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laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Angolo_convesso_e_concavo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Angolo_convesso_e_concavo"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Angolo convesso e concavo</span> </div> </a> <ul id="toc-Angolo_convesso_e_concavo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_misurazione_dell'ampiezza_degli_angoli_convessi_e_concavi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#La_misurazione_dell'ampiezza_degli_angoli_convessi_e_concavi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>La misurazione dell'ampiezza degli angoli convessi e concavi</span> </div> </a> <button aria-controls="toc-La_misurazione_dell'ampiezza_degli_angoli_convessi_e_concavi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione La misurazione dell'ampiezza degli angoli convessi e concavi</span> </button> <ul id="toc-La_misurazione_dell'ampiezza_degli_angoli_convessi_e_concavi-sublist" class="vector-toc-list"> <li id="toc-Considerazioni_preliminari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Considerazioni_preliminari"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Considerazioni preliminari</span> </div> </a> <ul id="toc-Considerazioni_preliminari-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dalla_misura_dell'angolo_alla_misura_dell'ampiezza_dell'angolo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dalla_misura_dell'angolo_alla_misura_dell'ampiezza_dell'angolo"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Dalla misura dell'angolo alla misura dell'ampiezza dell'angolo</span> </div> </a> <ul id="toc-Dalla_misura_dell'angolo_alla_misura_dell'ampiezza_dell'angolo-sublist" class="vector-toc-list"> <li id="toc-Sistemi_di_misurazione_dell'ampiezza_dell'angolo" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sistemi_di_misurazione_dell'ampiezza_dell'angolo"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Sistemi di misurazione dell'ampiezza dell'angolo</span> </div> </a> <ul id="toc-Sistemi_di_misurazione_dell'ampiezza_dell'angolo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conversioni_angolari" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conversioni_angolari"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Conversioni angolari</span> </div> </a> <ul id="toc-Conversioni_angolari-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ampiezze_di_angoli_particolari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ampiezze_di_angoli_particolari"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Ampiezze di angoli particolari</span> </div> </a> <ul id="toc-Ampiezze_di_angoli_particolari-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angoli_complementari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angoli_complementari"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Angoli complementari</span> </div> </a> <ul id="toc-Angoli_complementari-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angoli_opposti_al_vertice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angoli_opposti_al_vertice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Angoli opposti al vertice</span> </div> </a> <ul id="toc-Angoli_opposti_al_vertice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angoli_formati_da_rette_tagliate_da_una_trasversale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angoli_formati_da_rette_tagliate_da_una_trasversale"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Angoli formati da rette tagliate da una trasversale</span> </div> </a> <ul id="toc-Angoli_formati_da_rette_tagliate_da_una_trasversale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Somma_degli_angoli_interni" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Somma_degli_angoli_interni"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Somma degli angoli interni</span> </div> </a> <ul id="toc-Somma_degli_angoli_interni-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Angoli_con_segno" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Angoli_con_segno"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Angoli con segno</span> </div> </a> <ul id="toc-Angoli_con_segno-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angoli_solidi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Angoli_solidi"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Angoli solidi</span> </div> </a> <ul id="toc-Angoli_solidi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Indice" > <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="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Angolo</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 141 lingue" > <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-141" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">141 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/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"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Winkel_(Geometrie)" title="Winkel (Geometrie) - tedesco svizzero" lang="gsw" hreflang="gsw" data-title="Winkel (Geometrie)" data-language-autonym="Alemannisch" data-language-local-name="tedesco svizzero" class="interlanguage-link-target"><span>Alemannisch</span></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"><span>Aragonés</span></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) - arabo" lang="ar" hreflang="ar" data-title="زاوية (هندسة)" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></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) - aramaico" lang="arc" hreflang="arc" data-title="ܙܘܝܬܐ (ܡܚܪܘܬܐ)" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="aramaico" class="interlanguage-link-target"><span>ܐܪܡܝܐ</span></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 - arabo marocchino" lang="ary" hreflang="ary" data-title="زاوية" data-language-autonym="الدارجة" data-language-local-name="arabo marocchino" class="interlanguage-link-target"><span>الدارجة</span></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) - arabo egiziano" lang="arz" hreflang="arz" data-title="زاويه (هندسه)" data-language-autonym="مصرى" data-language-local-name="arabo egiziano" class="interlanguage-link-target"><span>مصرى</span></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"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81ngulu" title="Ángulu - asturiano" lang="ast" hreflang="ast" data-title="Ángulu" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></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"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bucaq" title="Bucaq - azerbaigiano" lang="az" hreflang="az" data-title="Bucaq" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" class="interlanguage-link-target"><span>Azərbaycanca</span></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"><span>تۆرکجه</span></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 - baschiro" lang="ba" hreflang="ba" data-title="Мөйөш" data-language-autonym="Башҡортса" data-language-local-name="baschiro" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Komps" title="Komps - samogitico" lang="sgs" hreflang="sgs" data-title="Komps" data-language-autonym="Žemaitėška" data-language-local-name="samogitico" class="interlanguage-link-target"><span>Žemaitėška</span></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"><span>Bikol Central</span></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 - bielorusso" lang="be" hreflang="be" data-title="Вугал" data-language-autonym="Беларуская" data-language-local-name="bielorusso" class="interlanguage-link-target"><span>Беларуская</span></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"><span>Беларуская (тарашкевіца)</span></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 - bulgaro" lang="bg" hreflang="bg" data-title="Ъгъл" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</span></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 - bengalese" lang="bn" hreflang="bn" data-title="কোণ" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Korn_(mentoniezh)" title="Korn (mentoniezh) - bretone" lang="br" hreflang="br" data-title="Korn (mentoniezh)" data-language-autonym="Brezhoneg" data-language-local-name="bretone" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ugao" title="Ugao - bosniaco" lang="bs" hreflang="bs" data-title="Ugao" data-language-autonym="Bosanski" data-language-local-name="bosniaco" class="interlanguage-link-target"><span>Bosanski</span></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"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Angle" title="Angle - catalano" lang="ca" hreflang="ca" data-title="Angle" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></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"><span>ᏣᎳᎩ</span></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 - curdo centrale" lang="ckb" hreflang="ckb" data-title="گۆشە" data-language-autonym="کوردی" data-language-local-name="curdo centrale" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Muyu%C5%9F" title="Muyuş - turco crimeo" lang="crh" hreflang="crh" data-title="Muyuş" data-language-autonym="Qırımtatarca" data-language-local-name="turco crimeo" class="interlanguage-link-target"><span>Qırımtatarca</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C3%9Ahel" title="Úhel - ceco" lang="cs" hreflang="cs" data-title="Úhel" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></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 - slavo ecclesiastico" lang="cu" hreflang="cu" data-title="Ѫгълъ" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="slavo ecclesiastico" class="interlanguage-link-target"><span>Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</span></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 - ciuvascio" lang="cv" hreflang="cv" data-title="Кĕтес" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ongl" title="Ongl - gallese" lang="cy" hreflang="cy" data-title="Ongl" data-language-autonym="Cymraeg" data-language-local-name="gallese" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Vinkel" title="Vinkel - danese" lang="da" hreflang="da" data-title="Vinkel" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Winkel" title="Winkel - tedesco" lang="de" hreflang="de" data-title="Winkel" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></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 - greco" lang="el" hreflang="el" data-title="Γωνία" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Angle" title="Angle - inglese" lang="en" hreflang="en" data-title="Angle" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></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"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81ngulo" title="Ángulo - spagnolo" lang="es" hreflang="es" data-title="Ángulo" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Nurk" title="Nurk - estone" lang="et" hreflang="et" data-title="Nurk" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Angelu_(geometria)" title="Angelu (geometria) - basco" lang="eu" hreflang="eu" data-title="Angelu (geometria)" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%DB%8C%D9%87" title="زاویه - persiano" lang="fa" hreflang="fa" data-title="زاویه" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kulma" title="Kulma - finlandese" lang="fi" hreflang="fi" data-title="Kulma" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Tutuni" title="Tutuni - figiano" lang="fj" hreflang="fj" data-title="Tutuni" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="figiano" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Angle" title="Angle - francese" lang="fr" hreflang="fr" data-title="Angle" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Winkel" title="Winkel - frisone settentrionale" lang="frr" hreflang="frr" data-title="Winkel" data-language-autonym="Nordfriisk" data-language-local-name="frisone settentrionale" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uillinn_(matamaitic)" title="Uillinn (matamaitic) - irlandese" lang="ga" hreflang="ga" data-title="Uillinn (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></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"><span>贛語</span></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"><span>Kriyòl gwiyannen</span></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) - gaelico scozzese" lang="gd" hreflang="gd" data-title="Ceàrn (Matamataig)" data-language-autonym="Gàidhlig" data-language-local-name="gaelico scozzese" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81ngulo" title="Ángulo - galiziano" lang="gl" hreflang="gl" data-title="Ángulo" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Takamby" title="Takamby - guaraní" lang="gn" hreflang="gn" data-title="Takamby" data-language-autonym="Avañe'ẽ" data-language-local-name="guaraní" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%96%D7%95%D7%95%D7%99%D7%AA" title="זווית - ebraico" lang="he" hreflang="he" data-title="זווית" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></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"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Angle" title="Angle - hindi figiano" lang="hif" hreflang="hif" data-title="Angle" data-language-autonym="Fiji Hindi" data-language-local-name="hindi figiano" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kut" title="Kut - croato" lang="hr" hreflang="hr" data-title="Kut" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Ang" title="Ang - creolo