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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> 3 Types </div> </a> <button aria-controls="toc-Types-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Types subsection </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Individual_angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Individual_angles"> <div class="vector-toc-text"> 3.1 Individual angles </div> </a> <ul id="toc-Individual_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vertical_and_adjacent_angle_pairs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vertical_and_adjacent_angle_pairs"> <div class="vector-toc-text"> 3.2 Vertical and adjacent angle pairs </div> </a> <ul id="toc-Vertical_and_adjacent_angle_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combining_angle_pairs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combining_angle_pairs"> <div class="vector-toc-text"> 3.3 Combining angle pairs </div> </a> <ul id="toc-Combining_angle_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polygon-related_angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polygon-related_angles"> <div class="vector-toc-text"> 3.4 Polygon-related angles </div> </a> <ul id="toc-Polygon-related_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plane-related_angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plane-related_angles"> <div class="vector-toc-text"> 3.5 Plane-related angles </div> </a> <ul id="toc-Plane-related_angles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measuring_angles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Measuring_angles"> <div class="vector-toc-text"> 4 Measuring angles </div> </a> <button aria-controls="toc-Measuring_angles-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Measuring angles subsection </button> <ul id="toc-Measuring_angles-sublist" class="vector-toc-list"> <li id="toc-Units" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> 4.1 Units </div> </a> <ul id="toc-Units-sublist" 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"> 4.2 Dimensional analysis </div> </a> <ul id="toc-Dimensional_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Signed_angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Signed_angles"> <div class="vector-toc-text"> 4.3 Signed angles </div> </a> <ul id="toc-Signed_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalent_angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalent_angles"> <div class="vector-toc-text"> 4.4 Equivalent angles </div> </a> <ul id="toc-Equivalent_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_quantities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_quantities"> <div class="vector-toc-text"> 4.5 Related quantities </div> </a> <ul id="toc-Related_quantities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Angles_between_curves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Angles_between_curves"> <div class="vector-toc-text"> 5 Angles between curves </div> </a> <ul id="toc-Angles_between_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bisecting_and_trisecting_angles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bisecting_and_trisecting_angles"> <div class="vector-toc-text"> 6 Bisecting and trisecting angles </div> </a> <ul id="toc-Bisecting_and_trisecting_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dot_product_and_generalisations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dot_product_and_generalisations"> <div class="vector-toc-text"> 7 Dot product and generalisations </div> </a> <button aria-controls="toc-Dot_product_and_generalisations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Dot product and generalisations subsection </button> <ul id="toc-Dot_product_and_generalisations-sublist" class="vector-toc-list"> <li id="toc-Inner_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inner_product"> <div class="vector-toc-text"> 7.1 Inner product </div> </a> <ul id="toc-Inner_product-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angles_between_subspaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angles_between_subspaces"> <div class="vector-toc-text"> 7.2 Angles between subspaces </div> </a> <ul id="toc-Angles_between_subspaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angles_in_Riemannian_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angles_in_Riemannian_geometry"> <div class="vector-toc-text"> 7.3 Angles in Riemannian geometry </div> </a> <ul id="toc-Angles_in_Riemannian_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperbolic_angle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hyperbolic_angle"> <div class="vector-toc-text"> 7.4 Hyperbolic angle </div> </a> <ul id="toc-Hyperbolic_angle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Angles_in_geography_and_astronomy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Angles_in_geography_and_astronomy"> <div class="vector-toc-text"> 8 Angles in geography and astronomy </div> </a> <ul id="toc-Angles_in_geography_and_astronomy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 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"> 10 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"> 11 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 12 Bibliography </div> </a> <ul id="toc-Bibliography-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"> 13 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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > 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">Angle</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 139 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-139" aria-hidden="true" > 139 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-kbd mw-list-item"><a href="https://kbd.wikipedia.org/wiki/%D0%9F%D0%BB%D3%80%D0%B0%D0%BD%D1%8D%D0%BF%D1%8D" title="ПлӀанэпэ – Kabardian" lang="kbd" hreflang="kbd" data-title="ПлӀанэпэ" data-language-autonym="Адыгэбзэ" data-language-local-name="Kabardian" class="interlanguage-link-target">Адыгэбзэ</a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Hoek_(meetkunde)" title="Hoek (meetkunde) – Afrikaans" lang="af" hreflang="af" data-title="Hoek (meetkunde)" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Winkel_(Geometrie)" title="Winkel (Geometrie) – Alemannic" lang="gsw" hreflang="gsw" data-title="Winkel (Geometrie)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%D9%8A%D8%A9_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9)" title="زاوية (هندسة) – Arabic" lang="ar" hreflang="ar" data-title="زاوية (هندسة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Anglo" title="Anglo – Aragonese" lang="an" hreflang="an" data-title="Anglo" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/%DC%99%DC%98%DC%9D%DC%AC%DC%90_(%DC%A1%DC%9A%DC%AA%DC%98%DC%AC%DC%90)" title="ܙܘܝܬܐ (ܡܚܪܘܬܐ) – Aramaic" lang="arc" hreflang="arc" data-title="ܙܘܝܬܐ (ܡܚܪܘܬܐ)" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="Aramaic" class="interlanguage-link-target">ܐܪܡܝܐ</a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D4%B1%D5%B6%D5%AF%D5%AB%D6%82%D5%B6" title="Անկիւն – Western Armenian" lang="hyw" hreflang="hyw" data-title="Անկիւն" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target">Արեւմտահայերէն</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%A3" title="কোণ – Assamese" lang="as" hreflang="as" data-title="কোণ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81ngulu" title="Ángulu – Asturian" lang="ast" hreflang="ast" data-title="Ángulu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Takamby" title="Takamby – Guarani" lang="gn" hreflang="gn" data-title="Takamby" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/K%27uchu" title="K'uchu – Aymara" lang="ay" hreflang="ay" data-title="K'uchu" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target">Aymar aru</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bucaq" title="Bucaq – Azerbaijani" lang="az" hreflang="az" data-title="Bucaq" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A2%DA%86%DB%8C" title="آچی – South Azerbaijani" lang="azb" hreflang="azb" data-title="آچی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%A3" title="কোণ – Bangla" lang="bn" hreflang="bn" data-title="কোণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Kak-t%C5%8D%CD%98" title="Kak-tō͘ – Minnan" lang="nan" hreflang="nan" data-title="Kak-tō͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D3%A9%D0%B9%D3%A9%D1%88" title="Мөйөш – Bashkir" lang="ba" hreflang="ba" data-title="Мөйөш" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D1%83%D0%B3%D0%B0%D0%BB" title="Вугал – Belarusian" lang="be" hreflang="be" data-title="Вугал" 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%9A%D1%83%D1%82" title="Кут – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Кут" 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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Anggulo" title="Anggulo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Anggulo" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%AA%D0%B3%D1%8A%D0%BB" title="Ъгъл – Bulgarian" lang="bg" hreflang="bg" data-title="Ъгъл" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ugao" title="Ugao – Bosnian" lang="bs" hreflang="bs" data-title="Ugao" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Korn_(mentoniezh)" title="Korn (mentoniezh) – Breton" lang="br" hreflang="br" data-title="Korn (mentoniezh)" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D2%AE%D0%BD%D1%81%D1%8D%D0%B3" title="Үнсэг – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Үнсэг" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Angle" title="Angle – Catalan" lang="ca" hreflang="ca" data-title="Angle" 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%9A%C4%95%D1%82%D0%B5%D1%81" title="Кĕтес – Chuvash" lang="cv" hreflang="cv" data-title="Кĕтес" 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/%C3%9Ahel" title="Úhel – Czech" lang="cs" hreflang="cs" data-title="Úhel" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Gonyo" title="Gonyo – Shona" lang="sn" hreflang="sn" data-title="Gonyo" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ongl" title="Ongl – Welsh" lang="cy" hreflang="cy" data-title="Ongl" 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/Vinkel" title="Vinkel – Danish" lang="da" hreflang="da" data-title="Vinkel" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%D9%8A%D8%A9" title="زاوية – Moroccan Arabic" lang="ary" hreflang="ary" data-title="زاوية" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Winkel" title="Winkel – German" lang="de" hreflang="de" data-title="Winkel" 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/Nurk" title="Nurk – Estonian" lang="et" hreflang="et" data-title="Nurk" 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%93%CF%89%CE%BD%CE%AF%CE%B1" title="Γωνία – Greek" lang="el" hreflang="el" data-title="Γωνία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%A3%D0%B6%D0%BE" title="Ужо – Erzya" lang="myv" hreflang="myv" data-title="Ужо" data-language-autonym="Эрзянь" data-language-local-name="Erzya" class="interlanguage-link-target">Эрзянь</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81ngulo" title="Ángulo – Spanish" lang="es" hreflang="es" data-title="Ángulo" 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/Angulo" title="Angulo – Esperanto" lang="eo" hreflang="eo" data-title="Angulo" 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/Angelu_(geometria)" title="Angelu (geometria) – Basque" lang="eu" hreflang="eu" data-title="Angelu (geometria)" 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%B2%D8%A7%D9%88%DB%8C%D9%87" title="زاویه – Persian" lang="fa" hreflang="fa" data-title="زاویه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Angle" title="Angle – Fiji Hindi" lang="hif" hreflang="hif" data-title="Angle" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Angle" title="Angle – French" lang="fr" hreflang="fr" data-title="Angle" 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/Uillinn_(matamaitic)" title="Uillinn (matamaitic) – Irish" lang="ga" hreflang="ga" data-title="Uillinn (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Ce%C3%A0rn_(Matamataig)" title="Ceàrn (Matamataig) – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Ceàrn (Matamataig)" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81ngulo" title="Ángulo – Galician" lang="gl" hreflang="gl" data-title="Ángulo" 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/%E8%A7%92" title="角 – Gan" lang="gan" hreflang="gan" data-title="角" 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/%EA%B0%81_(%EC%88%98%ED%95%99)" title="각 (수학) – Korean" lang="ko" hreflang="ko" data-title="각 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6" title="Անկյուն – Armenian" lang="hy" hreflang="hy" data-title="Անկյուն" 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%95%E0%A5%8B%E0%A4%A3" title="कोण – Hindi" lang="hi" hreflang="hi" data-title="कोण" 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/Kut" title="Kut – Croatian" lang="hr" hreflang="hr" data-title="Kut" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Angulo" title="Angulo – Ido" lang="io" hreflang="io" data-title="Angulo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Anggulo" title="Anggulo – Iloko" lang="ilo" hreflang="ilo" data-title="Anggulo" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target">Ilokano</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sudut_(geometri)" title="Sudut (geometri) – Indonesian" lang="id" hreflang="id" data-title="Sudut (geometri)" 