc">Minute and second of arc</a></li> <li><a href="/wiki/Steradian" title="Steradian">Steradian</a>, a higher-dimensional analog of the radian which measures solid angle</li> <li><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-22"><a href="#cite_ref-22">^</a> Other proposals include the abbreviation "rad" (<a href="#CITEREFBrinsmade1936">Brinsmade 1936</a>), the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \theta \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>θ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \theta \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9eda2f58c19644eb126786716dcf01b62ee19e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.9ex; height:2.843ex;" alt="{\displaystyle \langle \theta \rangle }"> (<a href="#CITEREFRomain1962">Romain 1962</a>), and the constants <a href="/wiki/%D7%9D" class="mw-redirect" title="ם">ם</a> (<a href="#CITEREFBrownstein1997">Brownstein 1997</a>), ◁ (<a href="#CITEREFLévy-Leblond1998">Lévy-Leblond 1998</a>), k (<a href="#CITEREFFoster2010">Foster 2010</a>), θC (<a href="#CITEREFQuincey2021">Quincey 2021</a>), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {C}}={\frac {2\pi }{\Theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi mathvariant="normal">Θ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cal {C}}={\frac {2\pi }{\Theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115315228178cfdcbeefbfc3873f9bd5ed6bb816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.668ex; height:5.343ex;" alt="{\displaystyle {\cal {C}}={\frac {2\pi }{\Theta }}}"> (<a href="#CITEREFMohrShirleyPhillipsTrott2022">Mohr et al. 2022</a>). </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Hall_1909-1">^ <a href="#cite_ref-Hall_1909_1-0">a</a> <a href="#cite_ref-Hall_1909_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHallFrink1909" class="citation book cs1 cs1-prop-location-test cs1-prop-long-vol">Hall, Arthur Graham; 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(February 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121019161705/http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html">"Biography of Roger Cotes"</a>. The MacTutor History of Mathematics. Archived from <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html">the original</a> on 2012-10-19. Retrieved 2006-04-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+MacTutor+History+of+Mathematics&rft.atitle=Biography+of+Roger+Cotes&rft.date=2005-02&rft.aulast=O%27Connor&rft.aufirst=J.+J.&rft.au=Robertson%2C+E.+F.&rft_id=http%3A%2F%2Fwww-groups.dcs.st-and.ac.uk%2F~history%2FPrintonly%2FCotes.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARadian" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCotes1722" class="citation book cs1 cs1-prop-foreign-lang-source">Cotes, Roger (1722). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J6BGAAAAcAAJ&pg=RA2-PA95">"Editoris notæ ad Harmoniam mensurarum"</a>. In Smith, Robert (ed.). Harmonia mensurarum (in Latin). Cambridge, England. pp. 94–95. <q>In Canone Logarithmico exhibetur Systema quoddam menfurarum numeralium, quæ Logarithmi dicuntur: atque hujus systematis Modulus is est Logarithmus, qui metitur Rationem Modularem in Corol. 6. definitam. Similiter in Canone Trigonometrico finuum & tangentium, exhibetur Systema quoddam menfurarum numeralium, quæ Gradus appellantur: atque hujus systematis Modulus is est Numerus Graduum, qui metitur Angulum Modularem modo definitun, hoc est, qui continetur in arcu Radio æquali. Eft autem hic Numerus ad Gradus 180 ut Circuli Radius ad Semicircuinferentiam, hoc eft ut 1 ad 3.141592653589 &c. Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. Cujus Reciprocus eft 0.0174532925 &c. Hujus moduli subsidio (quem in chartula quadam Auctoris manu descriptum inveni) commodissime computabis mensuras angulares, queinadmodum oftendam in Nota III.</q> [In the Logarithmic Canon there is presented a certain system of numerical measures called Logarithms: and the Modulus of this system is the Logarithm, which measures the Modular Ratio as defined in Corollary 6. Similarly, in the Trigonometrical Canon of sines and tangents, there is presented a certain system of numerical measures called Degrees: and the Modulus of this system is the Number of Degrees which measures the Modular Angle defined in the manner defined, that is, which is contained in an equal Radius arc. Now this Number is equal to 180 Degrees as the Radius of a Circle to the Semicircumference, this is as 1 to 3.141592653589 &c. Hence the Modulus of the Trigonometric Canon will be 57.2957795130 &c. Whose Reciprocal is 0.0174532925 &c. 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Metrologia. 53 (3): 991–997. <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/2016Metro..53..991M">2016Metro..53..991M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0026-1394%2F53%2F3%2F991">10.1088/0026-1394/53/3/991</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126032642">126032642</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrologia&rft.atitle=On+the+units+radian+and+cycle+for+the+quantity+plane+angle&rft.volume=53&rft.issue=3&rft.pages=991-997&rft.date=2016-06-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126032642%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0026-1394%2F53%2F3%2F991&rft_id=info%3Abibcode%2F2016Metro..53..991M&rft.aulast=Mills&rft.aufirst=Ian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARadian" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuincey2021" class="citation journal cs1">Quincey, Paul (1 October 2021). "Angles in the SI: a detailed proposal for solving the problem". Metrologia. 58 (5): 053002. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2108.05704">2108.05704</a>. <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/2021Metro..58e3002Q">2021Metro..58e3002Q</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1681-7575%2Fac023f">10.1088/1681-7575/ac023f</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:236547235">236547235</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrologia&rft.atitle=Angles+in+the+SI%3A+a+detailed+proposal+for+solving+the+problem&rft.volume=58&rft.issue=5&rft.pages=053002&rft.date=2021-10-01&rft_id=info%3Aarxiv%2F2108.05704&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A236547235%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1681-7575%2Fac023f&rft_id=info%3Abibcode%2F2021Metro..58e3002Q&rft.aulast=Quincey&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARadian" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeonard2021" class="citation journal cs1">Leonard, B P (1 October 2021). "Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI)". Metrologia. 