haitiano" lang="ht" hreflang="ht" data-title="Ang" data-language-autonym="Kreyòl ayisyen" data-language-local-name="creolo haitiano" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%B6g" title="Szög - ungherese" lang="hu" hreflang="hu" data-title="Szög" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6" title="Անկյուն - armeno" lang="hy" hreflang="hy" data-title="Անկյուն" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></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"><span>Արեւմտահայերէն</span></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"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sudut_(geometri)" title="Sudut (geometri) - indonesiano" lang="id" hreflang="id" data-title="Sudut (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Anggulo" title="Anggulo - ilocano" lang="ilo" hreflang="ilo" data-title="Anggulo" data-language-autonym="Ilokano" data-language-local-name="ilocano" class="interlanguage-link-target"><span>Ilokano</span></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"><span>Ido</span></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) - islandese" lang="is" hreflang="is" data-title="Horn (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></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 - giapponese" lang="ja" hreflang="ja" data-title="角度" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Hanggl" title="Hanggl - creolo giamaicano" lang="jam" hreflang="jam" data-title="Hanggl" data-language-autonym="Patois" data-language-local-name="creolo giamaicano" class="interlanguage-link-target"><span>Patois</span></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 - georgiano" lang="ka" hreflang="ka" data-title="კუთხე" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><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 - cabardino" lang="kbd" hreflang="kbd" data-title="ПлӀанэпэ" data-language-autonym="Адыгэбзэ" data-language-local-name="cabardino" class="interlanguage-link-target"><span>Адыгэбзэ</span></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) - kazako" lang="kk" hreflang="kk" data-title="Бұрыш (геометрия)" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></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"><span>ភាសាខ្មែរ</span></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"><span>ಕನ್ನಡ</span></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) - coreano" lang="ko" hreflang="ko" data-title="각 (수학)" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hoke" title="Hoke - curdo" lang="ku" hreflang="ku" data-title="Hoke" data-language-autonym="Kurdî" data-language-local-name="curdo" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Angle" title="Angle - cornico" lang="kw" hreflang="kw" data-title="Angle" data-language-autonym="Kernowek" data-language-local-name="cornico" class="interlanguage-link-target"><span>Kernowek</span></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 - kirghiso" lang="ky" hreflang="ky" data-title="Бурч" data-language-autonym="Кыргызча" data-language-local-name="kirghiso" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Angulus" title="Angulus - latino" lang="la" hreflang="la" data-title="Angulus" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Hook" title="Hook - limburghese" lang="li" hreflang="li" data-title="Hook" data-language-autonym="Limburgs" data-language-local-name="limburghese" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Angol" title="Angol - lombardo" lang="lmo" hreflang="lmo" data-title="Angol" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></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"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kampas" title="Kampas - lituano" lang="lt" hreflang="lt" data-title="Kampas" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></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 - lettone" lang="lv" hreflang="lv" data-title="Leņķis" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></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"><span>Мокшень</span></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) - malgascio" lang="mg" hreflang="mg" data-title="Zoro (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="malgascio" class="interlanguage-link-target"><span>Malagasy</span></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"><span>Олык марий</span></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 - macedone" lang="mk" hreflang="mk" data-title="Агол" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target"><span>Македонски</span></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"><span>മലയാളം</span></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) - mongolo" lang="mn" hreflang="mn" data-title="Өнцөг (геометр)" data-language-autonym="Монгол" data-language-local-name="mongolo" class="interlanguage-link-target"><span>Монгол</span></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"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sudut" title="Sudut - malese" lang="ms" hreflang="ms" data-title="Sudut" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" class="interlanguage-link-target"><span>Bahasa Melayu</span></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 - birmano" lang="my" hreflang="my" data-title="ထောင့်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></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"><span>Эрзянь</span></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) - mazandarani" lang="mzn" hreflang="mzn" data-title="سوک(هنّسه)" data-language-autonym="مازِرونی" data-language-local-name="mazandarani" class="interlanguage-link-target"><span>مازِرونی</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Winkel_(Geometrie)" title="Winkel (Geometrie) - basso tedesco" lang="nds" hreflang="nds" data-title="Winkel (Geometrie)" data-language-autonym="Plattdüütsch" data-language-local-name="basso tedesco" class="interlanguage-link-target"><span>Plattdüütsch</span></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 - nepalese" lang="ne" hreflang="ne" data-title="कोण" data-language-autonym="नेपाली" data-language-local-name="nepalese" class="interlanguage-link-target"><span>नेपाली</span></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"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hoek_(meetkunde)" title="Hoek (meetkunde) - olandese" lang="nl" hreflang="nl" data-title="Hoek (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vinkel" title="Vinkel - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Vinkel" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vinkel" title="Vinkel - norvegese bokmål" lang="nb" hreflang="nb" data-title="Vinkel" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></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"><span>ߒߞߏ</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Angle" title="Angle - occitano" lang="oc" hreflang="oc" data-title="Angle" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></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"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/K%C4%85t" title="Kąt - polacco" lang="pl" hreflang="pl" data-title="Kąt" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/%C3%80ngol" title="Àngol - piemontese" lang="pms" hreflang="pms" data-title="Àngol" data-language-autonym="Piemontèis" data-language-local-name="piemontese" class="interlanguage-link-target"><span>Piemontèis</span></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"><span>پنجابی</span></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"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%82ngulo" title="Ângulo - portoghese" lang="pt" hreflang="pt" data-title="Ângulo" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></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"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Unghi" title="Unghi - rumeno" lang="ro" hreflang="ro" data-title="Unghi" data-language-autonym="Română" data-language-local-name="rumeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D0%B3%D0%BE%D0%BB" title="Угол - russo" lang="ru" hreflang="ru" data-title="Угол" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></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 - sacha" lang="sah" hreflang="sah" data-title="Муннук" data-language-autonym="Саха тыла" data-language-local-name="sacha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/%C3%80nculu" title="Ànculu - siciliano" lang="scn" hreflang="scn" data-title="Ànculu" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ugao" title="Ugao - serbo-croato" lang="sh" hreflang="sh" data-title="Ugao" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></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 - singalese" lang="si" hreflang="si" data-title="කෝණය" data-language-autonym="සිංහල" data-language-local-name="singalese" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/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"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Uhol" title="Uhol - slovacco" lang="sk" hreflang="sk" data-title="Uhol" data-language-autonym="Slovenčina" data-language-local-name="slovacco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kot" title="Kot - sloveno" lang="sl" hreflang="sl" data-title="Kot" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></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"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Xagal" title="Xagal - somalo" lang="so" hreflang="so" data-title="Xagal" data-language-autonym="Soomaaliga" data-language-local-name="somalo" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/K%C3%ABndi" title="Këndi - albanese" lang="sq" hreflang="sq" data-title="Këndi" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target"><span>Shqip</span></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 - serbo" lang="sr" hreflang="sr" data-title="Угао" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></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"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Vinkel" title="Vinkel - svedese" lang="sv" hreflang="sv" data-title="Vinkel" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></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"><span>Kiswahili</span></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"><span>தமிழ்</span></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"><span>తెలుగు</span></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 - tagico" lang="tg" hreflang="tg" data-title="Кунҷ" data-language-autonym="Тоҷикӣ" data-language-local-name="tagico" class="interlanguage-link-target"><span>Тоҷикӣ</span></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 - thailandese" lang="th" hreflang="th" data-title="มุม" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></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"><span>Tagalog</span></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 - turco" lang="tr" hreflang="tr" data-title="Açı" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></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 - tataro" lang="tt" hreflang="tt" data-title="Почмак" data-language-autonym="Татарча / tatarça" data-language-local-name="tataro" class="interlanguage-link-target"><span>Татарча / tatarça</span></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 - ucraino" lang="uk" hreflang="uk" data-title="Кут" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></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"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Burchak" title="Burchak - uzbeco" lang="uz" hreflang="uz" data-title="Burchak" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/G%C3%B3c" title="Góc - vietnamita" lang="vi" hreflang="vi" data-title="Góc" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></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"><span>Winaray</span></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"><span>吴语</span></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 - mengrelio" lang="xmf" hreflang="xmf" data-title="კუნთხუ" data-language-autonym="მარგალური" data-language-local-name="mengrelio" class="interlanguage-link-target"><span>მარგალური</span></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"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%A7%92" title="角 - cinese" lang="zh" hreflang="zh" data-title="角" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></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 - cinese classico" lang="lzh" hreflang="lzh" data-title="角" data-language-autonym="文言" data-language-local-name="cinese classico" class="interlanguage-link-target"><span>文言</span></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 - min nan" lang="nan" hreflang="nan" data-title="Kak-tō͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></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) - cantonese" lang="yue" hreflang="yue" data-title="角 (幾何)" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target"><span>粵語</span></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"><span>IsiZulu</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11352#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Angolo" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussione:Angolo" rel="discussion" title="Vedi le discussioni 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r130658281">body:not(.skin-minerva) .mw-parser-output .hatnote.nota-disambigua{clear:both;margin-top:0;padding:.05em .5em}</style> <style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint nota-disambigua"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/18px-Nota_disambigua.