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/Angulo" title="Angulo – Interlingua" lang="ia" hreflang="ia" data-title="Angulo" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Ingoni" title="Ingoni – Zulu" lang="zu" hreflang="zu" data-title="Ingoni" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target">IsiZulu</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Horn_(r%C3%BAmfr%C3%A6%C3%B0i)" title="Horn (rúmfræði) – Icelandic" lang="is" hreflang="is" data-title="Horn (rúmfræði)" 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/Angolo" title="Angolo – Italian" lang="it" hreflang="it" data-title="Angolo" 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%96%D7%95%D7%95%D7%99%D7%AA" title="זווית – Hebrew" lang="he" hreflang="he" data-title="זווית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%8B%E0%B2%A8" title="ಕೋನ – Kannada" lang="kn" hreflang="kn" data-title="ಕೋನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%A3%E1%83%97%E1%83%AE%E1%83%94" title="კუთხე – Georgian" lang="ka" hreflang="ka" data-title="კუთხე" 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%91%D2%B1%D1%80%D1%8B%D1%88_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Бұрыш (геометрия) – Kazakh" lang="kk" hreflang="kk" data-title="Бұрыш (геометрия)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Angle" title="Angle – Cornish" lang="kw" hreflang="kw" data-title="Angle" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target">Kernowek</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Pembe_(jiometria)" title="Pembe (jiometria) – Swahili" lang="sw" hreflang="sw" data-title="Pembe (jiometria)" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Ang" title="Ang – Haitian Creole" lang="ht" hreflang="ht" data-title="Ang" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Ang" title="Ang – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Ang" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hoke" title="Hoke – Kurdish" lang="ku" hreflang="ku" data-title="Hoke" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D1%83%D1%80%D1%87" title="Бурч – Kyrgyz" lang="ky" hreflang="ky" data-title="Бурч" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Angulus" title="Angulus – Latin" lang="la" hreflang="la" data-title="Angulus" 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/Le%C5%86%C4%B7is" title="Leņķis – Latvian" lang="lv" hreflang="lv" data-title="Leņķis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kampas" title="Kampas – Lithuanian" lang="lt" hreflang="lt" data-title="Kampas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Hook" title="Hook – Limburgish" lang="li" hreflang="li" data-title="Hook" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Lit%C3%BAmu" title="Litúmu – Lingala" lang="ln" hreflang="ln" data-title="Litúmu" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target">Lingála</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Angol" title="Angol – Lombard" lang="lmo" hreflang="lmo" data-title="Angol" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%B6g" title="Szög – Hungarian" lang="hu" hreflang="hu" data-title="Szög" 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%90%D0%B3%D0%BE%D0%BB" title="Агол – Macedonian" lang="mk" hreflang="mk" data-title="Агол" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Zoro_(je%C3%B4metria)" title="Zoro (jeômetria) – Malagasy" lang="mg" hreflang="mg" data-title="Zoro (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8B%E0%B5%BA" title="കോൺ – Malayalam" lang="ml" hreflang="ml" data-title="കോൺ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%A8" title="कोन – Marathi" lang="mr" hreflang="mr" data-title="कोन" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%99%E1%83%A3%E1%83%9C%E1%83%97%E1%83%AE%E1%83%A3" title="კუნთხუ – Mingrelian" lang="xmf" hreflang="xmf" data-title="კუნთხუ" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target">მარგალური</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%D9%8A%D9%87_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="زاويه (هندسه) – Egyptian Arabic" lang="arz" hreflang="arz" data-title="زاويه (هندسه)" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%D8%B3%D9%88%DA%A9(%D9%87%D9%86%D9%91%D8%B3%D9%87)" title="سوک(هنّسه) – Mazanderani" lang="mzn" hreflang="mzn" data-title="سوک(هنّسه)" data-language-autonym="مازِرونی" data-language-local-name="Mazanderani" class="interlanguage-link-target">مازِرونی</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sudut" title="Sudut – Malay" lang="ms" hreflang="ms" data-title="Sudut" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%A3%D0%B6%D0%B5%D1%81%D1%8C" title="Ужесь – Moksha" lang="mdf" hreflang="mdf" data-title="Ужесь" data-language-autonym="Мокшень" data-language-local-name="Moksha" class="interlanguage-link-target">Мокшень</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D3%A8%D0%BD%D1%86%D3%A9%D0%B3_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80)" title="Өнцөг (геометр) – Mongolian" lang="mn" hreflang="mn" data-title="Өнцөг (геометр)" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%91%E1%80%B1%E1%80%AC%E1%80%84%E1%80%B7%E1%80%BA" title="ထောင့် – Burmese" lang="my" hreflang="my" data-title="ထောင့်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Tutuni" title="Tutuni – Fijian" lang="fj" hreflang="fj" data-title="Tutuni" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hoek_(meetkunde)" title="Hoek (meetkunde) – Dutch" lang="nl" hreflang="nl" data-title="Hoek (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%A3" title="कोण – Nepali" lang="ne" hreflang="ne" data-title="कोण" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%95%E0%A5%81%E0%A4%82" title="कुं – Newari" lang="new" hreflang="new" data-title="कुं" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A7%92%E5%BA%A6" title="角度 – Japanese" lang="ja" hreflang="ja" data-title="角度" 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-nqo mw-list-item"><a href="https://nqo.wikipedia.org/wiki/%DF%9E%DF%8B%DF%B2%DF%9B%DF%90%DF%B2" title="ߞߋ߲ߛߐ߲ – N’Ko" lang="nqo" hreflang="nqo" data-title="ߞߋ߲ߛߐ߲" data-language-autonym="ߒߞߏ" data-language-local-name="N’Ko" class="interlanguage-link-target">ߒߞߏ</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Winkel" title="Winkel – Northern Frisian" lang="frr" hreflang="frr" data-title="Winkel" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vinkel" title="Vinkel – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Vinkel" 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/Vinkel" title="Vinkel – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Vinkel" 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/Angle" title="Angle – Occitan" lang="oc" hreflang="oc" data-title="Angle" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9B%D1%83%D0%BA" title="Лук – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Лук" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target">Олык марий</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Burchak" title="Burchak – Uzbek" lang="uz" hreflang="uz" data-title="Burchak" 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%95%E0%A9%8B%E0%A8%A8" title="ਕੋਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੋਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%DB%8C%DB%81" title="زاویہ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="زاویہ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%D9%8A%D9%87" title="زاويه – Pashto" lang="ps" hreflang="ps" data-title="زاويه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Hanggl" title="Hanggl – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Hanggl" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%98%E1%9E%BB%E1%9F%86" title="មុំ – Khmer" lang="km" hreflang="km" data-title="មុំ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Winkel_(Geometrie)" title="Winkel (Geometrie) – Low German" lang="nds" hreflang="nds" data-title="Winkel (Geometrie)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/K%C4%85t" title="Kąt – Polish" lang="pl" hreflang="pl" data-title="Kąt" 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/%C3%82ngulo" title="Ângulo – Portuguese" lang="pt" hreflang="pt" data-title="Ângulo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Muyu%C5%9F" title="Muyuş – Crimean Tatar" lang="crh" hreflang="crh" data-title="Muyuş" data-language-autonym="Qırımtatarca" data-language-local-name="Crimean Tatar" class="interlanguage-link-target">Qırımtatarca</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Unghi" title="Unghi – Romanian" lang="ro" hreflang="ro" data-title="Unghi" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Chhuka" title="Chhuka – Quechua" lang="qu" hreflang="qu" data-title="Chhuka" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D0%B3%D0%BE%D0%BB" title="Угол – Russian" lang="ru" hreflang="ru" data-title="Угол" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9C%D1%83%D0%BD%D0%BD%D1%83%D0%BA" title="Муннук – Yakut" lang="sah" hreflang="sah" data-title="Муннук" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/K%C3%ABndi" title="Këndi – Albanian" lang="sq" hreflang="sq" data-title="Këndi" 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/%C3%80nculu" title="Ànculu – Sicilian" lang="scn" hreflang="scn" data-title="Ànculu" 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%9A%E0%B7%9D%E0%B6%AB%E0%B6%BA" title="කෝණය – Sinhala" lang="si" hreflang="si" data-title="කෝණය" 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/Angle" title="Angle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Angle" 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/Uhol" title="Uhol – Slovak" lang="sk" hreflang="sk" data-title="Uhol" 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/Kot" title="Kot – Slovenian" lang="sl" hreflang="sl" data-title="Kot" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D1%AA%D0%B3%D1%8A%D0%BB%D1%8A" title="Ѫгълъ – Church Slavic" lang="cu" hreflang="cu" data-title="Ѫгълъ" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="Church Slavic" class="interlanguage-link-target">Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Xagal" title="Xagal – Somali" lang="so" hreflang="so" data-title="Xagal" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%DB%86%D8%B4%DB%95" title="گۆشە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="گۆشە" 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%A3%D0%B3%D0%B0%D0%BE" title="Угао – Serbian" lang="sr" hreflang="sr" data-title="Угао" 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/Ugao" title="Ugao – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Ugao" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Juru_(%C3%A9lmu_ukur)" title="Juru (élmu ukur) – Sundanese" lang="su" hreflang="su" data-title="Juru (élmu ukur)" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kulma" title="Kulma – Finnish" lang="fi" hreflang="fi" data-title="Kulma" 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/Vinkel" title="Vinkel – Swedish" lang="sv" hreflang="sv" data-title="Vinkel" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Anggulo" title="Anggulo – Tagalog" lang="tl" hreflang="tl" data-title="Anggulo" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%AE%E0%AF%8D" title="கோணம் – Tamil" lang="ta" hreflang="ta" data-title="கோணம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9F%D0%BE%D1%87%D0%BC%D0%B0%D0%BA" title="Почмак – Tatar" lang="tt" hreflang="tt" data-title="Почмак" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B1%8B%E0%B0%A3%E0%B0%82" title="కోణం – Telugu" lang="te" hreflang="te" data-title="కోణం" 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%B8%A1%E0%B8%B8%E0%B8%A1" title="มุม – Thai" lang="th" hreflang="th" data-title="มุม" 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%9A%D1%83%D0%BD%D2%B7" title="Кунҷ – Tajik" lang="tg" hreflang="tg" data-title="Кунҷ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8E%A4%E1%8F%9C%E1%8F%85%E1%8F%9B_%E1%8E%A4%E1%8F%9E%E1%8F%B4%E1%8F%8D%E1%8F%9B" title="ᎤᏜᏅᏛ ᎤᏞᏴᏍᏛ – Cherokee" lang="chr" hreflang="chr" data-title="ᎤᏜᏅᏛ ᎤᏞᏴᏍᏛ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="Cherokee" class="interlanguage-link-target">ᏣᎳᎩ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/A%C3%A7%C4%B1" title="Açı – Turkish" lang="tr" hreflang="tr" data-title="Açı" 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%9A%D1%83%D1%82" title="Кут – Ukrainian" lang="uk" hreflang="uk" data-title="Кут" 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%B2%D8%A7%D9%88%DB%8C%DB%81" title="زاویہ – Urdu" lang="ur" hreflang="ur" data-title="زاویہ" 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/G%C3%B3c" title="Góc – Vietnamese" lang="vi" hreflang="vi" data-title="Góc" 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 href="https://zh-classical.wikipedia.org/wiki/%E8%A7%92" title="角 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="角" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Anggulo" title="Anggulo – Waray" lang="war" hreflang="war" data-title="Anggulo" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%A7%92" title="角 – Wu" lang="wuu" hreflang="wuu" data-title="角" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%95%D7%95%D7%99%D7%A0%D7%A7%D7%9C" title="ווינקל – Yiddish" lang="yi" hreflang="yi" data-title="ווינקל" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A7%92_(%E5%B9%BE%E4%BD%95)" title="角 (幾何) – 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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">Figure formed by two rays meeting at a common point</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">Not to be confused with <a href="/wiki/Angel" title="Angel">Angel</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">This article is about angles in geometry. For other uses, see <a href="/wiki/Angle_(disambiguation)" class="mw-disambig" title="Angle (disambiguation)">Angle (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Angle.svg" class="mw-file-description"><img alt="two line bent at a point" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Angle.svg/280px-Angle.svg.png" decoding="async" width="280" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Angle.svg/420px-Angle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Angle.svg/560px-Angle.