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class="vector-toc-numb">3</span> <span>Propriedades</span> </div> </a> <ul id="toc-Propriedades-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Como_uma_distribuição_estatística" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Como_uma_distribuição_estatística"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Como uma distribuição estatística</span> </div> </a> <button aria-controls="toc-Como_uma_distribuição_estatística-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>Alternar a subsecção Como uma distribuição estatística</span> </button> <ul id="toc-Como_uma_distribuição_estatística-sublist" class="vector-toc-list"> <li id="toc-Distância_média_até_um_ponto_interno_arbitrário" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distância_média_até_um_ponto_interno_arbitrário"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Distância média até um ponto interno arbitrário</span> </div> </a> <ul id="toc-Distância_média_até_um_ponto_interno_arbitrário-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distância_média_até_um_ponto_externo_arbitrário" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distância_média_até_um_ponto_externo_arbitrário"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Distância média até um ponto externo arbitrário</span> </div> </a> <ul id="toc-Distância_média_até_um_ponto_externo_arbitrário-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notas_e_referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notas_e_referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notas e referências</span> </div> </a> <button aria-controls="toc-Notas_e_referências-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>Alternar a subsecção Notas e referências</span> </button> <ul id="toc-Notas_e_referências-sublist" class="vector-toc-list"> <li id="toc-Notas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Notas</span> </div> </a> <ul id="toc-Notas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> </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="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Índice" > <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="Alternar o índice" > <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">Alternar o índice</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">Círculo</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="Ir para um artigo noutra língua. Disponível em 60 línguas" > <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-60" 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">60 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Cerclo" title="Cerclo — aragonês" lang="an" hreflang="an" data-title="Cerclo" data-language-autonym="Aragonés" data-language-local-name="aragonês" 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/%D9%82%D8%B1%D8%B5_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="قرص (رياضيات) — árabe" lang="ar" hreflang="ar" data-title="قرص (رياضيات)" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/C%C3%ADrculu" title="Círculu — asturiano" lang="ast" hreflang="ast" data-title="Círculu" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Dair%C9%99" title="Dairə — azerbaijano" lang="az" hreflang="az" data-title="Dairə" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" 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/%DA%86%D8%A6%D9%88%D8%B1%D9%87" title="چئوره — South Azerbaijani" lang="azb" hreflang="azb" data-title="چئوره" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B3" title="Круг — bielorrusso" lang="be" hreflang="be" data-title="Круг" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" 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%80%D1%83%D0%B3" title="Круг — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Круг" 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-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Krug" title="Krug — bósnio" lang="bs" hreflang="bs" data-title="Krug" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cercle" title="Cercle — catalão" lang="ca" hreflang="ca" data-title="Cercle" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kruh" title="Kruh — checo" lang="cs" hreflang="cs" data-title="Kruh" data-language-autonym="Čeština" data-language-local-name="checo" 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/%D0%9A%D1%80%D1%AB%D0%B3%D1%8A" title="Крѫгъ — eslavo eclesiástico" lang="cu" hreflang="cu" data-title="Крѫгъ" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="eslavo eclesiástico" 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/%C3%87%D0%B0%D0%B2%D1%80%D0%B0%D1%88%D0%BA%D0%B0" title="Çаврашка — chuvash" lang="cv" hreflang="cv" data-title="Çаврашка" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Kreisscheibe" title="Kreisscheibe — alemão" lang="de" hreflang="de" data-title="Kreisscheibe" data-language-autonym="Deutsch" data-language-local-name="alemão" 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%94%CE%AF%CF%83%CE%BA%CE%BF%CF%82_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Δίσκος (γεωμετρία) — grego" lang="el" hreflang="el" data-title="Δίσκος (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="grego" 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/Disk_(mathematics)" title="Disk (mathematics) — inglês" lang="en" hreflang="en" data-title="Disk (mathematics)" data-language-autonym="English" data-language-local-name="inglês" 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/Disko_(matematiko)" title="Disko (matematiko) — esperanto" lang="eo" hreflang="eo" data-title="Disko (matematiko)" 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/Disco_(topolog%C3%ADa)" title="Disco (topología) — espanhol" lang="es" hreflang="es" data-title="Disco (topología)" data-language-autonym="Español" data-language-local-name="espanhol" 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/Ring" title="Ring — estónio" lang="et" hreflang="et" data-title="Ring" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B1%D8%B5_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="قرص (ریاضی) — persa" lang="fa" hreflang="fa" data-title="قرص (ریاضی)" data-language-autonym="فارسی" data-language-local-name="persa" 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/Kiekko_(matematiikka)" title="Kiekko (matematiikka) — finlandês" lang="fi" hreflang="fi" data-title="Kiekko (matematiikka)" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Ts%C3%B5%C3%B5r_(geomeetri%C3%A4)" title="Tsõõr (geomeetriä) — Võro" lang="vro" hreflang="vro" data-title="Tsõõr (geomeetriä)" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Disque_(g%C3%A9om%C3%A9trie)" title="Disque (géométrie) — francês" lang="fr" hreflang="fr" data-title="Disque (géométrie)" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/C%C3%ADrculo" title="Círculo — galego" lang="gl" hreflang="gl" data-title="Círculo" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he badge-Q70894304 mw-list-item" title=""><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%99%D7%92%D7%95%D7%9C" title="עיגול — hebraico" lang="he" hreflang="he" data-title="עיגול" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Krug" title="Krug — croata" lang="hr" hreflang="hr" data-title="Krug" data-language-autonym="Hrvatski" data-language-local-name="croata" 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/Wou_(mekanik)" title="Wou (mekanik) — haitiano" lang="ht" hreflang="ht" data-title="Wou (mekanik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitiano" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%87%D6%80%D5%BB%D5%A1%D5%B6_(%D5%A5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6)" title="Շրջան (երկրաչափություն) — arménio" lang="hy" hreflang="hy" data-title="Շրջան (երկրաչափություն)" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Cakram_(matematika)" title="Cakram (matematika) — indonésio" lang="id" hreflang="id" data-title="Cakram (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Disko_(geometrio)" title="Disko (geometrio) — ido" lang="io" hreflang="io" data-title="Disko (geometrio)" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Cerchio" title="Cerchio — italiano" lang="it" hreflang="it" data-title="Cerchio" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%86%E6%9D%BF" title="円板 — japonês" lang="ja" hreflang="ja" data-title="円板" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AC%E1%83%A0%E1%83%94" title="წრე — georgiano" lang="ka" hreflang="ka" data-title="წრე" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%ED%8C%90" title="원판 — coreano" lang="ko" hreflang="ko" data-title="원판" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skritulys" title="Skritulys — lituano" lang="lt" hreflang="lt" data-title="Skritulys" 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/Ri%C5%86%C4%B7is" title="Riņķis — letão" lang="lv" hreflang="lv" data-title="Riņķis" data-language-autonym="Latviešu" data-language-local-name="letão" 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%A8%D0%B0%D1%80%D0%BA%D1%81%D1%81%D1%8C" title="Шаркссь — mocsa" lang="mdf" hreflang="mdf" data-title="Шаркссь" data-language-autonym="Мокшень" data-language-local-name="mocsa" 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/Kapila_(je%C3%B4metria)" title="Kapila (jeômetria) — malgaxe" lang="mg" hreflang="mg" data-title="Kapila (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="malgaxe" 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%A2%D1%8B%D1%80%D1%82%D1%8B%D1%88_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B5)" title="Тыртыш (математике) — Eastern Mari" lang="mhr" hreflang="mhr" data-title="Тыртыш (математике)" 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%9A%D1%80%D1%83%D0%B3" title="Круг — macedónio" lang="mk" hreflang="mk" data-title="Круг" data-language-autonym="Македонски" data-language-local-name="macedónio" 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%A1%E0%B4%BF%E0%B4%B8%E0%B5%8D%E0%B4%95%E0%B5%8D_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%B6%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82)" title="ഡിസ്ക് (ഗണിതശാസ്ത്രം) — malaiala" lang="ml" hreflang="ml" data-title="ഡിസ്ക് (ഗണിതശാസ്ത്രം)" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%9A%D0%B8%D1%80%D1%8C%D0%BA%D1%81" title="Кирькс — erzya" lang="myv" hreflang="myv" data-title="Кирькс" data-language-autonym="Эрзянь" data-language-local-name="erzya" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Schijf_(wiskunde)" title="Schijf (wiskunde) — neerlandês" lang="nl" hreflang="nl" data-title="Schijf (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Disk_(matematikk)" title="Disk (matematikk) — norueguês bokmål" lang="nb" hreflang="nb" data-title="Disk (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ko%C5%82o" title="Koło — polaco" lang="pl" hreflang="pl" data-title="Koło" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Disc_(geometrie)" title="Disc (geometrie) — romeno" lang="ro" hreflang="ro" data-title="Disc (geometrie)" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B3" title="Круг — russo" lang="ru" hreflang="ru" data-title="Круг" 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%A2%D3%A9%D0%B3%D2%AF%D1%80%D2%AF%D0%BA" title="Төгүрүк — sakha" lang="sah" hreflang="sah" data-title="Төгүрүк" data-language-autonym="Саха тыла" data-language-local-name="sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Krug" title="Krug — servo-croata" lang="sh" hreflang="sh" data-title="Krug" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kruh" title="Kruh — eslovaco" lang="sk" hreflang="sk" data-title="Kruh" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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/Krog" title="Krog — esloveno" lang="sl" hreflang="sl" data-title="Krog" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Qarku_(matematik%C3%AB)" title="Qarku (matematikë) — albanês" lang="sq" hreflang="sq" data-title="Qarku (matematikë)" data-language-autonym="Shqip" data-language-local-name="albanês" 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%9A%D1%80%D1%83%D0%B3" title="Круг — sérvio" lang="sr" hreflang="sr" data-title="Круг" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Cirkelskiva" title="Cirkelskiva — sueco" lang="sv" hreflang="sv" data-title="Cirkelskiva" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%81_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="வட்டு (கணிதம்) — tâmil" lang="ta" hreflang="ta" data-title="வட்டு (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Daire" title="Daire — turco" lang="tr" hreflang="tr" data-title="Daire" 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%A2%D2%AF%D0%B3%D3%99%D1%80%D3%99%D0%BA" title="Түгәрәк — tatar" lang="tt" hreflang="tt" data-title="Түгәрәк" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" 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%80%D1%83%D0%B3" title="Круг — ucraniano" lang="uk" hreflang="uk" data-title="Круг" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Doira_(shakl)" title="Doira (shakl) — usbeque" lang="uz" hreflang="uz" data-title="Doira (shakl)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" 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/H%C3%ACnh_tr%C3%B2n" title="Hình tròn — vietnamita" lang="vi" hreflang="vi" data-title="Hình tròn" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9C%86%E7%9B%98" title="圆盘 — chinês" lang="zh" hreflang="zh" data-title="圆盘" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</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/Q238231#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</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="Espaços nominais"> <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/C%C3%ADrculo" title="Ver a página de conteúdo [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discuss%C3%A3o:C%C3%ADrculo" rel="discussion" title="Discussão sobre o conteúdo da página [t]" accesskey="t"><span>Discussão</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Mudar a variante da língua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">português</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistas"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/C%C3%ADrculo"><span>Ler</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit" title="Editar esta página [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&action=edit" title="Editar o código-fonte desta página [e]" accesskey="e"><span>Editar código-fonte</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&action=history" title="Edições anteriores desta página. [h]" accesskey="h"><span>Ver histórico</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Ferramentas de página"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Ferramentas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Ferramentas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Ferramentas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ocultar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mais opções" > <div class="vector-menu-heading"> Operações </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/C%C3%ADrculo"><span>Ler</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit" title="Editar