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/27px-Nota_disambigua.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/36px-Nota_disambigua.svg.png 2x" data-file-width="200" data-file-height="200" /></span></span> <span class="hatnote-text"><a href="/wiki/Aiuto:Disambiguazione" title="Aiuto:Disambiguazione">Disambiguazione</a> – Se stai cercando altri significati, vedi <b><a href="/wiki/Angolo_(disambigua)" class="mw-disambig" title="Angolo (disambigua)">Angolo (disambigua)</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Angle_Symbol.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Angle_Symbol.svg/170px-Angle_Symbol.svg.png" decoding="async" width="170" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Angle_Symbol.svg/255px-Angle_Symbol.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Angle_Symbol.svg/340px-Angle_Symbol.svg.png 2x" data-file-width="198" data-file-height="200" /></a><figcaption>∠ è il simbolo dell'angolo</figcaption></figure> <p>Un <b>angolo</b> (dal <a href="/wiki/Lingua_latina" title="Lingua latina">latino</a> <i>angulus</i>, dal <a href="/wiki/Lingua_greca" title="Lingua greca">greco</a> ἀγκύλος (<i>ankýlos</i>), derivazione dalla radice <a href="/wiki/Lingue_indoeuropee" title="Lingue indoeuropee">indoeuropea</a> <i>ank</i>, piegare, curvare<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>), in <a href="/wiki/Matematica" title="Matematica">matematica</a>, indica ciascuna delle due porzioni di <a href="/wiki/Piano_(geometria)" title="Piano (geometria)">piano</a> comprese tra due <a href="/wiki/Semiretta" title="Semiretta">semirette</a> aventi la stessa origine. Si può definire anche <b>angolo piano</b> per distinguerlo dal concetto derivato di <a href="/wiki/Angolo_solido" title="Angolo solido">angolo solido</a>. Le semirette vengono dette lati dell'angolo, e la loro origine vertice dell'angolo. Il termine, così definito, riguarda nozioni di larghissimo uso, innanzitutto nella <a href="/wiki/Geometria" title="Geometria">geometria</a> e nella <a href="/wiki/Trigonometria" title="Trigonometria">trigonometria</a>. </p><p>A ogni angolo si associa un'ampiezza, la misura correlata alla posizione di una semiretta rispetto all'altra e pertanto alla conformazione della porzione di piano costituente l'angolo: essa si esprime in gradi <a href="/wiki/Grado_d%27arco" title="Grado d'arco">sessagesimali</a>, in gradi <a href="/wiki/Grado_sessadecimale" title="Grado sessadecimale">sessadecimali</a>, in <a href="/wiki/Grado_centesimale" title="Grado centesimale">gradi centesimali</a> o in <a href="/wiki/Radiante" title="Radiante">radianti</a>, sempre con valori <a href="/wiki/Numero_reale" title="Numero reale">reali</a>.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Associando all'angolo un verso si introducono le ampiezze degli angoli con segno, che consentono di definire <a href="/wiki/Funzione_trigonometrica" title="Funzione trigonometrica">funzioni trigonometriche</a> con argomenti reali anche negativi. Le ampiezze con segno forniscono contributi essenziali alle possibilità del <a href="/wiki/Calcolo_infinitesimale" title="Calcolo infinitesimale">calcolo infinitesimale</a> e alle applicazioni alla <a href="/wiki/Fisica_classica" title="Fisica classica">fisica classica</a> e alle conseguenti discipline quantitative. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Angolo_convesso_e_concavo">Angolo convesso e concavo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=1" title="Modifica la sezione Angolo convesso e concavo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=1" title="Edit section's source code: Angolo convesso e concavo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Angolo_convesso.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Angolo_convesso.png/240px-Angolo_convesso.png" decoding="async" width="240" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Angolo_convesso.png/360px-Angolo_convesso.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Angolo_convesso.png/480px-Angolo_convesso.png 2x" data-file-width="2370" data-file-height="1978" /></a><figcaption>Angolo convesso</figcaption></figure> <p>Si chiama angolo concavo l'angolo che contiene i prolungamenti delle semirette (lati) che lo formano. L'angolo convesso è la porzione di piano che non contiene i prolungamenti delle semirette che dividono il piano. Gli angoli convessi hanno ampiezza compresa tra 0 e 180 gradi sessagesimali, da 0 a 200 gradi centesimali, da 0 a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span> radianti; mentre l'ampiezza degli angoli concavi misura tra 180 e 360 gradi, da 200 a 400 gradi centesimali, da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" /></span> radianti. Le ampiezze sono sempre non negative. </p><p>Se le semirette sono diverse, ma appartengono alla stessa retta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f035e033d7d2c784a07e01448f7605945dfd435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.411ex; height:2.509ex;" alt="{\displaystyle R,}" /></span> ciascuno dei due semipiani definiti da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> muniti del vertice (che distingue le semirette) si dice <a href="/wiki/Angolo_piatto" title="Angolo piatto">angolo piatto</a>. </p><p>A parte il caso particolare dell'angolo piatto, il piano si tripartisce in tre insiemi: la <i>frontiera dell'angolo</i>, ossia l'insieme dei punti appartenenti alle due semirette <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ee86b741f542feb8f95f3c81fd53b043a25e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.283ex; height:2.509ex;" alt="{\displaystyle T,}" /></span> tra cui il vertice, e due <a href="/wiki/Spazio_connesso" title="Spazio connesso">insiemi connessi</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e1b324cf5b68f2729a8634ff76e396b634b75d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{2}}" /></span> e separati dai punti della frontiera. Di questi due insiemi, solo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}" /></span> è costituito da punti che appartengono a segmenti con un estremo su una semiretta e l'altro sull'altra; in altre parole solo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}" /></span> è un <a href="/wiki/Insieme_convesso" title="Insieme convesso">insieme convesso</a>. Il terzo insieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e1b324cf5b68f2729a8634ff76e396b634b75d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{2}}" /></span> non è convesso. Si definisce <i>angolo convesso</i> determinato da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> l'unione di questo insieme convesso e della frontiera, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}\cup S\cup T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∪</mo> <mi>S</mi> <mo>∪</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}\cup S\cup T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5938633669265c713aa7671e251c519e95d8d456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.328ex; height:2.509ex;" alt="{\displaystyle K_{1}\cup S\cup T}" /></span>. Si definisce <i>angolo concavo</i> determinato da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> l'unione del terzo insieme non convesso e della frontiera, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2}\cup S\cup T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∪</mo> <mi>S</mi> <mo>∪</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2}\cup S\cup T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174c4a3375d4f66fbf56c6b8007e32ae5affe96e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.328ex; height:2.509ex;" alt="{\displaystyle K_{2}\cup S\cup T}" /></span>. I due angoli definiti dalle due semirette si dicono <a href="/wiki/Angolo_esplementare" title="Angolo esplementare">angoli esplementari</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Angolo.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Angolo.gif/220px-Angolo.gif" decoding="async" width="220" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Angolo.gif/330px-Angolo.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Angolo.gif/440px-Angolo.gif 2x" data-file-width="555" data-file-height="510" /></a><figcaption>Angolo e triangolo ABC come suo sottoinsieme.</figcaption></figure> <p>Angoli convessi e concavi sono sottoinsiemi <a href="/wiki/Insieme_limitato" title="Insieme limitato">illimitati</a> del piano, quindi sono insiemi non misurabili attraverso la loro <a href="/wiki/Area" title="Area">area</a> che ha valore infinito. Spesso con angolo (convesso) si indica anche la parte di piano delimitata da due segmenti con un estremo in comune (vertice). Si può ricondurre questa definizione alla precedente prolungando i due segmenti dalla parte del loro estremo diverso dal vertice per ottenere le due semirette. Questa estensione della definizione rende lecito assegnare a ogni triangolo tre angoli (convessi) associati biunivocamente ai suoi tre vertici. </p><p>Tuttavia il triangolo, essendo un sottoinsieme chiuso e limitato del piano, ha area finita, infatti esso è l'intersezione degli angoli corrispondenti ai suoi tre vertici. </p> <div class="mw-heading mw-heading2"><h2 id="La_misurazione_dell'ampiezza_degli_angoli_convessi_e_concavi"><span id="La_misurazione_dell.27ampiezza_degli_angoli_convessi_e_concavi"></span>La misurazione dell'ampiezza degli angoli convessi e concavi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=2" title="Modifica la sezione La misurazione dell'ampiezza degli angoli convessi e concavi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=2" title="Edit section's source code: La misurazione dell'ampiezza degli angoli convessi e concavi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Considerazioni_preliminari">Considerazioni preliminari</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=3" title="Modifica la sezione Considerazioni preliminari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=3" title="Edit section's source code: Considerazioni preliminari"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>È naturale porsi il problema di "misurare un angolo": gli angoli possono servire per tante costruzioni e se a essi si associano misure numeriche ci si aspetta che per molte costruzioni possano essere utili calcoli numerici su queste misure. </p><p>Il problema della misura di un angolo non può essere risolto attraverso una misura della sua superficie che non è <a href="/wiki/Insieme_limitato" title="Insieme limitato">limitata</a> e che comunque non sarebbe significativa nemmeno nel caso di angoli sottesi da segmenti come nel caso del triangolo: si considerino per esempio triangoli simili. </p><p>Se si hanno due angoli convessi o concavi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> con lo stesso vertice e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> è sottoinsieme di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> (situazione che si determina solo se i lati di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> sono sottoinsiemi di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>) è ragionevole chiedere che la misura di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> sia maggiore della misura di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span>. </p><p>Dato un angolo convesso <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}" /></span> si dice <i>semiretta bisettrice</i> dell'angolo la semiretta avente il vertice di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> come estremo e i cui punti sono equidistanti dai lati di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>. Si può costruire facilmente la <a href="/wiki/Bisettrice" title="Bisettrice">bisettrice</a> con un compasso. La semiretta bisettrice di un angolo concavo si definisce come la semiretta avente come estremo il vertice dell'angolo allineata con la bisettrice del suo angolo (convesso) esplementare. </p><p>La semiretta bisettrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> di un angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> convesso o concavo e ciascuno dei suoi due lati determinano due angoli convessi. La riflessione rispetto alla retta contenente la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> scambia i due lati di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> e trasforma uno dei due angoli nell'altro. È quindi ragionevole attribuire ai due angoli determinati dalla bisettrice una misura che sia la metà della misura di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>. È altrettanto ragionevole considerare che le misure dei due angoli determinati dalla semiretta bisettrice siano la metà della misura dell'angolo di partenza. Il processo di dimezzamento di un angolo può essere ripetuto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span> volte con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span> grande a piacere. </p><p>Un angolo convesso si dice <i>angolo retto</i> se i suoi due lati sono ortogonali, cioè un <a href="/wiki/Angolo_retto" title="Angolo retto">angolo retto</a> è la metà di un angolo piatto. </p><p>Un angolo convesso contenuto in un angolo retto avente il suo stesso vertice si dice <i>angolo acuto</i>. Un angolo convesso contenente un angolo retto