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>A green angle formed by two red <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">rays</a> on the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></figcaption></figure> In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, an angle or plane angle is the figure formed by two <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">rays</a>, called the <a href="/wiki/Side_(plane_geometry)" class="mw-redirect" title="Side (plane geometry)">sides</a> of the angle, sharing a common endpoint, called the <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a> of the angle.<a href="#cite_note-1">[1]</a> Two intersecting <a href="/wiki/Curve" title="Curve">curves</a> may also define an angle, which is the angle of the rays lying <a href="/wiki/Tangent" title="Tangent">tangent</a> to the respective curves at their point of intersection. Angles are also formed by the intersection of two planes; these are called <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angles</a>. In any case, the resulting angle lies in a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> (spanned by the two rays or perpendicular to the line of <a href="/wiki/Plane-plane_intersection" class="mw-redirect" title="Plane-plane intersection">plane-plane intersection</a>). The <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> of an angle is called an angular measure or simply "angle". Two different angles may have the same measure, as in an <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangle</a>. "Angle" also denotes the angular sector, the infinite region of the plane bounded by the sides of an angle.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a><a href="#cite_note-7">[a]</a> <a href="/wiki/Angle_of_rotation" class="mw-redirect" title="Angle of rotation">Angle of rotation</a> is a <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a> conventionally defined as the ratio of a <a href="/wiki/Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">circular arc</a> length to its <a href="/wiki/Radius" title="Radius">radius</a>, and may be a <a href="/wiki/Negative_number" title="Negative number">negative number</a>. In the case of an ordinary angle, the arc is centered at the vertex and delimited by the sides. In the case of an angle of <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a>, the arc is centered at the center of the rotation and delimited by any other point and its <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> after the rotation. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History_and_etymology">History and etymology</h2>[<a href="/w/index.php?title=Angle&action=edit&section=1" title="Edit section: History and etymology">edit</a>]</div> The word angle comes from the <a href="/wiki/Latin" title="Latin">Latin</a> word angulus, meaning "corner". <a href="/wiki/Cognate" title="Cognate">Cognate</a> words include the <a href="/wiki/Greek_language" title="Greek language">Greek</a> ἀγκύλος (ankylοs) meaning "crooked, curved" and the <a href="/wiki/English_language" title="English language">English</a> word "<a href="/wiki/Ankle" title="Ankle">ankle</a>". Both are connected with the <a href="/wiki/Proto-Indo-European_language" title="Proto-Indo-European language">Proto-Indo-European</a> root *ank-, meaning "to bend" or "bow".<a href="#cite_note-8">[7]</a> <a href="/wiki/Euclid" title="Euclid">Euclid</a> defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician <a href="/wiki/Proclus" title="Proclus">Proclus</a>, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by <a href="/wiki/Eudemus_of_Rhodes" title="Eudemus of Rhodes">Eudemus of Rhodes</a>, who regarded an angle as a deviation from a <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a>; the second, angle as quantity, by <a href="/wiki/Carpus_of_Antioch" title="Carpus of Antioch">Carpus of Antioch</a>, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.<a href="#cite_note-9">[8]</a> <div class="mw-heading mw-heading2"><h2 id="Identifying_angles">Identifying angles</h2>[<a href="/w/index.php?title=Angle&action=edit&section=2" title="Edit section: Identifying angles">edit</a>]</div> In <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">mathematical expressions</a>, it is common to use <a href="/wiki/Greek_letter" class="mw-redirect" title="Greek letter">Greek letters</a> (<var>α</var>, <var>β</var>, <var>γ</var>, <var>θ</var>, <var>φ</var>, . . . ) as <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> denoting the size of some angle<a href="#cite_note-FOOTNOTEAboughantous201018-10">[9]</a> (the symbol <a href="/wiki/Pi_(letter)" title="Pi (letter)">π</a> is typically not used for this purpose to avoid confusion with the <a href="/wiki/Pi" title="Pi">constant denoted by that symbol</a>). Lower case Roman letters (a, b, c, . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples. The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">rays</a> AB and AC (that is, the half-lines from point A through points B and C) is denoted ∠BAC or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\rm {BAC}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">C</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\rm {BAC}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03586e0b96df959c0c91d4622d2dee499ebaf560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.067ex; height:3.176ex;" alt="{\displaystyle {\widehat {\rm {BAC}}}}">. Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A"). In other ways, an angle denoted as, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see <a href="#Signed_angles">§ Signed angles</a>). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, ∠BAC always refers to the anticlockwise (positive) angle from B to C about A and ∠CAB the anticlockwise (positive) angle from C to B about A. <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2>[<a href="/w/index.php?title=Angle&action=edit&section=3" title="Edit section: Types">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Oblique angle" redirects here. For the cinematographic technique, see <a href="/wiki/Dutch_angle" title="Dutch angle">Dutch angle</a>.</div> <div class="mw-heading mw-heading3"><h3 id="Individual_angles">Individual angles</h3>[<a href="/w/index.php?title=Angle&action=edit&section=4" title="Edit section: Individual angles">edit</a>]</div> There is some common terminology for angles, whose measure is always non-negative (see <a href="#Signed_angles">§ Signed angles</a>): <ul><li>An angle equal to 0° or not turned is called a zero angle.<a href="#cite_note-FOOTNOTEMoser197141-11">[10]</a></li> <li>An angle smaller than a right angle (less than 90°) is called an acute angle<a href="#cite_note-FOOTNOTEGodfreySiddons19199-12">[11]</a> ("acute" meaning "<a href="/wiki/Sharpness_(visual)" class="mw-redirect" title="Sharpness (visual)">sharp</a>").</li> <li>An angle equal to <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>⁠1/4⁠ <a href="/wiki/Turn_(angle)" title="Turn (angle)">turn</a> (90° or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠ <a href="/wiki/Radian" title="Radian">radians</a>) is called a <a href="/wiki/Right_angle" title="Right angle">right angle</a>. Two lines that form a right angle are said to be <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a>, <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a>, or <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a>.<a href="#cite_note-FOOTNOTEMoser197171-13">[12]</a></li> <li>An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle<a href="#cite_note-FOOTNOTEGodfreySiddons19199-12">[11]</a> ("obtuse" meaning "blunt").</li> <li>An angle equal to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ turn (180° or π radians) is called a straight angle.<a href="#cite_note-FOOTNOTEMoser197141-11">[10]</a></li> <li>An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a reflex angle.</li> <li>An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, round angle or perigon.</li> <li>An angle that is not a multiple of a right angle is called an oblique angle.</li></ul> The names, intervals, and measuring units are shown in the table below: <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:404px;max-width:404px"><div class="trow"><div class="tsingle" style="width:113px;max-width:113px"><div class="thumbimage"><a href="/wiki/File:Right_angle.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Right_angle.svg/111px-Right_angle.svg.png" decoding="async" width="111" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Right_angle.svg/167px-Right_angle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Right_angle.svg/222px-Right_angle.svg.png 2x" data-file-width="800" data-file-height="800" /></a></div><div class="thumbcaption"><a href="/wiki/Right_angle" title="Right angle">Right angle</a></div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><a href="/wiki/File:Angle_obtuse_acute_straight.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/200px-Angle_obtuse_acute_straight.svg.png" decoding="async" width="200" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/300px-Angle_obtuse_acute_straight.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/400px-Angle_obtuse_acute_straight.svg.png 2x" data-file-width="800" data-file-height="445" /></a></div><div class="thumbcaption">Acute (<var>a</var>), obtuse (<var>b</var>), and straight (<var>c</var>) angles. The acute and obtuse angles are also known as oblique angles.</div></div><div class="tsingle" style="width:83px;max-width:83px"><div class="thumbimage"><a href="/wiki/File:Reflex_angle.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Reflex_angle.svg/81px-Reflex_angle.svg.png" decoding="async" width="81" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Reflex_angle.svg/122px-Reflex_angle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Reflex_angle.svg/162px-Reflex_angle.svg.png 2x" data-file-width="64" data-file-height="88" /></a></div><div class="thumbcaption">Reflex angle</div></div></div></div></div> <table class="wikitable" style="text-align:center;"> <tbody><tr> <td style="background:#f2f2f2; text-align:center;">Name   </td> <td style="width:3em;">zero angle </td> <td style="width:3em;">acute angle </td> <td style="width:3em;">right angle </td> <td style="width:3em;">obtuse angle </td> <td style="width:3em;">straight angle </td> <td style="width:3em;">reflex angle </td> <td style="width:3em;">perigon </td></tr> <tr> <th>Unit</th> <th colspan="10"><a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">Interval</a> </th></tr> <tr> <td style="background:#f2f2f2; text-align:center;"><a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turn</a>   </td> <td style="width:3em;">0 turn </td> <td style="width:3em;">(0, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠) turn </td> <td style="width:3em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ turn </td> <td style="width:3em;">(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠) turn </td> <td style="width:3em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ turn </td> <td style="width:3em;">(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, 1) turn </td> <td style="width:3em;">1 turn </td></tr> <tr> <td style="background:#f2f2f2; text-align:center;"><a href="/wiki/Radian" title="Radian">radian</a> </td> <td>0 rad </td> <td>(0, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠π) rad </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠π rad </td> <td>(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠π, π) rad </td> <td>π rad </td> <td>(π, 2π) rad </td> <td>2π rad </td></tr> <tr> <td style="background:#f2f2f2; text-align:center;"><a href="/wiki/Degree_(angle)" title="Degree (angle)">degree</a>   </td> <td style="width:3em;">0° </td> <td style="width:3em;">(0, 90)° </td> <td style="width:3em;">90° </td> <td style="width:3em;">(90, 180)° </td> <td style="width:3em;">180° </td> <td style="width:3em;">(180, 360)° </td> <td style="width:3em;">360° </td></tr> <tr> <td style="background:#f2f2f2; text-align:center;"><a href="/wiki/Gradian" title="Gradian">gon</a>   </td> <td style="width:3em;">0g </td> <td style="width:3em;">(0, 100)g </td> <td style="width:3em;">100g </td> <td style="width:3em;">(100, 200)g </td> <td style="width:3em;">200g </td> <td style="width:3em;">(200, 400)g </td> <td style="width:3em;">400g </td></tr> </tbody></table> <div class="mw-heading mw-heading3"><h3 id="Vertical_and_adjacent_angle_pairs">Vertical and <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>adjacent angle pairs</h3>[<a href="/w/index.php?title=Angle&action=edit&section=5" title="Edit section: Vertical and adjacent angle pairs">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vertical_Angles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Vertical_Angles.svg/150px-Vertical_Angles.svg.png" decoding="async" width="150" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Vertical_Angles.svg/225px-Vertical_Angles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Vertical_Angles.svg/300px-Vertical_Angles.svg.png 2x" data-file-width="223" data-file-height="291" /></a><figcaption>Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. <a href="/wiki/Hatch_mark#Congruency_notation" title="Hatch mark">Hatch marks</a> are used here to show angle equality.