58 (5): 052001. <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/2021Metro..58e2001L">2021Metro..58e2001L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1681-7575%2Fabe0fc">10.1088/1681-7575/abe0fc</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:234036217">234036217</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrologia&rft.atitle=Proposal+for+the+dimensionally+consistent+treatment+of+angle+and+solid+angle+by+the+International+System+of+Units+%28SI%29&rft.volume=58&rft.issue=5&rft.pages=052001&rft.date=2021-10-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A234036217%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1681-7575%2Fabe0fc&rft_id=info%3Abibcode%2F2021Metro..58e2001L&rft.aulast=Leonard&rft.aufirst=B+P&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARadian" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMohrShirleyPhillipsTrott2022" class="citation journal cs1">Mohr, Peter J; Shirley, Eric L; Phillips, William D; Trott, Michael (23 June 2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1681-7575%2Fac7bc2">"On the dimension of angles and their units"</a>. Metrologia. 59 (5): 053001. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2203.12392">2203.12392</a>. <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/2022Metro..59e3001M">2022Metro..59e3001M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1681-7575%2Fac7bc2">10.1088/1681-7575/ac7bc2</a>.</cite><span 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.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"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Trigonometry/Radian_and_degree_measures" class="extiw" title="wikibooks:Trigonometry/Radian and degree measures">Trigonometry/Radian and degree measures</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, 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title="SI base unit">Base units</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ampere" title="Ampere">ampere</a></li> <li><a href="/wiki/Candela" title="Candela">candela</a></li> <li><a href="/wiki/Kelvin" title="Kelvin">kelvin</a></li> <li><a href="/wiki/Kilogram" title="Kilogram">kilogram</a></li> <li><a href="/wiki/Metre" title="Metre">metre</a></li> <li><a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a></li> <li><a href="/wiki/Second" title="Second">second</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/SI_derived_unit" title="SI derived unit">Derived units with special names</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Becquerel" title="Becquerel">becquerel</a></li> <li><a href="/wiki/Coulomb" title="Coulomb">coulomb</a></li> <li><a href="/wiki/Celsius" title="Celsius">degree Celsius</a></li> <li><a href="/wiki/Farad" title="Farad">farad</a></li> <li><a href="/wiki/Gray_(unit)" title="Gray (unit)">gray</a></li> <li><a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a></li> <li><a href="/wiki/Hertz" title="Hertz">hertz</a></li> <li><a href="/wiki/Joule" title="Joule">joule</a></li> <li><a href="/wiki/Katal" title="Katal">katal</a></li> <li><a href="/wiki/Lumen_(unit)" title="Lumen (unit)">lumen</a></li> <li><a href="/wiki/Lux" title="Lux">lux</a></li> <li><a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a></li> <li><a href="/wiki/Ohm" title="Ohm">ohm</a></li> <li><a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascal</a></li> <li><a class="mw-selflink selflink">radian</a></li> <li><a href="/wiki/Siemens_(unit)" title="Siemens (unit)">siemens</a></li> <li><a href="/wiki/Sievert" title="Sievert">sievert</a></li> <li><a href="/wiki/Steradian" title="Steradian">steradian</a></li> <li><a href="/wiki/Tesla_(unit)" title="Tesla (unit)">tesla</a></li> <li><a href="/wiki/Volt" title="Volt">volt</a></li> <li><a href="/wiki/Watt" title="Watt">watt</a></li> <li><a href="/wiki/Weber_(unit)" title="Weber (unit)">weber</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Non-SI_units_mentioned_in_the_SI" title="Non-SI units mentioned in the SI">Other accepted units</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomical_unit" title="Astronomical unit">astronomical unit</a></li> <li><a href="/wiki/Dalton_(unit)" title="Dalton (unit)">dalton</a></li> <li><a href="/wiki/Day" title="Day">day</a></li> <li><a href="/wiki/Decibel" title="Decibel">decibel</a></li> <li><a href="/wiki/Degree_(angle)" title="Degree (angle)">degree of arc</a></li> <li><a href="/wiki/Electronvolt" title="Electronvolt">electronvolt</a></li> <li><a href="/wiki/Hectare" title="Hectare">hectare</a></li> <li><a href="/wiki/Hour" title="Hour">hour</a></li> <li><a href="/wiki/Litre" title="Litre">litre</a></li> <li><a href="/wiki/Minute" title="Minute">minute</a></li> <li><a href="/wiki/Minute_and_second_of_arc" title="Minute and second of arc">minute and second of arc</a></li> <li><a href="/wiki/Neper" title="Neper">neper</a></li> <li><a href="/wiki/Tonne" title="Tonne">tonne</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Conversion_of_units" title="Conversion of units">Conversion of units</a></li> <li><a href="/wiki/Metric_prefix" title="Metric prefix">Metric prefixes</a></li> <li><a href="/wiki/Historical_definitions_of_the_SI_base_units" title="Historical definitions of the SI base units">Historical definitions of the SI base units</a></li> <li><a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">2019 revision of the SI</a></li> <li><a href="/wiki/System_of_units_of_measurement" title="System of units of measurement">System of units of measurement</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:SI_units" title="Category:SI units">Category</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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