avente lo stesso vertice si dice <i>angolo ottuso</i>. </p><p>Due angoli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> che hanno in comune solo una semiretta e non hanno alcun punto interno in comune si dicono <i>angoli consecutivi</i>. Se due angoli consecutivi hanno le semirette non in comune opposte (cioè la loro unione è una retta) allora si dicono <i>angoli adiacenti</i>. Per quanto riguarda gli angoli consecutivi, se questi sono angoli convessi la loro unione è un angolo che potrebbe essere convesso o concavo: si tratta dell'angolo definito dalle due semirette che sono i lati di uno solo dei due angoli. A questo angolo unione è ragionevole assegnare come misura la somma delle misure degli angoli consecutivi. L'angolo unione si dice "somma" dei due angoli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span>. </p><p>In base alle considerazioni precedenti è lecito attribuire agli angoli misure costituite da numeri reali. </p><p>Due angoli trasformabili l'uno nell'altro mediante <a href="/wiki/Isometria_del_piano" title="Isometria del piano">isometrie</a> si dicono congruenti. Evidentemente una misura degli angoli invariante per le isometrie costituisce uno strumento con molti vantaggi: in particolare consente di individuare le classi di congruenza degli angoli. Quindi si chiede una misura degli angoli a valori reali e invariante per congruenza. </p> <div class="mw-heading mw-heading3"><h3 id="Dalla_misura_dell'angolo_alla_misura_dell'ampiezza_dell'angolo"><span id="Dalla_misura_dell.27angolo_alla_misura_dell.27ampiezza_dell.27angolo"></span>Dalla misura dell'angolo alla misura dell'ampiezza dell'angolo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=4" title="Modifica la sezione Dalla misura dell'angolo alla misura dell'ampiezza dell'angolo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=4" title="Edit section's source code: Dalla misura dell'angolo alla misura dell'ampiezza dell'angolo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se l'angolo è definito come la porzione del piano tra due semirette, la sua unità di misura dovrebbe essere una lunghezza al quadrato, ma questa misura non ha né significato né utilità pratica. Si è quindi pensato di considerare non la misura dell'angolo in sé, ma quella dell'ampiezza <i>del movimento</i> che porta una delle semirette a sovrapporsi all'altra. </p><p>Come giungere a determinare l'ampiezza di un angolo ha certamente chiesto maggiori sforzi all'intelletto umano di quanti ne abbia richiesti la misurazione di lunghezze e superfici. <a href="/wiki/Misurazione" title="Misurazione">Misurare</a> significa esprimere una <a href="/wiki/Grandezza_fisica" title="Grandezza fisica">grandezza</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> in rapporto a un'altra grandezza data, a essa omogenea, che funge da <a href="/wiki/Unit%C3%A0_di_misura" title="Unità di misura">unità di misura</a>. Se questo processo sorge abbastanza spontaneo per le grandezze spaziali, per cui basta ripetere un segmento o affiancare un quadrato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></span> per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> volte fino all'esaurimento della lunghezza o della superficie (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=nU}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>n</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=nU}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4eeb839aa38fa200272d606efcefb628104812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.019ex; height:2.176ex;" alt="{\displaystyle A=nU}" /></span>), lo stesso diventa meno intuitivo per le grandezze angolari, dove pure la stessa elaborazione mentale di un'unità di misura adatta richiede un maggior grado di astrazione. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Angoli_retti.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Angoli_retti.png/300px-Angoli_retti.png" decoding="async" width="300" height="288" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/48/Angoli_retti.png 1.5x" data-file-width="315" data-file-height="302" /></a><figcaption></figcaption></figure> <p>Si prendano in considerazione i quattro angoli di ampiezza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> della figura. Volendoli quantificare con l'area delimitata dai lati in verde, prolungando i lati a infinito nel caso <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> si ottiene un'area infinita e nei restanti casi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075d661417b8ca5a991a2a7bd4991cc1ab856d9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.411ex; height:2.509ex;" alt="{\displaystyle B,}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab9cb9bd7fcb5f2a543b128c1b876019b158c0fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.571ex; height:2.509ex;" alt="{\displaystyle D,}" /></span> considerando solo le superfici entro le linee tratteggiate, tre aree determinate e quindi misurabili, ma visibilmente diverse fra loro, seppur originate dal medesimo angolo. Si presuma inoltre di dividere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> esattamente in due angoli uguali, in modo che sia esprimibile in rapporto a questi ultimi, come <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =2\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>2</mn> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =2\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e0813077f6c8e68cf1cc3036e2dd96aefc4c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.081ex; height:2.509ex;" alt="{\displaystyle \alpha =2\beta }" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span>. Per quanto detto sopra, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> può quindi essere considerato un'<i>unità di misura</i> e, se ora se ne considera l'area, l'uguaglianza sarà soddisfatta soltanto dai casi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab9cb9bd7fcb5f2a543b128c1b876019b158c0fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.571ex; height:2.509ex;" alt="{\displaystyle D,}" /></span> ma non da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075d661417b8ca5a991a2a7bd4991cc1ab856d9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.411ex; height:2.509ex;" alt="{\displaystyle B,}" /></span> dove i due triangoli hanno aree diverse, pur trattandosi di due angoli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> perfettamente sovrapponibili. Ne discende che l'angolo non può essere misurato idoneamente in termini di <a href="/wiki/Area" title="Area">area</a>. </p><p>Si immagini quindi una semiretta che partendo dalla posizione verticale giri attorno al proprio estremo fino a diventare orizzontale; la semiretta ha compiuto un angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> e nel suo movimento ha coperto la superficie compresa tra le due semirette. Sovrapponendo idealmente le immagini <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" /></span> si nota che, come in un <a href="/wiki/Compasso_(strumento)" title="Compasso (strumento)">compasso</a>, allontanandosi dal centro di rotazione ogni punto traccia sul piano un <a href="/wiki/Arco_(geometria)" title="Arco (geometria)">arco</a> più lungo, pur mantenendo immutato il rapporto fra lunghezza di quest'ultimo e il raggio. Inoltre se la semiretta compisse soltanto l'angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> la lunghezza degli archi prodotti sarebbe invariabilmente la metà della lunghezza degli archi loro omologhi in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>. </p><p>Si consideri ora una rotazione completa che riporta la semiretta alla posizione di partenza, cioè un angolo di massima ampiezza. In questo caso la semiretta copre l'intera superficie del piano tracciando infinite circonferenze; prendendo una qualunque di queste e segmentandola in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> parti uguali, si possono individuare per ogni arco altrettante porzioni di piano equipollenti, in pratica una generica unità di misura per l'angolo. Dunque soltanto capendo che la misurazione dell'angolo non può essere avvenire quantificando un'area si comprende che bisogna astrarre il concetto di angolo come parte del piano e considerarlo invece cinematicamente come una porzione di superficie coperta da una semiretta in rotazione sul proprio estremo. Solo in questo modo è possibile misurarlo. </p><p>Sebbene questa nozione non sia immediata, deve comunque trattarsi di una conquista concettuale antica, se il sistema per la misurazione degli angoli comunemente più utilizzato ancora oggi, il <a href="/wiki/Sistema_sessagesimale" class="mw-redirect" title="Sistema sessagesimale">sistema sessagesimale</a>, è giunto sino noi dall'antica <a href="/wiki/Civilt%C3%A0_babilonese" class="mw-redirect" title="Civiltà babilonese">civiltà babilonese</a> invariato nei secoli.<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Sistemi_di_misurazione_dell'ampiezza_dell'angolo"><span id="Sistemi_di_misurazione_dell.27ampiezza_dell.27angolo"></span>Sistemi di misurazione dell'ampiezza dell'angolo</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=5" title="Modifica la sezione Sistemi di misurazione dell'ampiezza dell'angolo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=5" title="Edit section's source code: Sistemi di misurazione dell'ampiezza dell'angolo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel sistema sessagesimale l'angolo completo o <i>angolo giro</i> è suddiviso in 360 spicchi, equivalenti all'unità di misura convenzionale denominata <a href="/wiki/Grado_d%27arco" title="Grado d'arco">grado sessagesimale</a>, indicata col simbolo <a href="/wiki/Grado_(simbolo)" title="Grado (simbolo)">°</a>. La divisione in 360 parti dell'angolo giro è riconducibile all'uso <a href="/wiki/Astronomia" title="Astronomia">astronomico</a> babilonese<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>: il Sole compie un giro completo sulla <a href="/wiki/Sfera_celeste" title="Sfera celeste">volta celeste</a> nell'arco di un anno, a quel tempo stimato di circa 360 giorni, quindi un grado corrisponde pressappoco allo spostamento del Sole sull'<a href="/wiki/Eclittica" title="Eclittica">eclittica</a> in un giorno. </p><p>Il nome "grado sessagesimale" deriva dal fatto che le sottounità del grado, il <a href="/wiki/Primo_(geometria)" title="Primo (geometria)">minuto</a> e il <a href="/wiki/Secondo_(geometria)" title="Secondo (geometria)">secondo</a>, sono divise in sessantesimi; perciò, come nell'orologio, ogni grado è diviso in 60 minuti primi indicati col simbolo <i>'</i> e chiamati semplicemente minuti, e ogni minuto è diviso in 60 minuti secondi indicati col simbolo <i>''</i> e chiamati semplicemente secondi. Ulteriori suddivisioni del secondo seguono invece il comune sistema decimale. Questa suddivisione deriva dal fatto che nell'antica Babilonia era in auge un <a href="/wiki/Matematica_babilonese" title="Matematica babilonese">sistema numerico</a> su base <a href="/wiki/60_(numero)" title="60 (numero)">sessagesimale</a>, giunto sino a noi quale retaggio storico nell'orologio e nei <a href="/wiki/Goniometro" title="Goniometro">goniometri</a>. </p><p>L'ampiezza di un angolo è espresso in una forma tipo: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 57^{\circ }\;17'\;44{,}8''.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>57</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="thickmathspace"></mspace> <msup> <mn>17</mn> <mo>′</mo> </msup> <mspace width="thickmathspace"></mspace> <mn>44</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <msup> <mn>8</mn> <mo>″</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 57^{\circ }\;17'\;44{,}8''.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4071cd12698a0adebb7931ece968256b439a5b14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.597ex; height:2.843ex;" alt="{\displaystyle 57^{\circ }\;17'\;44{,}8''.}" /></span> </p><p>Nel tempo sono poi stati adottati altri <a href="/wiki/Sistemi_di_misurazione" title="Sistemi di misurazione">sistemi di misurazione</a> nel tentativo di rendere più agevole la misura. Alla fine del Settecento si tentò di razionalizzare il sistema sessagesimale: venne proposto un <a href="/w/index.php?title=Sistema_centesimale&action=edit&redlink=1" class="new" title="Sistema centesimale (la pagina non esiste)">sistema centesimale</a>, basato appunto sul <a href="/wiki/Grado_centesimale" title="Grado centesimale">grado centesimale</a>, una centesima parte nell'angolo retto, sostituendo il 90 con il tondo 100. Trovò utilizzo pratico soltanto attorno al 1850 quando <a href="/wiki/Ignazio_Porro" title="Ignazio Porro">Ignazio Porro</a><sup id="cite_ref-Ignazio_porro_4-0" class="reference"><a href="#cite_note-Ignazio_porro-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> lo usò per costruire i suoi primi strumenti a divisione centesimale. Con questo sistema l'angolo giro viene diviso in 400 spicchi uguali con sottomultipli a frazioni decimali. È quindi una unita di misura convenzionale non motivata da ragioni matematiche. </p><p>Tramite l'<a href="/wiki/Analisi_infinitesimale" class="mw-redirect" title="Analisi infinitesimale">analisi infinitesimale</a> si giunse ad una nuova unità, per certi aspetti più "motivata" o "naturale": il <i><a href="/wiki/Radiante" title="Radiante">radiante</a></i>, definito come il rapporto tra la lunghezza di un <a href="/wiki/Arco_(geometria)" title="Arco (geometria)">arco</a> di <a href="/wiki/Circonferenza" title="Circonferenza">circonferenza</a> e il raggio della circonferenza stessa. Questo rapporto non dipende dal raggio, ma solo dall'angolo compreso. In questo modo l'angolo giro misura 2<a href="/wiki/Pi_greco" title="Pi greco">π</a>, cioè il rapporto tra la lunghezza della circonferenza e il suo raggio. </p><p>Riepilogando, per misurare l'ampiezza dell'angolo i sistemi di misura più attestati sono<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>: </p> <ul><li>il <i>sistema centesimale</i>, con unità di misura il <a href="/wiki/Grado_centesimale" title="Grado centesimale">grado centesimale</a>;</li> <li>il <i>sistema sessagesimale</i>, con unità di misura il <a href="/wiki/Grado_d%27arco" title="Grado d'arco">grado sessagesimale</a>;</li> <li>il <i>sistema sessadecimale</i>, con unità di misura il <a href="/wiki/Grado_sessadecimale" title="Grado sessadecimale">grado sessadecimale</a>. È una variante del precedente con divisione dell'<a href="/wiki/Angolo_giro" title="Angolo giro">angolo giro</a> in 360 parti in cui i sottomultipli dei gradi sono espressi in forma decimale;</li> <li>il <i>sistema radiante</i>, o <i>sistema matematico</i>, con unità di misura il <a href="/wiki/Radiante" title="Radiante">radiante</a>.