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Vertical angle" redirects here. Not to be confused with <a href="/wiki/Zenith_angle" class="mw-redirect" title="Zenith angle">Zenith angle</a>.</div> When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other. <div><ul><li>A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as vert. opp. ∠s.<a href="#cite_note-tb-14">[13]</a> <div class="paragraphbreak" style="margin-top:0.5em"></div> The equality of vertically opposite angles is called the vertical angle theorem. <a href="/wiki/Eudemus_of_Rhodes" title="Eudemus of Rhodes">Eudemus of Rhodes</a> attributed the proof to <a href="/wiki/Thales" class="mw-redirect" title="Thales">Thales of Miletus</a>.<a href="#cite_note-15">[14]</a><a href="#cite_note-FOOTNOTEShuteShirkPorter196025–27-16">[15]</a> The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,<a href="#cite_note-FOOTNOTEShuteShirkPorter196025–27-16">[15]</a> when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: <ul><li>All straight angles are equal.</li> <li>Equals added to equals are equal.</li> <li>Equals subtracted from equals are equal.</li></ul> When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x, the measure of angle C would be 180° − x. Similarly, the measure of angle D would be 180° − x. Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180° − (180° − x) = 180° − 180° + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure. <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Adjacentangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Adjacentangles.svg/225px-Adjacentangles.svg.png" decoding="async" width="225" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Adjacentangles.svg/338px-Adjacentangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Adjacentangles.svg/450px-Adjacentangles.svg.png 2x" data-file-width="85" data-file-height="73" /></a><figcaption>Angles A and B are adjacent.</figcaption></figure></li><li>Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called complementary, supplementary, and explementary angles (see <a href="#Combining_angle_pairs">§ Combining angle pairs</a> below).</li></ul></div> A <a href="/wiki/Transversal_(geometry)" title="Transversal (geometry)">transversal</a> is a line that intersects a pair of (often parallel) lines and is associated with exterior angles, interior angles, alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles.<a href="#cite_note-FOOTNOTEJacobs1974255-17">[16]</a> <div class="mw-heading mw-heading3"><h3 id="Combining_angle_pairs">Combining angle pairs</h3>[<a href="/w/index.php?title=Angle&action=edit&section=6" title="Edit section: Combining angle pairs">edit</a>]</div> The angle addition postulate states that if B is in the interior of angle AOC, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">C</mi> </mrow> <mo>=</mo> <mi>m</mi> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">B</mi> </mrow> <mo>+</mo> <mi>m</mi> <mi mathvariant="normal">∠</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d164e93f371ea784134a169a0d552f9f9da2a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:32.652ex; height:2.343ex;" alt="{\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} }"> I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. Three special angle pairs involve the summation of angles: <figure typeof="mw:File/Thumb"><a href="/wiki/File:Complement_angle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Complement_angle.svg/150px-Complement_angle.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Complement_angle.svg/225px-Complement_angle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Complement_angle.svg/300px-Complement_angle.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>The complementary angles <var>a</var> and <var>b</var> (<var>b</var> is the complement of <var>a</var>, and <var>a</var> is the complement of <var>b</var>.)</figcaption></figure> <div><ul><li>Complementary angles are angle pairs whose measures sum to one right angle (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ turn, 90°, or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠ radians).<a href="#cite_note-18">[17]</a> If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary because the sum of internal angles of a <a href="/wiki/Triangle" title="Triangle">triangle</a> is 180 degrees, and the right angle accounts for 90 degrees. <div class="paragraphbreak" style="margin-top:0.5em"></div> The adjective complementary is from the Latin complementum, associated with the verb complere, "to fill up". An acute angle is "filled up" by its complement to form a right angle. <div class="paragraphbreak" style="margin-top:0.5em"></div> The difference between an angle and a right angle is termed the complement of the angle.<a href="#cite_note-Chisholm_1911-19">[18]</a> <div class="paragraphbreak" style="margin-top:0.5em"></div> If angles A and B are complementary, the following relationships hold: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>A</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>B</mi> <mo>=</mo> <mn>1</mn> </mtd> <mtd /> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>A</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>B</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mi>B</mi> </mtd> <mtd /> <mtd> <mi>sec</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>B</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47aa6b35b8e53468f73d7ec051d55371da314fba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:44.312ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}}"> <div class="paragraphbreak" style="margin-top:0.5em"></div> (The <a href="/wiki/Tangent" title="Tangent">tangent</a> of an angle equals the <a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent">cotangent</a> of its complement, and its secant equals the <a href="/wiki/Cosecant" class="mw-redirect" title="Cosecant">cosecant</a> of its complement.) <div class="paragraphbreak" style="margin-top:0.5em"></div> The <a href="/wiki/Prefix" title="Prefix">prefix</a> "<a href="/wiki/Co_(function_prefix)" class="mw-redirect" title="Co (function prefix)">co-</a>" in the names of some trigonometric ratios refers to the word "complementary". <div style="clear:right;" class=""></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Angle_obtuse_acute_straight.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/300px-Angle_obtuse_acute_straight.svg.png" decoding="async" width="300" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/450px-Angle_obtuse_acute_straight.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Angle_obtuse_acute_straight.svg/600px-Angle_obtuse_acute_straight.svg.png 2x" data-file-width="800" data-file-height="445" /></a><figcaption>The angles <var>a</var> and <var>b</var> are supplementary angles.</figcaption></figure></li><li>Two angles that sum to a straight angle (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ turn, 180°, or π radians) are called supplementary angles.<a href="#cite_note-20">[19]</a> <div class="paragraphbreak" style="margin-top:0.5em"></div> If the two supplementary angles are <a href="#adjacent">adjacent</a> (i.e., have a common <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a> and share just one side), their non-shared sides form a <a href="/wiki/Line_(geometry)" title="Line (geometry)">straight line</a>. Such angles are called a linear pair of angles.<a href="#cite_note-FOOTNOTEJacobs197497-21">[20]</a> However, supplementary angles do not have to be on the same line and can be separated in space. For example, adjacent angles of a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> are supplementary, and opposite angles of a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a> (one whose vertices all fall on a single circle) are supplementary. <div class="paragraphbreak" style="margin-top:0.5em"></div> If a point P is exterior to a circle with center O, and if the <a href="/wiki/Tangent_lines_to_circles" title="Tangent lines to circles">tangent lines</a> from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary. <div class="paragraphbreak" style="margin-top:0.5em"></div> The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs. <div class="paragraphbreak" style="margin-top:0.5em"></div> In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle. <div style="clear:right;" class=""></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Conjugate_Angles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Conjugate_Angles.svg/220px-Conjugate_Angles.svg.png" decoding="async" width="220" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Conjugate_Angles.svg/330px-Conjugate_Angles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Conjugate_Angles.svg/440px-Conjugate_Angles.svg.png 2x" data-file-width="413" data-file-height="366" /></a><figcaption>Angles AOB and COD are conjugate as they form a complete angle. Considering magnitudes, 45° + 315° = 360°.</figcaption></figure></li><li>Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are called explementary angles or conjugate angles.<a href="#cite_note-22">[21]</a> <div class="paragraphbreak" style="margin-top:0.5em"></div> The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle. <div style="clear:right;" class=""></div></li></ul></div> <div class="mw-heading mw-heading3"><h3 id="Polygon-related_angles">Polygon-related angles</h3>[<a href="/w/index.php?title=Angle&action=edit&section=7" title="Edit section: Polygon-related angles">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ExternalAngles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/ExternalAngles.svg/300px-ExternalAngles.svg.png" decoding="async" width="300" height="237" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/ExternalAngles.svg/450px-ExternalAngles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/ExternalAngles.svg/600px-ExternalAngles.svg.png 2x" data-file-width="744" data-file-height="588" /></a><figcaption>Internal and external angles</figcaption></figure> <ul><li>An angle that is part of a <a href="/wiki/Simple_polygon" title="Simple polygon">simple polygon</a> is called an <a href="/wiki/Interior_angle" class="mw-redirect" title="Interior angle">interior angle</a> if it lies on the inside of that simple polygon. A simple <a href="/wiki/Concave_polygon" title="Concave polygon">concave polygon</a> has at least one interior angle, that is, a reflex angle. <div class="paragraphbreak" style="margin-top:0.5em"></div> In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the measures of the interior angles of a <a href="/wiki/Triangle" title="Triangle">triangle</a> add up to π radians, 180°, or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ turn; the measures of the interior angles of a simple <a href="/wiki/Convex_polygon" title="Convex polygon">convex</a> <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex <a href="/wiki/Polygon" title="Polygon">polygon</a> with n sides add up to (n − 2)π radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2)<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ turn.</li> <li>The supplement of an interior angle is called an <a href="/wiki/Exterior_angle" class="mw-redirect" title="Exterior angle">exterior angle</a>; that is, an interior angle and an exterior angle form a <a href="#Linear_pair_of_angles">linear pair of angles</a>. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon.<a href="#cite_note-FOOTNOTEHendersonTaimina2005104-23">[22]</a> If the corresponding interior angle is a reflex angle, the exterior angle should be considered <a href="/wiki/Negative_number" title="Negative number">negative</a>. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an <a href="/wiki/Orientation_(space)" class="mw-redirect" title="Orientation (space)">orientation</a> of the <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> (or <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a>) to decide the sign of the exterior angle measure. <div class="paragraphbreak" style="margin-top:0.5em"></div> In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in <a href="/wiki/Logo_(programming_language)" title="Logo (programming language)">Logo Turtle programs</a> when drawing regular polygons.</li> <li>In a <a href="/wiki/Triangle" title="Triangle">triangle</a>, the <a href="/wiki/Bisection" title="Bisection">bisectors</a> of two exterior angles and the bisector of the other interior angle are <a href="/wiki/Concurrent_lines" title="Concurrent lines">concurrent</a> (meet at a single point).<a href="#cite_note-Johnson-24">[23]</a>: 149 </li> <li>In a triangle, three intersection points, each of an external angle bisector with the opposite <a href="/wiki/Extended_side" title="Extended side">extended side</a>, are <a href="/wiki/Collinearity" title="Collinearity">collinear</a>.<a href="#cite_note-Johnson-24">[23]</a>: 149 </li> <li>In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.<a href="#cite_note-Johnson-24">[23]</a>: 149 </li> <li>Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle.<a href="#cite_note-25">[24]</a> This conflicts with the above usage.