</li> <li>in ambito militare si usa anche il <a href="/wiki/Millesimo_di_radiante" title="Millesimo di radiante">millesimo di radiante</a>, detto comunemente "millesimo", che viene impiegato per determinare gli scarti e relative correzioni nei tiri con l'artiglieria. Su una circonferenza avente raggio un km equivale a una corda lunga un metro. Per esempio, per correggere un colpo caduto 100 metri a destra di un bersaglio posto alla distanza di 10 km bisognerà apportare una correzione di 10°° (millesimi) rosso. La <a href="/wiki/Scala_graduata" title="Scala graduata">scala graduata</a> che si osserva all'interno di alcuni binocoli è espressa in millesimi di radianti, il colore rosso significa rotazione verso sinistra mentre il colore verde significa rotazione verso destra.</li></ul> <p>Il primo viene più che altro usato in ambito strettamente <a href="/wiki/Topografia" title="Topografia">topografico</a>, il secondo e il terzo per consuetudine e il quarto per una maggiore semplicità dei calcoli nelle formule matematiche e il quinto sono in un campo ristretto. La relazione che lega il sistema radiante e il sistema sessagesimale e permette il passaggio da uno all'altro è </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {180^{\circ }}{\alpha }}={\frac {\pi }{x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mi>α</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {180^{\circ }}{\alpha }}={\frac {\pi }{x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1264fc6f84bed216e15b532641c7a64dae0ea46b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.291ex; height:5.343ex;" alt="{\displaystyle {\frac {180^{\circ }}{\alpha }}={\frac {\pi }{x}},}" /></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> è la misura dell'ampiezza dell'angolo espresso in gradi e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> è la misura espressa in radianti. Più in generale, il rapporto tra un angolo e il rispettivo angolo giro in una unità di misura è costante rispetto ad un altro sistema. </p> <div class="mw-heading mw-heading4"><h4 id="Conversioni_angolari">Conversioni angolari</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=6" title="Modifica la sezione Conversioni angolari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=6" title="Edit section's source code: Conversioni angolari"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Indicando l'ampiezza di un angolo con: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }\;p^{\prime }\;s^{\prime \prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="thickmathspace"></mspace> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mspace width="thickmathspace"></mspace> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }\;p^{\prime }\;s^{\prime \prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0173c28f4a4bd6797cb73ddbef6ba68934444fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.914ex; height:2.843ex;" alt="{\displaystyle \alpha ^{\circ }\;p^{\prime }\;s^{\prime \prime }}" /></span> nel <a href="/wiki/Grado_d%27arco" title="Grado d'arco">sistema sessagesimale</a>, dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,p,s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,p,s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a05fc04c3e1987f27fc91d489bc720ca8f587dc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.815ex; height:2.009ex;" alt="{\displaystyle \alpha ,p,s}" /></span> sono rispettivamente i gradi, primi e secondi d'arco (numeri interi)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58c1a8a9b0e73db4ff404c5b1421a7e98885a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.343ex;" alt="{\displaystyle \alpha ^{\circ }}" /></span> nel <a href="/wiki/Grado_sessadecimale" title="Grado sessadecimale">sistema sessadecimale</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{gon}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{gon}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad404d33de6a75a9badafa85cb5651d9ec955f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.293ex; height:2.343ex;" alt="{\displaystyle \alpha ^{gon}}" /></span> nel <a href="/wiki/Grado_centesimale" title="Grado centesimale">sistema centesimale</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{rad}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{rad}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc9c964bba00506c8e6a292efd2141a67621ba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.191ex; height:2.676ex;" alt="{\displaystyle \alpha ^{rad}}" /></span> nel <a href="/wiki/Radiante" title="Radiante">sistema matematico</a>,</dd></dl> <p>indicando con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Int[\;]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>n</mi> <mi>t</mi> <mo stretchy="false">[</mo> <mspace width="thickmathspace"></mspace> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Int[\;]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f270d340d149cf4f9a14573f6ac8b700c7921b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.345ex; height:2.843ex;" alt="{\displaystyle Int[\;]}" /></span> la <a href="/wiki/Parte_intera" title="Parte intera">parte intera</a> di un numero reale e ricordando che vale la proporzione generale </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha ^{rad}}{\pi }}={\frac {\alpha ^{\circ }}{180}}={\frac {\alpha ^{gon}}{200}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>d</mi> </mrow> </msup> <mi>π</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mn>180</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> <mn>200</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha ^{rad}}{\pi }}={\frac {\alpha ^{\circ }}{180}}={\frac {\alpha ^{gon}}{200}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72d2d47550c125cbffae118148e2754cbe78e8cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.676ex; height:5.676ex;" alt="{\displaystyle {\frac {\alpha ^{rad}}{\pi }}={\frac {\alpha ^{\circ }}{180}}={\frac {\alpha ^{gon}}{200}}}" /></span></dd></dl> <p>valgono le seguenti formule di conversione da un sistema di misura all'altro </p> <table class="wikitable"> <tbody><tr> <th>Conversione da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mspace width="1em"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow \quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cdbd70038662afe04ac22e9baabbe753ee1285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.485ex; height:2.509ex;" alt="{\displaystyle \downarrow \quad }" /></span> a<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \longrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⟶</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \longrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ffb6a294b21bebe64570c4088d77a884dec95ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.806ex; height:1.843ex;" alt="{\displaystyle \longrightarrow }" /></span></th> <th>Sessagesimale</th> <th>Sessadecimale</th> <th>Centesimale</th> <th>Matematico </th></tr> <tr> <td>Sessagesimale</td> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }\;=\;\alpha +{\frac {p}{60}}+{\frac {s}{3600}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="thickmathspace"></mspace> <mo>=</mo> <mspace width="thickmathspace"></mspace> <mi>α</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>60</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mn>3600</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }\;=\;\alpha +{\frac {p}{60}}+{\frac {s}{3600}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08864f5a5b1050f4e8ee5090d84d0429ee93cd65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.746ex; height:4.843ex;" alt="{\displaystyle \alpha ^{\circ }\;=\;\alpha +{\frac {p}{60}}+{\frac {s}{3600}}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{gon}={\frac {10}{9}}\alpha ^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>9</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{gon}={\frac {10}{9}}\alpha ^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dcedb143bc68e417666a8f55f77d5a36502c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.094ex; height:5.176ex;" alt="{\displaystyle \alpha ^{gon}={\frac {10}{9}}\alpha ^{\circ }}" /></span> dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58c1a8a9b0e73db4ff404c5b1421a7e98885a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.343ex;" alt="{\displaystyle \alpha ^{\circ }}" /></span> è calcolato con la formula precedente </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{rad}={\frac {\pi }{180}}\alpha ^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>d</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{rad}={\frac {\pi }{180}}\alpha ^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c49e28988df7016875b6f9b8e146d88cda06f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.155ex; height:4.676ex;" alt="{\displaystyle \alpha ^{rad}={\frac {\pi }{180}}\alpha ^{\circ }}" /></span> dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58c1a8a9b0e73db4ff404c5b1421a7e98885a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.343ex;" alt="{\displaystyle \alpha ^{\circ }}" /></span> è calcolato con la formula precedente </td></tr> <tr> <td>Sessadecimale</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =Int[\alpha ^{\circ }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>I</mi> <mi>n</mi> <mi>t</mi> <mo stretchy="false">[</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =Int[\alpha ^{\circ }]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abef4f85232d5f4206c193fa6ffa4a3564830493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.828ex; height:2.843ex;" alt="{\displaystyle \alpha =Int[\alpha ^{\circ }]}" /></span><br /> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=Int[(\alpha ^{\circ }-\alpha )\cdot 60]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>I</mi> <mi>n</mi> <mi>t</mi> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>60</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=Int[(\alpha ^{\circ }-\alpha )\cdot 60]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e751c169c8a18b4f6e159a49af9856605c3f07a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:21.74ex; height:2.843ex;" alt="{\displaystyle p=Int[(\alpha ^{\circ }-\alpha )\cdot 60]}" /></span><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=(((\alpha ^{\circ }-\alpha )\cdot 60)-p)\cdot 60}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>60</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=(((\alpha ^{\circ }-\alpha )\cdot 60)-p)\cdot 60}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da96eee255d2d3a15298bef0e878a66d9849872f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.505ex; height:2.843ex;" alt="{\displaystyle s=(((\alpha ^{\circ }-\alpha )\cdot 60)-p)\cdot 60}" /></span> </p> </td> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{gon}={\frac {10}{9}}\alpha ^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>9</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{gon}={\frac {10}{9}}\alpha ^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dcedb143bc68e417666a8f55f77d5a36502c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.094ex; height:5.176ex;" alt="{\displaystyle \alpha ^{gon}={\frac {10}{9}}\alpha ^{\circ }}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{rad}={\frac {\pi }{180}}\alpha ^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>d</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>180</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{rad}={\frac {\pi }{180}}\alpha ^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c49e28988df7016875b6f9b8e146d88cda06f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.155ex; height:4.676ex;" alt="{\displaystyle \alpha ^{rad}={\frac {\pi }{180}}\alpha ^{\circ }}" /></span> </td></tr> <tr> <td>Centesimale</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }={\frac {9}{10}}\alpha ^{gon}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>10</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }={\frac {9}{10}}\alpha ^{gon}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64cf9018846fd8f43328853e4b68c58b06d0789c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.094ex; height:5.176ex;" alt="{\displaystyle \alpha ^{\circ }={\frac {9}{10}}\alpha ^{gon}}" /></span><br /> <p>quindi si applicano le formule precedenti per la conversione da sessadecimale a sessagesimale </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }={\frac {9}{10}}\alpha ^{gon}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>10</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }={\frac {9}{10}}\alpha ^{gon}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64cf9018846fd8f43328853e4b68c58b06d0789c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.094ex; height:5.176ex;" alt="{\displaystyle \alpha ^{\circ }={\frac {9}{10}}\alpha ^{gon}}" /></span> </td> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{rad}={\frac {\pi }{200}}\alpha ^{gon}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>d</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>200</mn> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{rad}={\frac {\pi }{200}}\alpha ^{gon}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69ede33163903fb06ba8ac48b01138753aca6d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.905ex; height:4.676ex;" alt="{\displaystyle \alpha ^{rad}={\frac {\pi }{200}}\alpha ^{gon}}" /></span> </td></tr> <tr> <td>Matematico</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }={\frac {180}{\pi }}\alpha ^{rad}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>180</mn> <mi>π</mi> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }={\frac {180}{\pi }}\alpha ^{rad}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e821e45ec89693bfaf1d765c8c905594514e26be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.155ex; height:5.176ex;" alt="{\displaystyle \alpha ^{\circ }={\frac {180}{\pi }}\alpha ^{rad}}" /></span><br /> <p>quindi si applicano le formule precedenti per la conversione da sessadecimale a sessagesimale </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{\circ }={\frac {180}{\pi }}\alpha ^{rad}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>180</mn> <mi>π</mi> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{\circ }={\frac {180}{\pi }}\alpha ^{rad}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e821e45ec89693bfaf1d765c8c905594514e26be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.155ex; height:5.176ex;" alt="{\displaystyle \alpha ^{\circ }={\frac {180}{\pi }}\alpha ^{rad}}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{gon}={\frac {200}{\pi }}\alpha ^{rad}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>o</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>200</mn> <mi>π</mi> </mfrac> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{gon}={\frac {200}{\pi }}\alpha ^{rad}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7365b6624904b87fbe48e5abce10f7eb1b347c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.905ex; height:5.176ex;" alt="{\displaystyle \alpha ^{gon}={\frac {200}{\pi }}\alpha ^{rad}}" /></span> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Ampiezze_di_angoli_particolari">Ampiezze di angoli particolari</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=7" title="Modifica la sezione Ampiezze di angoli particolari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=7" title="Edit section's source code: Ampiezze di angoli particolari"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988" /> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Angolo_acuto" title="Angolo acuto">Angolo acuto</a></b>, <b><a href="/wiki/Angolo_ottuso" title="Angolo ottuso">Angolo ottuso</a></b>, <b><a href="/wiki/Angolo_retto" title="Angolo retto">Angolo retto</a></b>, <b><a href="/wiki/Angolo_piatto" title="Angolo piatto">Angolo piatto</a></b> e <b><a href="/wiki/Angolo_giro" title="Angolo giro">Angolo giro</a></b>.</span></div> </div> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Angolo_retto_piano_giro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Angolo_retto_piano_giro.svg/377px-Angolo_retto_piano_giro.svg.png" decoding="async" width="377" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Angolo_retto_piano_giro.svg/566px-Angolo_retto_piano_giro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Angolo_retto_piano_giro.svg/754px-Angolo_retto_piano_giro.svg.png 2x" data-file-width="377" data-file-height="220" /></a><figcaption></figcaption></figure> <ul><li>Un <i>angolo acuto</i> ha ampiezza inferiore a quella di un angolo retto, ossia</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }<\alpha <90^{\circ }(\pi /2).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>α</mi> <mo><</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }<\alpha <90^{\circ }(\pi /2).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f95041e2ee90052b12b96ae01d38e67fc7fcacf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.393ex; height:2.843ex;" alt="{\displaystyle 0^{\circ }<\alpha <90^{\circ }(\pi /2).}" /></span></dd></dl> <ul><li>Un <i>angolo retto</i> ha l'ampiezza uguale a un quarto dell'ampiezza di un angolo giro, ossia</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =90^{\circ }(\pi /2).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =90^{\circ }(\pi /2).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d71c217189426f9b77f1828fe45336fc77b3fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.078ex; height:2.843ex;" alt="{\displaystyle \alpha =90^{\circ }(\pi /2).}" /></span></dd></dl> <ul><li>Un <i>angolo ottuso</i> ha l'ampiezza compresa fra quelle di un angolo retto e di un angolo piatto, ossia</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }(\pi /2)<\alpha <180^{\circ }(\pi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo><</mo> <mi>α</mi> <mo><</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }(\pi /2)<\alpha <180^{\circ }(\pi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81cf3ad6e0311c3fa2ca549628ddb314a0d6d7fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.86ex; height:2.843ex;" alt="{\displaystyle 90^{\circ }(\pi /2)<\alpha <180^{\circ }(\pi ).}" /></span></dd></dl> <ul><li>Un <i>angolo piatto</i> ha ampiezza pari a metà di quella di un angolo giro, ossia</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =180^{\circ }(\pi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =180^{\circ }(\pi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecbf463abff375220cc3f6783a9341dc3ecf61f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.916ex; height:2.843ex;" alt="{\displaystyle \alpha =180^{\circ }(\pi ).}" /></span></dd></dl> <ul><li>Un <i>angolo giro</i> ha ampiezza uguale a</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =360^{\circ }(2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =360^{\circ }(2\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e0ff70b2333bb2b50e64eaa0c936564d8891b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.431ex; height:2.843ex;" alt="{\displaystyle \alpha =360^{\circ }(2\pi )}" /></span></dd> <dd>e corrisponde a una rotazione completa di una semiretta intorno al suo estremo.</dd></dl> <ul><li>Un <i>angolo concavo</i> ha ampiezza maggiore di quella di un angolo piatto, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >180^{\circ }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >180^{\circ }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f097d59fcb0a8f66e7da817f68ecf5be01a3ab2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.775ex; height:2.343ex;" alt="{\displaystyle \alpha >180^{\circ }.}" /></span></li> <li>Un <i>angolo convesso</i> ha ampiezza minore di quella di un angolo piatto, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <180^{\circ }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <180^{\circ }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84564f4129d5594be1b11961388a081bfe386736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.775ex; height:2.343ex;" alt="{\displaystyle \alpha <180^{\circ }.}" /></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Angoli_complementari">Angoli complementari</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=8" title="Modifica la sezione Angoli complementari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=8" title="Edit section's source code: Angoli complementari"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nella nomenclatura degli angoli di ampiezza compresa tra 0 e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 360^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 360^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb451712d2b1f32ec132e9cf56eca04a64cad60e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 360^{\circ }}" /></span> si è soliti usare aggettivi particolari per gli angoli associati a un angolo dato in quanto suoi "angoli di complemento" rispetto agli angoli fondamentali retto, piatto e giro. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Angoli_di_completamento.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Angoli_di_completamento.png/300px-Angoli_di_completamento.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/1e/Angoli_di_completamento.png 1.5x" data-file-width="301" data-file-height="241" /></a><figcaption></figcaption></figure> <p>Si dice <a href="/wiki/Angolo_complementare" title="Angolo complementare">complementare</a> di un angolo di ampiezza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> ogni angolo avente come ampiezza la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> "mancante" per ottenere un angolo retto, cioè tale che sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =90^{\circ }-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =90^{\circ }-\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067b4a8514874a94c24ce5112f26ed8046afa7b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.138ex; height:2.676ex;" alt="{\displaystyle \beta =90^{\circ }-\alpha }" /></span>. Da questa definizione segue che due angoli complementari devono essere entrambi acuti e che ha senso attribuire un complementare solo a un <a href="/wiki/Angolo_acuto" title="Angolo acuto">angolo acuto</a>. </p><p>Si dice <a href="/wiki/Angolo_supplementare" title="Angolo supplementare">supplementare</a> di un angolo di ampiezza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> ogni angolo avente come ampiezza la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> "mancante" per ottenere un angolo piatto, cioè tale che sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =180^{\circ }-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =180^{\circ }-\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b162a1dd4841ec4ab602aa668ea1d94bcf165a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.3ex; height:2.676ex;" alt="{\displaystyle \beta =180^{\circ }-\alpha }" /></span>. Da questa definizione segue che ogni supplementare di un angolo acuto è un <a href="/wiki/Angolo_ottuso" title="Angolo ottuso">angolo ottuso</a> e viceversa, mentre ogni supplementare di un angolo retto è anch'esso un angolo retto. Quando due angoli supplementari sono anche <i>consecutivi</i>, cioè hanno in comune solo una semiretta, vengono detti anche angoli <i>adiacenti</i>. </p><p>Si dice <a href="/wiki/Angolo_esplementare" title="Angolo esplementare">esplementare</a> di un angolo di ampiezza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> ogni angolo avente come ampiezza la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> "mancante" per ottenere un angolo giro, cioè tale che sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =360^{\circ }-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =360^{\circ }-\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0449fbc3d5535ad8ef8b0bd355303fa76063bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.3ex; height:2.676ex;" alt="{\displaystyle \beta =360^{\circ }-\alpha }" /></span>. Ne segue che ogni esplementare di un angolo concavo è un angolo convesso e viceversa, mentre ogni esplementare di un angolo piatto è anch'esso piatto. </p> <div class="mw-heading mw-heading3"><h3 id="Angoli_opposti_al_vertice">Angoli opposti al vertice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=9" title="Modifica la sezione Angoli opposti al vertice" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=9" title="Edit section's source code: Angoli opposti al vertice"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File"><a href="/wiki/File:Angoli_opposti_ai_vertici.