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Plane-related_angles">Plane-related angles</h3>[<a href="/w/index.php?title=Angle&action=edit&section=8" title="Edit section: Plane-related angles">edit</a>]</div> <ul><li>The angle between two <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">planes</a> (such as two adjacent faces of a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a>) is called a <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a>.<a href="#cite_note-Chisholm_1911-19">[18]</a> It may be defined as the acute angle between two lines <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a> to the planes.</li> <li>The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and the <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a> to the plane.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Measuring_angles">Measuring angles</h2>[<a href="/w/index.php?title=Angle&action=edit&section=9" title="Edit section: Measuring angles">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Angle_measuring_instrument" class="mw-redirect" title="Angle measuring instrument">Angle measuring instrument</a></div> The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be equal congruent or equal in measure. In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full <a href="/wiki/Turn_(angle)" title="Turn (angle)">turn</a> are effectively equivalent. In other contexts, such as identifying a point on a <a href="/wiki/Spiral" title="Spiral">spiral</a> curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Angle_measure.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Angle_measure.svg/220px-Angle_measure.svg.png" decoding="async" width="220" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Angle_measure.svg/330px-Angle_measure.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Angle_measure.svg/440px-Angle_measure.svg.png 2x" data-file-width="800" data-file-height="563" /></a><figcaption>The measure of angle θ is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠s/r⁠ radians.</figcaption></figure> To measure an angle <var><a href="/wiki/Theta" title="Theta">θ</a></var>, a <a href="/wiki/Circular_arc" title="Circular arc">circular arc</a> centered at the vertex of the angle is drawn, e.g., with a pair of <a href="/wiki/Compasses_(drafting)" class="mw-redirect" title="Compasses (drafting)">compasses</a>. The ratio of the length <var>s</var> of the arc by the radius <var>r</var> of the circle is the number of <a href="/wiki/Radian" title="Radian">radians</a> in the angle:<a href="#cite_note-SIBrochure9thEd-26">[25]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .}"> <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> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b50b3fa7d9bdeb54337bc0d171fa86ba9fc448" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.516ex; height:4.676ex;" alt="{\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .}"> Conventionally, in mathematics and the <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a>, the radian is treated as being equal to the <a href="/wiki/Dimensionless_unit" class="mw-redirect" title="Dimensionless unit">dimensionless unit</a> 1, thus being normally omitted. The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠k/2π⁠, where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a> or 400 grad for <a href="/wiki/Gradian" title="Gradian">gradians</a>): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\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>k</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>r</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/342f35b8b904a88834a2e09d275a15bc57120fbd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.772ex; height:5.343ex;" alt="{\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.}"> The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio s/r is unaltered.<a href="#cite_note-28">[nb 1]</a> <div class="mw-heading mw-heading3"><h3 id="Units">Units</h3>[<a href="/w/index.php?title=Angle&action=edit&section=10" title="Edit section: Units">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Angle_radian.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Angle_radian.svg/150px-Angle_radian.svg.png" decoding="async" width="150" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Angle_radian.svg/225px-Angle_radian.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Angle_radian.svg/300px-Angle_radian.svg.png 2x" data-file-width="800" data-file-height="773" /></a><figcaption>Definition of 1 radian</figcaption></figure> Throughout history, angles have been <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measured</a> in various <a href="/wiki/Unit_(measurement)" class="mw-redirect" title="Unit (measurement)">units</a>. These are known as angular units, with the most contemporary units being the <a href="/wiki/Degree_(angle)" title="Degree (angle)">degree</a> ( ° ), the <a href="/wiki/Radian" title="Radian">radian</a> (rad), and the <a href="/wiki/Gradian" title="Gradian">gradian</a> (grad), though many others have been used throughout <a href="/wiki/History_of_Mathematics" class="mw-redirect" title="History of Mathematics">history</a>.<a href="#cite_note-29">[27]</a> Most units of angular measurement are defined such that one <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turn</a> (i.e., the angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n. Two exceptions are the radian (and its decimal submultiples) and the diameter part. In the <a href="/wiki/International_System_of_Quantities" title="International System of Quantities">International System of Quantities</a>, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>. The following table lists some units used to represent angles. <table class="wikitable"> <tbody><tr> <th>Name</th> <th>Number in one turn</th> <th>In degrees</th> <th>Description </th></tr> <tr> <td><a href="/wiki/Radian" title="Radian">radian</a></td> <td>2π</td> <td>≈57°17′45″</td> <td>The radian is determined by the circumference of a circle that is equal in length to the radius of the circle (n = 2π = 6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is rad. One turn is 2π radians, and one radian is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠180°/π⁠, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unit rad being omitted. The radian is used in virtually all mathematical work beyond simple, practical geometry due, for example, to the pleasing and "natural" properties that the <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a>. </td></tr> <tr> <td><a href="/wiki/Degree_(angle)" title="Degree (angle)">degree</a></td> <td>360</td> <td>1°</td> <td>The degree, denoted by a small superscript circle (°), is 1/360 of a turn, so one turn is 360°. One advantage of this old <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the <a href="/wiki/Minute_and_second_of_arc" title="Minute and second of arc">"minute" and "second"</a> sexagesimal subunits of the "degree–minute–second" system (discussed next) are also in use, especially for <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">geographical coordinates</a> and in <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> and <a href="/wiki/Ballistics" title="Ballistics">ballistics</a> (n = 360) </td></tr> <tr> <td><a href="/wiki/Arcminute" class="mw-redirect" title="Arcminute">arcminute</a></td> <td>21,600</td> <td>0°1′</td> <td>The minute of arc (or MOA, arcminute, or just minute) is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/60⁠ of a degree = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/21,600⁠ turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠30/60⁠ = 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3° 5.72′ = 3 + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5.72/60⁠ degrees. A <a href="/wiki/Nautical_mile" title="Nautical mile">nautical mile</a> was historically defined as an arcminute along a <a href="/wiki/Great_circle" title="Great circle">great circle</a> of the Earth. (n = 21,600). </td></tr> <tr> <td><a href="/wiki/Arcsecond" class="mw-redirect" title="Arcsecond">arcsecond</a></td> <td>1,296,000</td> <td>0°0′1″</td> <td>The second of arc (or arcsecond, or just second) is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/60⁠ of a minute of arc and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3600⁠ of a degree (n = 1,296,000). It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/60⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠30/3600⁠ degrees, or 3.125 degrees. The arcsecond is the angle used to measure a <a href="/wiki/Parsec" title="Parsec">parsec</a> </td></tr> <tr> <td><a href="/wiki/Grad_(angle)" class="mw-redirect" title="Grad (angle)">grad</a></td> <td>400</td> <td>0°54′</td> <td>The grad, also called grade, <a href="/wiki/Gradian" title="Gradian">gradian</a>, or gon. It is a decimal subunit of the quadrant. A right angle is 100 grads. A <a href="/wiki/Kilometre" title="Kilometre">kilometre</a> was historically defined as a <a href="/wiki/Centi" class="mw-redirect" title="Centi">centi</a>-grad of arc along a <a href="/wiki/Meridian_(geography)" title="Meridian (geography)">meridian</a> of the Earth, so the kilometer is the decimal analog to the <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> <a href="/wiki/Nautical_mile" title="Nautical mile">nautical mile</a> (n = 400). The grad is used mostly in <a href="/wiki/Triangulation_(surveying)" title="Triangulation (surveying)">triangulation</a> and continental <a href="/wiki/Surveying" title="Surveying">surveying</a>. </td></tr> <tr> <td><a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turn</a></td> <td>1</td> <td>360°</td> <td>The turn is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2π or <a href="/wiki/Turn_(angle)#Proposals_for_a_single_letter_to_represent_2π" title="Turn (angle)">𝜏 (tau)</a> radians. </td></tr> <tr> <td><a href="/wiki/Hour_angle" title="Hour angle">hour angle</a></td> <td>24</td> <td>15°</td> <td>The astronomical hour angle is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/24⁠ turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/12⁠ rad = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠ quad = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/24⁠ turn = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠16+2/3⁠ grad. </td></tr> <tr> <td><a href="/wiki/Points_of_the_compass" title="Points of the compass">(compass) point</a></td> <td>32</td> <td>11°15′</td> <td>The point or wind, used in <a href="/wiki/Navigation" title="Navigation">navigation</a>, is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/32⁠ of a turn. 1 point = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠ of a right angle = 11.25° = 12.5 grad. Each point is subdivided into four quarter points, so one turn equals 128. </td></tr> <tr> <td><a href="/wiki/Milliradian" title="Milliradian">milliradian</a></td> <td>2000π</td> <td>≈0.057°</td> <td>The true milliradian is defined as a thousandth of a radian, which means that a rotation of one <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turn</a> would equal exactly 2000π mrad (or approximately 6283.185 mrad). Almost all <a href="/wiki/Telescopic_sight" title="Telescopic sight">scope sights</a> for <a href="/wiki/Firearm" title="Firearm">firearms</a> are calibrated to this definition. In addition, three other related definitions are used for artillery and navigation, often called a 'mil', which are approximately equal to a milliradian. Under these three other definitions, one turn makes up for exactly 6000, 6300, or 6400 mils, spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the milliradian is approximately 0.05729578 degrees (3.43775 minutes). One "<a href="/wiki/NATO" title="NATO">NATO</a> mil" is defined as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6400⁠ of a turn. Just like with the milliradian, each of the other definitions approximates the milliradian's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2π/6400⁠ = 0.0009817... ≈ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/1000⁠). </td></tr> <tr> <td><a href="/wiki/Binary_angular_measurement" title="Binary angular measurement">binary degree</a></td> <td>256</td> <td>1°33′45″</td> <td>The binary degree, also known as the <a href="/wiki/Binary_radian" class="mw-redirect" title="Binary radian">binary radian</a> or brad or binary angular measurement (BAM).<a href="#cite_note-ooPIC-30">[28]</a> The binary degree is used in computing so that an angle can be efficiently represented in a single <a href="/wiki/Byte" title="Byte">byte</a> (albeit to limited precision). Other measures of the angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n. <a href="#cite_note-Hargreaves_2010-31">[29]</a> It is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/256⁠ of a turn.<a href="#cite_note-ooPIC-30">[28]</a> </td></tr> <tr> <td>π radian</td> <td>2</td> <td>180°</td> <td>The multiples of π radians (MULπ) unit is implemented in the <a href="/wiki/Reverse_Polish_Notation" class="mw-redirect" title="Reverse Polish Notation">RPN</a> scientific calculator <a href="/wiki/WP_43S" class="mw-redirect" title="WP 43S">WP 43S</a>.<a href="#cite_note-Bonin_2016-32">[30]</a> See also: <a href="/wiki/IEEE_754_recommended_operations" class="mw-redirect" title="IEEE 754 recommended operations">IEEE 754 recommended operations</a> </td></tr> <tr> <td><a href="/wiki/Circular_sector" title="Circular sector">quadrant</a></td> <td>4</td> <td>90°</td> <td>One quadrant is a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ turn and also known as a <a href="/wiki/Right_angle" title="Right angle">right angle</a>. The quadrant is the unit in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>. In German, the symbol ∟ has been used to denote a quadrant. 