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/93/Angoli_opposti_ai_vertici.png" decoding="async" width="163" height="287" class="mw-file-element" data-file-width="163" data-file-height="287" /></a><figcaption></figcaption></figure> <p>Due <a href="/wiki/Retta" title="Retta">rette</a> che si intersecano dividono il piano in quattro angoli; considerato uno qualsiasi di questi angoli: due degli altri gli sono adiacenti mentre il terzo, con cui condivide solo il vertice, è detto <i>angolo opposto al vertice</i>. Due angoli sono tra loro opposti al vertice se i prolungamenti dei lati di uno risultano essere i lati dell'altro. </p> <style data-mw-deduplicate="TemplateStyles:r144108411">.mw-parser-output .itwiki-template-approfondimento{border:1px solid var(--border-color-subtle,#c8ccd1);background-color:var(--background-color-interactive-subtle,#f8f9fa);padding:2px;box-sizing:border-box}@media all and (max-width:720px){.mw-parser-output .itwiki-template-approfondimento{width:100%!important}}@media all and (min-width:720px){.mw-parser-output .itwiki-template-approfondimento-sinistra{clear:left;float:left;margin-right:10px;margin-left:0}.mw-parser-output .itwiki-template-approfondimento-centro{margin-left:auto;margin-right:auto}.mw-parser-output .itwiki-template-approfondimento-destra{clear:right;float:right;margin-left:10px}}.mw-parser-output .itwiki-template-approfondimento-intestazione{background-color:#C3D0DF;padding:2px;text-align:center}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-approfondimento-intestazione{background-color:var(--background-color-neutral)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-approfondimento-intestazione{background-color:var(--background-color-neutral)}}</style><div role="complementary" class="itwiki-template-approfondimento itwiki-template-approfondimento-destra" style="width:50%;" id="Teorema_degli_angoli_opposti_al_vertice"> <div class="itwiki-template-approfondimento-intestazione"><b><a href="/wiki/Teorema_degli_angoli_opposti_al_vertice" title="Teorema degli angoli opposti al vertice">Teorema degli angoli opposti al vertice</a></b></div> <div style="margin: 0.4em 0; font-size:95%"> <p>Due angoli opposti al vertice sono sempre congruenti. </p><p><b>Dimostrazione</b> </p><p>Per definizione, due angoli adiacenti equivalgono a un angolo piatto, per cui valgono le seguenti uguaglianze </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta =180^{\circ }\qquad \beta +\gamma =180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="2em"></mspace> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta =180^{\circ }\qquad \beta +\gamma =180^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcc4cb99b97498359b22fb884a298b24f339cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.02ex; height:2.843ex;" alt="{\displaystyle \alpha +\beta =180^{\circ }\qquad \beta +\gamma =180^{\circ }}" /></span></dd></dl> <p>da cui </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta =\beta +\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>=</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta =\beta +\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022609f60ac5ed396ab99c22ae19a6aa7f8f4613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.193ex; height:2.676ex;" alt="{\displaystyle \alpha +\beta =\beta +\gamma }" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e181d8bf69abe745a52c3f2732edc1234c3e1319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle \alpha =\gamma }" /></span> <a href="/wiki/Come_volevasi_dimostrare" title="Come volevasi dimostrare">cvd</a>.</dd></dl> </div> </div> <p>Sono adiacenti gli angoli delle coppie <span class="nowrap">(α, β)</span>, <span class="nowrap">(β, γ)</span>, <span class="nowrap">(γ, δ)</span> e <span class="nowrap">(α, δ)</span>. </p><p>Sono invece opposti al vertice gli angoli delle coppie <span class="nowrap">(α, γ)</span> e <span class="nowrap">(β, δ)</span>. </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Angoli_formati_da_rette_tagliate_da_una_trasversale">Angoli formati da rette tagliate da una trasversale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=10" title="Modifica la sezione Angoli formati da rette tagliate da una trasversale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=10" title="Edit section's source code: Angoli formati da rette tagliate da una trasversale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r139255918">.mw-parser-output .avviso-mini{border:1px solid #aaa;background-color:#fbfbfb;margin-bottom:.5em;padding:0 2px 0 3px;margin:5px 10%;font-size:90%}.mw-parser-output .avviso-mini>div:first-of-type{margin:2px 2px 2px 0}.mw-parser-output .avviso-mini .mw-collapsible-content{padding:2px 0 0 7px}.mw-parser-output .avviso-mini-informazioni{border-left:10px solid #1e90ff}.mw-parser-output .avviso-mini-contenuto{border-left:10px solid #f28500}.mw-parser-output .avviso-mini-stile{border-left:10px solid #f4c430}.mw-parser-output .avviso-mini-statico{border-left:10px solid limegreen}.mw-parser-output .avviso-mini-struttura{border-left:10px solid #9932cc}.mw-parser-output .avviso-mini-generico{border-left:10px solid #bba}body.skin-minerva .mw-parser-output .avviso-mini{border:none;margin-bottom:1px;padding:inherit}body.skin-minerva .mw-parser-output .avviso-mini .mbox-text-div{font-style:normal}html.client-js body.skin-minerva .mw-parser-output .avviso-mini>div{padding:8px 8px 8px 32px!important;position:relative}</style><div class="ambox avviso-mini noprint metadata avviso-mini-generico plainlinks"> <div class="mbox-text"> <div class="mbox-text-div" style="display:flex; flex-direction:row; align-items:center; column-gap:5px;"><span class="mbox-image"><span typeof="mw:File"><span title="Abbozzo matematica"><img alt="Abbozzo matematica" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></span></span></span><span style="width:100%"><b>Questa sezione  sull'argomento matematica è solo un <a href="/wiki/Aiuto:Abbozzo" title="Aiuto:Abbozzo">abbozzo</a></b>. <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Angolo&action=edit">Contribuisci</a> a migliorarla secondo le <a href="/wiki/Aiuto:Manuale_di_stile" title="Aiuto:Manuale di stile">convenzioni di Wikipedia</a>. Segui i suggerimenti del <a href="/wiki/Progetto:Matematica" title="Progetto:Matematica">progetto di riferimento</a>.</span></div> </div> </div> <p>Quando sul piano due rette distinte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /></span> vengono tagliate da un trasversale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span> (incidente sia a<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> che a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /></span>), si originano otto angoli ognuno dei quali è posto in relazione con quelli che non hanno lo stesso vertice. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/File:Rette_intersecante_da_strasversale.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6c/Rette_intersecante_da_strasversale.png" decoding="async" width="206" height="217" class="mw-file-element" data-file-width="206" data-file-height="217" /></a><figcaption></figcaption></figure> <p>Con riferimento ai due semipiani separati dalla trasversale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea3ad87830a1055c7b85c04cf940cfd3b847ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}" /></span> sono definiti <i>coniugati</i> due angoli con vertici distinti disposti sullo stesso <a href="/wiki/Semipiano" title="Semipiano">semipiano</a>. Rispetto alle rette <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5748cdb81bf00075de8e7e6828c343687513830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.737ex; height:2.009ex;" alt="{\displaystyle s,}" /></span> invece, sono definiti <i>esterni</i> due angoli con vertici distinti che non intersecano la retta su cui giace un lato dell'altro angolo, mentre sono considerati <i>interni</i> due angoli con vertici distinti che intersecano la retta su cui giace un lato dell'altro angolo. Sono inoltre definiti <i>corrispondenti</i> due angoli coniugati tali che un lato di uno dei due angoli è contenuto in un lato dell'altro angolo. Con riferimento alla figura si ha la seguente esemplificazione. </p> <ul><li>Sono corrispondenti le coppie:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\alpha ';\quad \beta ,\beta ';\quad \gamma ,\gamma ';\quad \delta ,\delta '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <msup> <mi>α</mi> <mo>′</mo> </msup> <mo>;</mo> <mspace width="1em"></mspace> <mi>β</mi> <mo>,</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>;</mo> <mspace width="1em"></mspace> <mi>γ</mi> <mo>,</mo> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo>;</mo> <mspace width="1em"></mspace> <mi>δ</mi> <mo>,</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\alpha ';\quad \beta ,\beta ';\quad \gamma ,\gamma ';\quad \delta ,\delta '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe26c98feb3bfbb2617cdaa940b61cc055973a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.88ex; height:3.009ex;" alt="{\displaystyle \alpha ,\alpha ';\quad \beta ,\beta ';\quad \gamma ,\gamma ';\quad \delta ,\delta '.}" /></span></dd></dl> <ul><li>Sono coniugati interni le coppie:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ,\alpha ';\quad \gamma ,\delta '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>,</mo> <msup> <mi>α</mi> <mo>′</mo> </msup> <mo>;</mo> <mspace width="1em"></mspace> <mi>γ</mi> <mo>,</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ,\alpha ';\quad \gamma ,\delta '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/076186b9d1ba1e16f0105966546fad3c748ffac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.576ex; height:3.009ex;" alt="{\displaystyle \beta ,\alpha ';\quad \gamma ,\delta '.}" /></span></dd></dl> <ul><li>Sono coniugati esterni le coppie:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ';\quad \delta ,\gamma '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>;</mo> <mspace width="1em"></mspace> <mi>δ</mi> <mo>,</mo> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ';\quad \delta ,\gamma '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5d9fd2a97b4502d24fb9e65e09fe10e7a1a57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.594ex; height:3.009ex;" alt="{\displaystyle \alpha ,\beta ';\quad \delta ,\gamma '.}" /></span></dd></dl> <ul><li>Sono alterni interni le coppie:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ,\delta ';\quad \gamma ,\alpha '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>,</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>;</mo> <mspace width="1em"></mspace> <mi>γ</mi> <mo>,</mo> <msup> <mi>α</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ,\delta ';\quad \gamma ,\alpha '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a44f789caff662ce00228f66aee3df5199c856b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.576ex; height:3.009ex;" alt="{\displaystyle \beta ,\delta ';\quad \gamma ,\alpha '.}" /></span></dd></dl> <ul><li>Sono alterni esterni le coppie:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\gamma ';\quad \delta ,\beta '.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo>;</mo> <mspace width="1em"></mspace> <mi>δ</mi> <mo>,</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\gamma ';\quad \delta ,\beta '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6edb0591768953dc18818a9a4dbc5a35b213a64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.594ex; height:3.009ex;" alt="{\displaystyle \alpha ,\gamma ';\quad \delta ,\beta '.}" /></span></dd></dl> <p>Nel caso in cui le due rette <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /></span> siano <a href="/wiki/Parallelismo_(geometria)" title="Parallelismo (geometria)">parallele</a> gli angoli corrispondenti e gli angoli alterni, dello stesso tipo, sono congruenti. Invece gli angoli coniugati, anch'essi dello stesso tipo, sono <a href="/wiki/Angolo_supplementare" title="Angolo supplementare">supplementari</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Somma_degli_angoli_interni">Somma degli angoli interni</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=11" title="Modifica la sezione Somma degli angoli interni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=11" title="Edit section's source code: Somma degli angoli interni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nella <a href="/wiki/Geometria_euclidea" title="Geometria euclidea">geometria euclidea</a> la somma degli angoli interni di un triangolo è sempre di 180 gradi. Più in generale, data una qualunque <a href="/wiki/Figura_(geometria)" title="Figura (geometria)">figura geometrica</a> convessa di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> lati, la somma di tutti i suoi angoli interni è uguale a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-2)\times 180}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>180</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-2)\times 180}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad60d0c74815867f2599373cdb08674392b1538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.535ex; height:2.843ex;" alt="{\displaystyle (n-2)\times 180}" /></span> gradi. Quindi, per esempio, la somma totale di tutti gli angoli interni di un quadrilatero è uguale a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4-2)\times 180=2\times 180=360}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>180</mn> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mn>180</mn> <mo>=</mo> <mn>360</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4-2)\times 180=2\times 180=360}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8f6fbaa7745c7d1cc99bd69563f063c1990beb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.477ex; height:2.843ex;" alt="{\displaystyle (4-2)\times 180=2\times 180=360}" /></span> gradi. Un caso particolare è dato dal quadrato, che ha quattro angoli retti, la cui somma è infatti 360 gradi. Analogamente, la somma di tutti gli angoli interni di un pentagono, regolare o meno, è uguale a 540 gradi. </p><p>In altre <a href="/wiki/Geometria" title="Geometria">geometrie</a>, dette <a href="/wiki/Geometria_non_euclidea" title="Geometria non euclidea">non euclidee</a>, la somma degli angoli interni di un triangolo può assumere sia valori maggiori sia valori minori di 180 gradi. </p> <div class="mw-heading mw-heading2"><h2 id="Angoli_con_segno">Angoli con segno</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=12" title="Modifica la sezione Angoli con segno" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=12" title="Edit section's source code: Angoli con segno"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Molti problemi portano ad ampliare la nozione di angolo in modo da disporre di un'entità a cui si possa attribuire un'ampiezza data da un numero reale e quindi anche superiore a 360 gradi e negativa. Per questo occorre abbandonare l'associazione angolo - sottoinsieme del piano. Si dice che un angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> è maggiore di un angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> quando una parte di angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> è congruente all'angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span>. Un angolo convesso o concavo può essere descritto cinematicamente come la parte di piano "spazzata" da una semiretta mobile che ruota mantenendo fisso il suo estremo; questo è il vertice dell'angolo e le posizioni iniziale e finale della semiretta sono i lati dell'angolo. Questa descrizione porta a distinguere due versi del movimento rotatorio. Si definisce <i>verso negativo</i> o <i>verso orario</i> il verso della rotazione che, osservata dal di sopra del piano, corrisponde al movimento delle lancette di un orologio tradizionale; si definisce <i>verso positivo</i> o <i>verso antiorario</i> il verso opposto (ad esempio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -135^{\circ }\;=-{\frac {3}{4}}\pi =225^{\circ }\;={\frac {5}{4}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mn>135</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="thickmathspace"></mspace> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mi>π</mi> <mo>=</mo> <msup> <mn>225</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="thickmathspace"></mspace> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -135^{\circ }\;=-{\frac {3}{4}}\pi =225^{\circ }\;={\frac {5}{4}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9f26e8889ed27ef9a919346d1da99508680279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.946ex; height:5.176ex;" alt="{\displaystyle -135^{\circ }\;=-{\frac {3}{4}}\pi =225^{\circ }\;={\frac {5}{4}}\pi }" /></span>). </p><p>Per sviluppare considerazioni quantitative si considera una circonferenza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" /></span> il cui centro ha il ruolo del vertice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" /></span> per gli angoli che si prendono in considerazione. Il raggio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> di questa circonferenza può essere scelto ad arbitrio e talora risulta comodo avere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}" /></span>; quando si riferisce il piano a una coppia di assi cartesiani risulta comodo porre il vertice degli angoli nell'origine, in modo che la circonferenza corrisponda all'equazione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dd4f282df84a83620f71dc52345122e0e3a514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.64ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=r^{2}}" /></span>. </p><p>Ogni angolo di vertice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" /></span> determina un arco sulla circonferenza. Si consideri ora un movimento di una semiretta con estremo in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" /></span> in un verso o nell'altro da una posizione iniziale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> fino a una posizione finale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span>: esso determina sulla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" /></span> un <i>arco orientato</i> che ha come estremo iniziale il punto in cui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" /></span> viene intersecata dalla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> e come estremo finale il punto in cui viene intersecato dalla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span>. Si può pensare l'arco orientato come se fosse "tracciato" dalla penna di un compasso avente l'altro braccio nel punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}" /></span> Gli archi orientati con verso positivo si possono chiamare semplicemente archi (di circonferenza) positivi, quelli con verso negativo archi negativi. </p><p>Si può estendere la nozione di arco orientato pensando che il compasso possa compiere più di un giro, in verso positivo o negativo. </p><p>Si possono identificare gli angoli convessi con gli angoli relativi agli archi positivi interamente contenuti in una semicirconferenza; gli angoli concavi con gli archi positivi che contengono una semicirconferenza e sono contenuti in una circonferenza. </p><p>A questo punto si possono definire come <i>angoli con segno</i> di vertice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" /></span> le entità che generalizzano gli angoli convessi e concavi con vertice in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" /></span> e sono associate biunivocamente agli archi orientati sulla circonferenza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" /></span>. </p><p>Gli angoli con segno possono essere sommati senza le restrizioni degli angoli associati a parti di piano e gli archi relativi risultano essere giustapposti; angolo opposto a un angolo dato corrisponde all'arco considerato con il verso opposto. Di conseguenza agli angoli con segno si attribuisce un'ampiezza rappresentata da un numero reale tale che alla somma di due angoli con segno corrisponda la <a href="/wiki/Somma_algebrica" title="Somma algebrica">somma algebrica</a> delle ampiezze. </p><p>A questo punto si è indotti naturalmente ad associare all'ampiezza di un angolo con segno la lunghezza con segno del corrispondente arco. Questo richiede di precisare cosa si intenda per <a href="/wiki/Lunghezza_di_un_arco" title="Lunghezza di un arco">lunghezza di un arco</a> e più in particolare richiede di definire la lunghezza di una circonferenza </p><p>Le considerazioni sulla rettificazione di una circonferenza portano alla definizione del numero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span> e, sul piano computazionale, alle valutazioni del suo valore. </p> <div class="mw-heading mw-heading2"><h2 id="Angoli_solidi">Angoli solidi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=13" title="Modifica la sezione Angoli solidi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=13" title="Edit section's source code: Angoli solidi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988" /> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Angolo_solido" title="Angolo solido">Angolo solido</a></b>.</span></div> </div> <p>Un angolo solido è un'estensione allo <a href="/wiki/Spazio_tridimensionale" class="mw-redirect" title="Spazio tridimensionale">spazio tridimensionale</a> del concetto di angolo. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=14" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=14" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> Jonathan Slocum, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100627012240/http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html"><span style="font-style:italic;">Indo-European Lexicon: PIE Etymon and IE Reflexes</span></a>, su <span style="font-style:italic;">utexas.edu</span>, Linguistics Research Center, University of Texas, Austin, 4 marzo 2010. <small>URL consultato il 28 maggio 2024</small> <small>(archiviato dall'<abbr title="http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html">url originale</abbr> il 27 giugno 2010)</small>.</cite></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-:0_2-0">a</a></i></sup> <sup><i><a href="#cite_ref-:0_2-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.thefreedictionary.com/angular+unit"><span style="font-style:italic;">Angular Unit</span></a>, su <span style="font-style:italic;">The Free Dictionary</span>.</cite></span> </li> <li id="cite_note-:1-3"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-:1_3-0">a</a></i></sup> <sup><i><a href="#cite_ref-:1_3-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Carl B. Boyer e Uta C. Merzbach, <span style="font-style:italic;">A History of Mathematics</span>, Terza Edizione, pp. 21-36.</cite></span> </li> <li id="cite_note-Ignazio_porro-4"><a href="#cite_ref-Ignazio_porro_4-0"><b>^</b></a> <span class="reference-text"><cite class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060609081238/http://www.sullacrestadellonda.it/strumenti/glossario_topografico.htm"><span style="font-style:italic;">Strumenti navali</span></a> <small>(archiviato dall'<abbr title="http://www.sullacrestadellonda.it/strumenti/glossario_topografico.htm">url originale</abbr> il 9 giugno 2006)</small>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=15" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=15" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Angolo_solido" title="Angolo solido">Angolo solido</a></li> <li><a href="/wiki/Geometria" title="Geometria">Geometria</a></li> <li><a href="/wiki/Goniometro" title="Goniometro">Goniometro</a></li> <li><a href="/wiki/Pendenza_topografica" title="Pendenza topografica">Pendenza topografica</a></li> <li><a href="/wiki/Trigometro" title="Trigometro">Trigometro</a></li> <li><a href="/wiki/Zenit" title="Zenit">Zenit</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=16" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=16" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wiktionary.org/wiki/angolo" class="extiw" title="wikt:angolo">Wikizionario</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Angles (geometry)"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Angles_(geometry)?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/20px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/40px-Wiktionary_small.svg.png 1.5x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/angolo" class="extiw" title="wikt:angolo">angolo</a></b>»</li> <li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sull'<b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Angles_(geometry)?uselang=it">angolo</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Angolo&veaction=edit&section=17" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Angolo&action=edit&section=17" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFTreccani.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/angolo"><span style="font-style:italic;">Angolo</span></a>, su <span style="font-style:italic;">Treccani.it – Enciclopedie on line</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P3365" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_Italiana" class="citation libro" style="font-style:normal"> Raffaello Niccoli, Federigo Enriques e Luigi Volta, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/angolo_(Enciclopedia-Italiana)/"><span style="font-style:italic;">ANGOLO</span></a>, in <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Enciclopedia Italiana</a></span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 1929.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P4223" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFVocabolario_Treccani_angolo" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/vocabolario/angolo"><span style="font-style:italic;">Àngolo</span></a>, su <span style="font-style:italic;"><a href="/wiki/Vocabolario_Treccani" title="Vocabolario Treccani">Vocabolario Treccani</a></span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P5844" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFVocabolario_Treccani_2018" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/vocabolario/angolo_res-81e0c266-adad-11eb-94e0-00271042e8d9"><span style="font-style:italic;">Angolo</span></a>, su <span style="font-style:italic;"><a href="/wiki/Vocabolario_Treccani" title="Vocabolario Treccani">Vocabolario Treccani</a></span>, Thesaurus, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2018.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P5844" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFSapere.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.sapere.it/enciclopedia/àngolo+(geometria).html"><span style="font-style:italic;">àngolo (geometria)</span></a>, su <span style="font-style:italic;">sapere.it</span>, <a href="/wiki/De_Agostini" title="De Agostini">De Agostini</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P6706" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_della_Matematica" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/angolo_(Enciclopedia-della-Matematica)/"><span style="font-style:italic;">Angolo</span></a>, in <span style="font-style:italic;">Enciclopedia della Matematica</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2013.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P9621" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/angle-mathematics"><span style="font-style:italic;">angle</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Angle.html"><span style="font-style:italic;">Angle</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFSpringerEOM" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Angle"><span style="font-style:italic;">Angle</span></a>, su <span style="font-style:italic;"><a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></span>, Springer e European Mathematical Society.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11352#P7554" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; 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