1 quad = 90° = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠ rad = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ turn = 100 grad. </td></tr> <tr> <td><a href="/wiki/Circular_sector" title="Circular sector">sextant</a></td> <td>6</td> <td>60°</td> <td>The sextant was the unit used by the <a href="/wiki/Babylonians" class="mw-redirect" title="Babylonians">Babylonians</a>,<a href="#cite_note-Jeans_1947-33">[31]</a><a href="#cite_note-Murnaghan_1946-34">[32]</a> The degree, minute of arc and second of arc are <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> subunits of the Babylonian unit. It is straightforward to construct with ruler and compasses. It is the angle of the <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> or is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠ turn. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad. </td></tr> <tr> <td>hexacontade</td> <td>60</td> <td>6°</td> <td>The hexacontade is a unit used by <a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a>. It equals 6°, so a whole turn was divided into 60 hexacontades. </td></tr> <tr> <td><a href="/w/index.php?title=Pechus&action=edit&redlink=1" class="new" title="Pechus (page does not exist)">pechus</a></td> <td>144 to 180</td> <td>2° to 2°30′</td> <td>The pechus was a <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian</a> unit equal to about 2° or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2+1/2⁠°. </td></tr> <tr> <td>diameter part</td> <td>≈376.991</td> <td>≈0.95493°</td> <td>The diameter part (occasionally used in Islamic mathematics) is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/60⁠ radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn. </td></tr> <tr> <td>zam</td> <td>224</td> <td>≈1.607°</td> <td>In old Arabia, a <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turn</a> was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam so that a <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turn</a> is 224 zam. </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Dimensional_analysis">Dimensional analysis</h3>[<a href="/w/index.php?title=Angle&action=edit&section=11" title="Edit section: Dimensional analysis">edit</a>]</div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Radian#Dimensional_analysis" title="Radian">Radian § Dimensional analysis</a>.[<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Radian&action=edit#Dimensional_analysis">edit</a>]</div><div class="excerpt"> 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-35">[33]</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_Measures2019151-36">[34]</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-37">[35]</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-38">[36]</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_Measures2019137-39">[37]</a><a href="#cite_note-40">[38]</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-41">[39]</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-42">[40]</a> Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".<a href="#cite_note-43">[41]</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-44">[42]</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-45">[43]</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-46">[44]</a><a href="#cite_note-FOOTNOTEMohrPhillips2015-47">[45]</a><a href="#cite_note-Radian_Quincey-48">[46]</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-FOOTNOTEQuincey2016-49">[47]</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-Radian_Quincey-48">[46]</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-FOOTNOTEQuincey2016-49">[47]</a><a href="#cite_note-50">[b]</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-Radian_Quincey-48">[46]</a><a href="#cite_note-FOOTNOTETorrens1986-51">[48]</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-FOOTNOTETorrens1986-51">[48]</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-FOOTNOTEMohrShirleyPhillipsTrott20226-52">[49]</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-Radian_Quincey-48">[46]</a><a href="#cite_note-FOOTNOTEMohrShirleyPhillipsTrott20228–9-53">[50]</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-FOOTNOTEQuincey2021-54">[51]</a> Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal.<a href="#cite_note-55">[52]</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-56">[53]</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-FOOTNOTEMohrShirleyPhillipsTrott20223-57">[54]</a><a href="#cite_note-58">[55]</a></div></div> <div class="mw-heading mw-heading3"><h3 id="Signed_angles">Signed angles </h3>[<a href="/w/index.php?title=Angle&action=edit&section=12" title="Edit section: Signed angles">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Angle_of_rotation" class="mw-redirect" title="Angle of rotation">Angle of rotation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Sign_(mathematics)#Angles" title="Sign (mathematics)">Sign (mathematics) § Angles</a>, and <a href="/wiki/Euclidean_space#Angle" title="Euclidean space">Euclidean space § Angle</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Angles_on_the_unit_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Angles_on_the_unit_circle.svg/220px-Angles_on_the_unit_circle.svg.png" decoding="async" width="220" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Angles_on_the_unit_circle.svg/330px-Angles_on_the_unit_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Angles_on_the_unit_circle.svg/440px-Angles_on_the_unit_circle.svg.png 2x" data-file-width="409" data-file-height="380" /></a><figcaption>Measuring from the <a href="/wiki/X-axis" class="mw-redirect" title="X-axis">x-axis</a>, angles on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> count as positive in the <a href="/wiki/Counterclockwise" class="mw-redirect" title="Counterclockwise">counterclockwise</a> direction, and negative in the <a href="/wiki/Clockwise" title="Clockwise">clockwise</a> direction.</figcaption></figure> It is frequently helpful to impose a convention that allows positive and negative angular values to represent <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">orientations</a> and/or <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a> in opposite directions or "sense" relative to some reference. In a two-dimensional <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive <a href="/wiki/X-axis" class="mw-redirect" title="X-axis">x-axis</a>, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive <a href="/wiki/Y-axis" class="mw-redirect" title="Y-axis">y-axis</a> and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented by standard position, defined by the x-axis rightward and the y-axis upward, positive rotations are <a href="/wiki/Anticlockwise" class="mw-redirect" title="Anticlockwise">anticlockwise</a>, and negative cycles are <a href="/wiki/Clockwise" title="Clockwise">clockwise</a>. In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an <a href="/wiki/Orientability" title="Orientability">orientation</a>, which is typically determined by a <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal vector</a> passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In <a href="/wiki/Navigation" title="Navigation">navigation</a>, <a href="/wiki/Bearing_(navigation)" title="Bearing (navigation)">bearings</a> or <a href="/wiki/Azimuth" title="Azimuth">azimuth</a> are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°. <div class="mw-heading mw-heading3"><h3 id="Equivalent_angles">Equivalent angles</h3>[<a href="/w/index.php?title=Angle&action=edit&section=13" title="Edit section: Equivalent angles">edit</a>]</div> <ul><li>Angles that have the same measure (i.e., the same magnitude) are said to be equal or <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a>. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all right angles are equal in measure).</li> <li>Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.</li> <li>The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative).<a href="#cite_note-59">[56]</a><a href="#cite_note-60">[57]</a> Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude <a href="/wiki/Modulo" title="Modulo">modulo</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ turn, 180°, or π radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Related_quantities">Related quantities</h3>[<a href="/w/index.php?title=Angle&action=edit&section=14" title="Edit section: Related quantities">edit</a>]</div> For an angular unit, it is definitional that the <a href="/wiki/Angle_addition_postulate" class="mw-redirect" title="Angle addition postulate">angle addition postulate</a> holds. Some quantities related to angles where the angle addition postulate does not hold include: <ul><li>The <a href="/wiki/Slope" title="Slope">slope</a> or gradient is equal to the <a href="/wiki/Tangent_(trigonometric_function)" class="mw-redirect" title="Tangent (trigonometric function)">tangent</a> of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction.</li> <li>The <a href="/wiki/Spread_(rational_trigonometry)" class="mw-redirect" title="Spread (rational trigonometry)">spread</a> between two lines is defined in <a href="/wiki/Rational_geometry" class="mw-redirect" title="Rational geometry">rational geometry</a> as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.</li> <li>Although done rarely, one can report the direct results of <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, such as the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> of the angle.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Angles_between_curves">Angles between curves</h2>[<a href="/w/index.php?title=Angle&action=edit&section=15" title="Edit section: Angles between curves">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Curve_angles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Curve_angles.svg/220px-Curve_angles.svg.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Curve_angles.svg/330px-Curve_angles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Curve_angles.svg/440px-Curve_angles.svg.png 2x" data-file-width="800" data-file-height="495" /></a><figcaption>The angle between the two curves at P is defined as the angle between the tangents <var>A</var> and <var>B</var> at <var>P</var>.</figcaption></figure> The angle between a line and a <a href="/wiki/Curve" title="Curve">curve</a> (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the <a href="/wiki/Tangent" title="Tangent">tangents</a> at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.<a href="#cite_note-61">[58]</a> <div class="mw-heading mw-heading2"><h2 id="Bisecting_and_trisecting_angles">Bisecting and trisecting angles</h2>[<a href="/w/index.php?title=Angle&action=edit&section=16" title="Edit section: Bisecting and trisecting angles">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Bisection#Angle_bisector" title="Bisection">Bisection § Angle bisector</a>, and <a href="/wiki/Angle_trisection" title="Angle trisection">Angle trisection</a></div> The <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greek mathematicians</a> knew how to bisect an angle (divide it into two angles of equal measure) using only a <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a> but could only trisect certain angles. In 1837, <a href="/wiki/Pierre_Wantzel" title="Pierre Wantzel">Pierre Wantzel</a> showed that this construction could not be performed for most angles. <div class="mw-heading mw-heading2"><h2 id="Dot_product_and_generalisations">Dot product and generalisations</h2>[<a href="/w/index.php?title=Angle&action=edit&section=17" title="Edit section: Dot product and generalisations">edit</a>]</div> In the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, the angle θ between two <a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vectors</a> u and v is related to their <a href="/wiki/Dot_product" title="Dot product">dot product</a> and their lengths by the formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a18878af580581a1a8c122ab324e722efd5a12f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.039ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}"> This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal vectors</a> and between <a href="/wiki/Skew_lines" title="Skew lines">skew lines</a> from their vector equations. <div class="mw-heading mw-heading3"><h3 id="Inner_product">Inner product</h3>[<a href="/w/index.php?title=Angle&action=edit&section=18" title="Edit section: Inner product">edit</a>]</div> To define angles in an abstract real <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>, we replace the Euclidean dot product ( · ) by the inner product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }">, i.e. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2ca506e5af6ea05b0ce72517cabaae9b31df82" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.784ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}"> In a complex <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>, the expression for the cosine above may give non-real values, so it is replaced with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39e31b9f9e22e6be0fde65116719d8a11b185ef6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.755ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}"> or, more commonly, using the absolute value, with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb6028c88ccd8fe54d81c245a6e17a49c711e94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.79ex; height:2.843ex;" alt="{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}"> The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {span} (\mathbf {u} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (\mathbf {u} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627f7185adb8f4ba1802de7db03ef9f42474ac5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.958ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (\mathbf {u} )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {span} (\mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (\mathbf {v} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00563c64123d9bffafc970072a0654965a408045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.884ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (\mathbf {v} )}"> spanned by the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> correspondingly. <div class="mw-heading mw-heading3"><h3 id="Angles_between_subspaces">Angles between subspaces</h3>[<a href="/w/index.php?title=Angle&action=edit&section=19" title="Edit section: Angles between subspaces">edit</a>]</div> The definition of the angle between one-dimensional subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {span} (\mathbf {u} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (\mathbf {u} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627f7185adb8f4ba1802de7db03ef9f42474ac5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.958ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (\mathbf {u} )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {span} (\mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (\mathbf {v} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00563c64123d9bffafc970072a0654965a408045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.884ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (\mathbf {v} )}"> given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd7ea2ad54f96dd2e65c143057f25c0c4ff6fd31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.756ex; height:2.843ex;" alt="{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|}"> in a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> can be extended to subspaces of finite dimensions. Given two subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}}"> <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">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e63ea009de5efbca2fc285b8550daaed577c6b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.038ex; width:1.635ex; height:2.176ex;" alt="{\displaystyle {\mathcal {U}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {W}}}"> <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">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {W}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1cc103563219127f59aec7ed9327a3595566dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle {\mathcal {W}}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>k</mi> <mo>≤</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">W</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a218c81681f0dd7cc51db9fe26ccc524603a764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.864ex; height:2.843ex;" alt="{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}">, this leads to a definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> angles called canonical or <a href="/wiki/Principal_angles" class="mw-redirect" title="Principal angles">principal angles</a> between subspaces. <div class="mw-heading mw-heading3"><h3 id="Angles_in_Riemannian_geometry">Angles in Riemannian geometry</h3>[<a href="/w/index.php?title=Angle&action=edit&section=20" title="Edit section: Angles in Riemannian geometry">edit</a>]</div> In <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> is used to define the angle between two <a href="/wiki/Tangent" title="Tangent">tangents</a>. Where U and V are tangent vectors and gij are the components of the metric tensor G, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <msqrt> <mrow> <mo>|</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mo>|</mo> </mrow> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9685961ef6102dc031001a0c97cd7953eeee7b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:30.576ex; height:8.676ex;" alt="{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}"> <div class="mw-heading mw-heading3"><h3 id="Hyperbolic_angle">Hyperbolic angle</h3>[<a href="/w/index.php?title=Angle&action=edit&section=21" title="Edit section: Hyperbolic angle">edit</a>]</div> A <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a> is an <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a> of a <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic function</a> just as the circular angle is the argument of a <a href="/wiki/Circular_function" class="mw-redirect" title="Circular function">circular function</a>. The comparison can be visualized as the size of the openings of a <a href="/wiki/Hyperbolic_sector" title="Hyperbolic sector">hyperbolic sector</a> and a <a href="/wiki/Circular_sector" title="Circular sector">circular sector</a> since the <a href="/wiki/Area" title="Area">areas</a> of these sectors correspond to the angle magnitudes in each case.<a href="#cite_note-62">[59]</a> Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> in their angle argument, the circular ones are just <a href="/wiki/Alternating_series" title="Alternating series">alternating series</a> forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in <a href="/wiki/Introduction_to_the_Analysis_of_the_Infinite" class="mw-redirect" title="Introduction to the Analysis of the Infinite">Introduction to the Analysis of the Infinite</a> (1748). <div class="mw-heading mw-heading2"><h2 id="Angles_in_geography_and_astronomy">Angles in geography and astronomy</h2>[<a href="/w/index.php?title=Angle&action=edit&section=22" title="Edit section: Angles in geography and astronomy">edit</a>]</div> In <a href="/wiki/Geography" title="Geography">geography</a>, the location of any point on the Earth can be identified using a <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">geographic coordinate system</a>. This system specifies the <a href="/wiki/Latitude" title="Latitude">latitude</a> and <a href="/wiki/Longitude" title="Longitude">longitude</a> of any location in terms of angles subtended at the center of the Earth, using the <a href="/wiki/Equator" title="Equator">equator</a> and (usually) the <a href="/wiki/Greenwich_meridian" class="mw-redirect" title="Greenwich meridian">Greenwich meridian</a> as references. In <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, a given point on the <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> (that is, the apparent position of an astronomical object) can be identified using any of several <a href="/wiki/Astronomical_coordinate_systems" title="Astronomical coordinate systems">astronomical coordinate systems</a>, where the references vary according to the particular system. Astronomers measure the <a href="/wiki/Angular_separation" class="mw-redirect" title="Angular separation">angular separation</a> of two <a href="/wiki/Star" title="Star">stars</a> by imagining two lines through the center of the <a href="/wiki/Earth" title="Earth">Earth</a>, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured. In both geography and astronomy, a sighting direction can be specified in terms of a <a href="/wiki/Vertical_angle" class="mw-redirect" title="Vertical angle">vertical angle</a> such as <a href="/wiki/Altitude_angle" class="mw-redirect" title="Altitude angle">altitude</a> /<a href="/wiki/Elevation_angle" class="mw-redirect" title="Elevation angle">elevation</a> with respect to the <a href="/wiki/Horizon" title="Horizon">horizon</a> as well as the <a href="/wiki/Azimuth" title="Azimuth">azimuth</a> with respect to <a href="/wiki/North" title="North">north</a>. Astronomers also measure objects' apparent size as an <a href="/wiki/Angular_diameter" title="Angular diameter">angular diameter</a>. For example, the <a href="/wiki/Full_moon" title="Full moon">full moon</a> has an angular diameter of approximately 0.5° when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The <a href="/wiki/Small-angle_formula" class="mw-redirect" title="Small-angle formula">small-angle formula</a> can convert such an angular measurement into a distance/size ratio. Other astronomical approximations include: <ul><li>0.5° is the approximate diameter of the <a href="/wiki/Sun" title="Sun">Sun</a> and of the <a href="/wiki/Moon" title="Moon">Moon</a> as viewed from Earth.</li> <li>1° is the approximate width of the <a href="/wiki/Little_finger" title="Little finger">little finger</a> at arm's length.</li> <li>10° is the approximate width of a closed fist at arm's length.</li> <li>20° is the approximate width of a handspan at arm's length.</li></ul> These measurements depend on the individual subject, and the above should be treated as rough <a href="/wiki/Rule_of_thumb" title="Rule of thumb">rule of thumb</a> approximations only. In astronomy, <a href="/wiki/Right_ascension" title="Right ascension">right ascension</a> and <a href="/wiki/Declination" title="Declination">declination</a> are usually measured in angular units, expressed in terms of time, based on a 24-hour day. <table class="wikitable"> <tbody><tr> <th>Unit</th> <th><a href="/wiki/Sexagesimal" title="Sexagesimal">Symbol</a></th> <th>Degrees</th> <th>Radians</th> <th>Turns</th> <th>Other </th></tr> <tr> <th>Hour </th> <td>h</td> <td>15°</td> <td><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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>π⁄12 rad</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄24 turn</td> <td> </td></tr> <tr> <th>Minute </th> <td>m</td> <td>0°15′</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">π⁄720 rad</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄1,440 turn</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄60 hour </td></tr> <tr> <th>Second </th> <td>s</td> <td>0°0′15″</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">π⁄43200 rad</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄86,400 turn</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄60 minute </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Angle&action=edit&section=23" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Angle_measuring_instrument" class="mw-redirect" title="Angle measuring instrument">Angle measuring instrument</a></li> <li><a href="/wiki/Angles_between_flats" title="Angles between flats">Angles between flats</a></li> <li><a href="/wiki/Angular_statistics" class="mw-redirect" title="Angular statistics">Angular statistics</a> (<a href="/wiki/Angular_mean" class="mw-redirect" title="Angular mean">mean</a>, <a href="/wiki/Angular_standard_deviation" class="mw-redirect" title="Angular standard deviation">standard deviation</a>)</li> <li><a href="/wiki/Bisection#Angle_bisector" title="Bisection">Angle bisector</a></li> <li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a></li> <li><a href="/wiki/Angular_diameter" title="Angular diameter">Angular diameter</a></li> <li><a href="/wiki/Angular_velocity" title="Angular velocity">Angular velocity</a></li> <li><a href="/wiki/Argument_(complex_analysis)" title="Argument (complex analysis)">Argument (complex analysis)</a></li> <li><a href="/wiki/Astrological_aspect" title="Astrological aspect">Astrological aspect</a></li> <li><a href="/wiki/Central_angle" title="Central angle">Central angle</a></li> <li><a href="/wiki/Clock_angle_problem" title="Clock angle problem">Clock angle problem</a></li> <li><a href="/wiki/Decimal_degrees" title="Decimal degrees">Decimal degrees</a></li> <li><a href="/wiki/Dihedral_angle" title="Dihedral angle">Dihedral angle</a></li> <li><a href="/wiki/Exterior_angle_theorem" title="Exterior angle theorem">Exterior angle theorem</a></li> <li><a href="/wiki/Golden_angle" title="Golden angle">Golden angle</a></li> <li><a href="/wiki/Great_circle_distance" class="mw-redirect" title="Great circle distance">Great circle distance</a></li> <li><a href="/wiki/Horn_angle" title="Horn angle">Horn angle</a></li> <li><a href="/wiki/Inscribed_angle" title="Inscribed angle">Inscribed angle</a></li> <li><a href="/wiki/Irrational_angle" class="mw-redirect" title="Irrational angle">Irrational angle</a></li> <li><a href="/wiki/Phase_(waves)" title="Phase (waves)">Phase (waves)</a></li> <li><a href="/wiki/Protractor" class="mw-redirect" title="Protractor">Protractor</a></li> <li><a href="/wiki/Solid_angle" title="Solid angle">Solid angle</a></li> <li><a href="/wiki/Spherical_angle" class="mw-redirect" title="Spherical angle">Spherical angle</a></li> <li><a href="/wiki/Subtended_angle" title="Subtended angle">Subtended angle</a></li> <li><a href="/wiki/Tangential_angle" title="Tangential angle">Tangential angle</a></li> <li><a href="/wiki/Transcendent_angle" class="mw-redirect" title="Transcendent angle">Transcendent angle</a></li> <li><a href="/wiki/Trisection" class="mw-redirect" title="Trisection">Trisection</a></li> <li><a href="/wiki/Zenith_angle" class="mw-redirect" title="Zenith angle">Zenith angle</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Angle&action=edit&section=24" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-28"><a href="#cite_ref-28">^</a> This approach requires, however, an additional proof that the measure of the angle does not change with changing radius r, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance.<a href="#cite_note-Dimitric_2012-27">[26]</a> </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-7"><a href="#cite_ref-7">^</a> An angular sector can be constructed by the combination of two rotated <a href="/wiki/Half-plane" class="mw-redirect" title="Half-plane">half-planes</a>, either their intersection or union (in the case of acute or obtuse angles, respectively).<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> It corresponds to a <a href="/wiki/Circular_sector" title="Circular sector">circular sector</a> of infinite radius and a flat <a href="/wiki/Pencil_of_half-lines" class="mw-redirect" title="Pencil of half-lines">pencil of half-lines</a>.<a href="#cite_note-6">[6]</a> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</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>[<a href="/w/index.php?title=Angle&action=edit&section=25" title="Edit section: References">edit</a>]</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-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFSidorov2001">Sidorov 2001</a> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</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="CITEREFEvgrafov2019" class="citation book cs1">Evgrafov, M. A. (2019-09-18). <a rel="nofollow" class="external text" href="https://www.google.com.br/books/edition/Analytic_Functions/N8-wDwAAQBAJ?hl=en&gbpv=1&dq=angle+and+%2522angular+sector%2522+domain&pg=PA126&printsec=frontcover">Analytic Functions</a>. Courier Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-84366-7" title="Special:BookSources/978-0-486-84366-7"><bdi>978-0-486-84366-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+Functions&rft.pub=Courier+Dover+Publications&rft.date=2019-09-18&rft.isbn=978-0-486-84366-7&rft.aulast=Evgrafov&rft.aufirst=M.+A.&rft_id=https%3A%2F%2Fwww.google.com.br%2Fbooks%2Fedition%2FAnalytic_Functions%2FN8-wDwAAQBAJ%3Fhl%3Den%26gbpv%3D1%26dq%3Dangle%2Band%2B%252522angular%2Bsector%252522%2Bdomain%26pg%3DPA126%26printsec%3Dfrontcover&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPapadopoulos2012" class="citation book cs1">Papadopoulos, Athanase (2012). <a rel="nofollow" class="external text" href="https://www.google.com.br/books/edition/Strasbourg_Master_Class_on_Geometry/f6yZeVMqhNEC?hl=en&gbpv=1&dq=angle+and+%2522angular+sector%2522+region&pg=PA12&printsec=frontcover">Strasbourg Master Class on Geometry</a>. European Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-105-7" title="Special:BookSources/978-3-03719-105-7"><bdi>978-3-03719-105-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Strasbourg+Master+Class+on+Geometry&rft.pub=European+Mathematical+Society&rft.date=2012&rft.isbn=978-3-03719-105-7&rft.aulast=Papadopoulos&rft.aufirst=Athanase&rft_id=https%3A%2F%2Fwww.google.com.br%2Fbooks%2Fedition%2FStrasbourg_Master_Class_on_Geometry%2Ff6yZeVMqhNEC%3Fhl%3Den%26gbpv%3D1%26dq%3Dangle%2Band%2B%252522angular%2Bsector%252522%2Bregion%26pg%3DPA12%26printsec%3Dfrontcover&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD'Andrea2023" class="citation book cs1">D'Andrea, Francesco (2023-08-19). <a rel="nofollow" class="external text" href="https://www.google.com.br/books/edition/A_Guide_to_Penrose_Tilings/BszREAAAQBAJ?hl=en&gbpv=1&dq=angle+and+%2522angular+sector%2522&pg=PA68&printsec=frontcover">A Guide to Penrose Tilings</a>. Springer Nature. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-28428-1" title="Special:BookSources/978-3-031-28428-1"><bdi>978-3-031-28428-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+to+Penrose+Tilings&rft.pub=Springer+Nature&rft.date=2023-08-19&rft.isbn=978-3-031-28428-1&rft.aulast=D%27Andrea&rft.aufirst=Francesco&rft_id=https%3A%2F%2Fwww.google.com.br%2Fbooks%2Fedition%2FA_Guide_to_Penrose_Tilings%2FBszREAAAQBAJ%3Fhl%3Den%26gbpv%3D1%26dq%3Dangle%2Band%2B%252522angular%2Bsector%252522%26pg%3DPA68%26printsec%3Dfrontcover&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBulboacǎJoshiGoswami2019" class="citation book cs1">Bulboacǎ, Teodor; Joshi, Santosh B.; Goswami, Pranay (2019-07-08). <a rel="nofollow" class="external text" href="https://www.google.com.br/books/edition/Complex_Analysis/r0miDwAAQBAJ?hl=en&gbpv=1&dq=angle+and+%2522angular+sector%2522+half-planes&pg=PT22&printsec=frontcover">Complex Analysis: Theory and Applications</a>. Walter de Gruyter GmbH & Co KG. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-065803-3" title="Special:BookSources/978-3-11-065803-3"><bdi>978-3-11-065803-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis%3A+Theory+and+Applications&rft.pub=Walter+de+Gruyter+GmbH+%26+Co+KG&rft.date=2019-07-08&rft.isbn=978-3-11-065803-3&rft.aulast=Bulboac%C7%8E&rft.aufirst=Teodor&rft.au=Joshi%2C+Santosh+B.&rft.au=Goswami%2C+Pranay&rft_id=https%3A%2F%2Fwww.google.com.br%2Fbooks%2Fedition%2FComplex_Analysis%2Fr0miDwAAQBAJ%3Fhl%3Den%26gbpv%3D1%26dq%3Dangle%2Band%2B%252522angular%2Bsector%252522%2Bhalf-planes%26pg%3DPT22%26printsec%3Dfrontcover&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRedei2014" class="citation book cs1">Redei, L. 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Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4832-8270-1" title="Special:BookSources/978-1-4832-8270-1"><bdi>978-1-4832-8270-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundation+of+Euclidean+and+Non-Euclidean+Geometries+according+to+F.+Klein&rft.pub=Elsevier&rft.date=2014-07-15&rft.isbn=978-1-4832-8270-1&rft.aulast=Redei&rft.aufirst=L.&rft_id=https%3A%2F%2Fwww.google.com.br%2Fbooks%2Fedition%2FFoundation_of_Euclidean_and_Non_Euclidea%2FXMTSBQAAQBAJ%3Fhl%3Den%26gbpv%3D1%26dq%3Dhalf-pencil%2Bof%2Blines%26pg%3DPA45%26printsec%3Dfrontcover&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFSlocum2007">Slocum 2007</a> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFChisholm1911">Chisholm 1911</a>; <a href="#CITEREFHeiberg1908">Heiberg 1908</a>, pp. 177–178 </li> <li id="cite_note-FOOTNOTEAboughantous201018-10"><a href="#cite_ref-FOOTNOTEAboughantous201018_10-0">^</a> <a href="#CITEREFAboughantous2010">Aboughantous 2010</a>, p. 18. </li> <li id="cite_note-FOOTNOTEMoser197141-11">^ <a href="#cite_ref-FOOTNOTEMoser197141_11-0">a</a> <a href="#cite_ref-FOOTNOTEMoser197141_11-1">b</a> <a href="#CITEREFMoser1971">Moser 1971</a>, p. 41. </li> <li id="cite_note-FOOTNOTEGodfreySiddons19199-12">^ <a href="#cite_ref-FOOTNOTEGodfreySiddons19199_12-0">a</a> <a href="#cite_ref-FOOTNOTEGodfreySiddons19199_12-1">b</a> <a href="#CITEREFGodfreySiddons1919">Godfrey & Siddons 1919</a>, p. 9. </li> <li id="cite_note-FOOTNOTEMoser197171-13"><a href="#cite_ref-FOOTNOTEMoser197171_13-0">^</a> <a href="#CITEREFMoser1971">Moser 1971</a>, p. 71. </li> <li id="cite_note-tb-14"><a href="#cite_ref-tb_14-0">^</a> <a href="#CITEREFWongWong2009">Wong & Wong 2009</a>, pp. 161–163 </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuclid" class="citation book cs1"><a href="/wiki/Euclid" title="Euclid">Euclid</a>. <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">The Elements</a>.</cite> Proposition I:13. </li> <li id="cite_note-FOOTNOTEShuteShirkPorter196025–27-16">^ <a href="#cite_ref-FOOTNOTEShuteShirkPorter196025–27_16-0">a</a> <a href="#cite_ref-FOOTNOTEShuteShirkPorter196025–27_16-1">b</a> <a href="#CITEREFShuteShirkPorter1960">Shute, Shirk & Porter 1960</a>, pp. 25–27. </li> <li id="cite_note-FOOTNOTEJacobs1974255-17"><a href="#cite_ref-FOOTNOTEJacobs1974255_17-0">^</a> <a href="#CITEREFJacobs1974">Jacobs 1974</a>, p. 255. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/geometry/complementary-angles.html">"Complementary Angles"</a>. www.mathsisfun.com. Retrieved 2020-08-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Complementary+Angles&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fgeometry%2Fcomplementary-angles.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"> </li> <li id="cite_note-Chisholm_1911-19">^ <a href="#cite_ref-Chisholm_1911_19-0">a</a> <a href="#cite_ref-Chisholm_1911_19-1">b</a> <a href="#CITEREFChisholm1911">Chisholm 1911</a> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/geometry/supplementary-angles.html">"Supplementary Angles"</a>. www.mathsisfun.com. 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(2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Angle">"Angle"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Angle&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Sidorov&rft.aufirst=L.+A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAngle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlocum2007" class="citation cs2">Slocum, Jonathan (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100627012240/http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html">Preliminary Indo-European lexicon — Pokorny PIE data</a>, <a href="/wiki/Linguistics_Research_Center_at_UT_Austin" title="Linguistics Research Center at UT Austin">University of Texas research department: linguistics research center</a>, archived from <a rel="nofollow" class="external text" href="http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html">the original</a> on 27 June 2010, retrieved 2 Feb 2010</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Preliminary+Indo-European+lexicon+%E2%80%94+Pokorny+PIE+data&rft.pub=University+of+Texas+research+department%3A+linguistics+research+center&rft.date=2007&rft.aulast=Slocum&rft.aufirst=Jonathan&rft_id=http%3A%2F%2Fwww.utexas.edu%2Fcola%2Fcenters%2Flrc%2Fielex%2FX%2FP0089.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShuteShirkPorter1960" class="citation cs2">Shute, William G.; Shirk, William W.; Porter, George F. (1960), Plane and Solid Geometry, American Book Company, pp. 25–27</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plane+and+Solid+Geometry&rft.pages=%3Cspan+class%3D%22nowrap%22%3E25-%3C%2Fspan%3E27&rft.pub=American+Book+Company&rft.date=1960&rft.aulast=Shute&rft.aufirst=William+G.&rft.au=Shirk%2C+William+W.&rft.au=Porter%2C+George+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTorrens1986" class="citation journal cs1">Torrens, A B (1 January 1986). "On Angles and Angular Quantities". Metrologia. 22 (1): 1–7. <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/1986Metro..22....1T">1986Metro..22....1T</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%2F22%2F1%2F002">10.1088/0026-1394/22/1/002</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:250801509">250801509</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+Angles+and+Angular+Quantities&rft.volume=22&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E7&rft.date=1986-01-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250801509%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0026-1394%2F22%2F1%2F002&rft_id=info%3Abibcode%2F1986Metro..22....1T&rft.aulast=Torrens&rft.aufirst=A+B&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWongWong2009" class="citation cs2">Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines", New Century Mathematics, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-800177-5" title="Special:BookSources/978-0-19-800177-5"><bdi>978-0-19-800177-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Angles+in+Intersecting+and+Parallel+Lines&rft.btitle=New+Century+Mathematics&rft.place=Hong+Kong&rft.pages=%3Cspan+class%3D%22nowrap%22%3E161-%3C%2Fspan%3E163&rft.edition=1&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0-19-800177-5&rft.aulast=Wong&rft.aufirst=Tak-wah&rft.au=Wong%2C+Ming-sim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"></li></ul> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png" decoding="async" width="12" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/18px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/24px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /> This article incorporates text from a publication now in the <a href="/wiki/Public_domain" title="Public domain">public domain</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChisholm1911" class="citation encyclopaedia cs2"><a href="/wiki/Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a>, ed. (1911), "<a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Angle" class="extiw" title="s:1911 Encyclopædia Britannica/Angle">Angle</a>", <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a>, vol. 2 (11th ed.), Cambridge University Press, p. 14</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Angle&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.pages=14&rft.edition=11th&rft.pub=Cambridge+University+Press&rft.date=1911&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngle" class="Z3988"> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Angle&action=edit&section=27" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-d8647bfd6-cv6l9","wgBackendResponseTime":195,"wgPageParseReport":{"limitreport":{"cputime":"1.554","walltime":"1.998","ppvisitednodes":{"value":10812,"limit":1000000},"postexpandincludesize":{"value":200665,"limit":2097152},"templateargumentsize":{"value":12231,"limit":2097152},"expansiondepth":{"value":15,"limit":100},"expensivefunctioncount":{"value":15,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":236031,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1693.081 1 -total"," 28.92% 489.666 3 Template:Reflist"," 16.63% 281.570 1 Template:Excerpt"," 9.09% 153.851 12 Template:Cite_book"," 7.88% 133.431 20 Template:Cite_journal"," 7.05% 119.441 11 Template:Citation"," 7.02% 118.892 4 Template:Lang"," 6.01% 101.813 1 Template:Authority_control"," 5.82% 98.557 22 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