esta página [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&action=edit" title="Editar o código-fonte desta página [e]" accesskey="e"><span>Editar código-fonte</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&action=history"><span>Ver histórico</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Geral </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginas_afluentes/C%C3%ADrculo" title="Lista de todas as páginas que contêm hiperligações para esta [j]" accesskey="j"><span>Páginas afluentes</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Altera%C3%A7%C3%B5es_relacionadas/C%C3%ADrculo" rel="nofollow" title="Mudanças recentes nas páginas para as quais esta contém hiperligações [k]" accesskey="k"><span>Alterações relacionadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="//pt.wikipedia.org/wiki/Wikipedia:Carregar_ficheiro" title="Carregar ficheiros [u]" accesskey="u"><span>Carregar ficheiro</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&oldid=69057706" title="Hiperligação permanente para esta revisão desta página"><span>Hiperligação permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&action=info" title="Mais informações sobre esta página"><span>Informações da página</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=C%C3%ADrculo&id=69057706&wpFormIdentifier=titleform" title="Informação sobre como citar esta página"><span>Citar esta página</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Fpt.wikipedia.org%2Fwiki%2FC%25C3%25ADrculo"><span>Obter URL encurtado</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fpt.wikipedia.org%2Fwiki%2FC%25C3%25ADrculo"><span>Descarregar código QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimir/exportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Livro&bookcmd=book_creator&referer=C%C3%ADrculo"><span>Criar um livro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=C%C3%ADrculo&action=show-download-screen"><span>Descarregar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=C%C3%ADrculo&printable=yes" title="Versão para impressão desta página [p]" accesskey="p"><span>Versão para impressão</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Noutros projetos </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Discs" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link 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lang="pt" dir="ltr"><div class="hatnote"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/20px-Disambig_grey.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/30px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/40px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> <b>Nota:</b> "Circular" redireciona para este artigo. Para outros significados, veja <a href="/wiki/Circular_(desambigua%C3%A7%C3%A3o)" class="mw-disambig" title="Circular (desambiguação)">Circular (desambiguação)</a>.</div> <div class="hatnote"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/20px-Disambig_grey.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/30px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/40px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> <b>Nota:</b> Este artigo é sobre a reunião da circunferência com sua região interna. Para o contorno de um círculo, veja <a href="/wiki/Circunfer%C3%AAncia" title="Circunferência">circunferência</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Circle-withsegments.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/220px-Circle-withsegments.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/330px-Circle-withsegments.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/440px-Circle-withsegments.svg.png 2x" data-file-width="726" data-file-height="726" /></a><figcaption>Uma <a href="/wiki/Circunfer%C3%AAncia" title="Circunferência">circunferência</a> com <span style="margin:0px; padding-bottom:1px; font-size:100%; display:block;"><span style="border-width:0; border-top:2px dotted #000; border-top:black solid 3px; font-size:40%;">                    </span> comprimento da circunferência <i>C</i></span> <span style="margin:0px; padding-bottom:1px; font-size:100%; display:block;"><span style="border-width:0; border-top:2px dotted #000; border-top:blue solid 2px; font-size:40%;">                    </span> diâmetro <i>D</i></span> <span style="margin:0px; padding-bottom:1px; font-size:100%; display:block;"><span style="border-width:0; border-top:2px dotted #000; border-top:red solid 2px; font-size:40%;">                    </span> raio <i>R</i></span> <span style="margin:0px; padding-bottom:1px; font-size:100%; display:block;"><span style="border-width:0; border-top:2px dotted #000; border-top:green solid 2px; font-size:40%;">                    </span> centro ou origem <i>O</i></span></figcaption></figure> <p>Na <a href="/wiki/Geometria" title="Geometria">geometria</a>, um <b>círculo</b>, por vezes chamado de <b>disco</b>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>nota 1<span class="cite-bracket">]</span></a></sup> é a região em um plano delimitada por uma <a href="/wiki/Circunfer%C3%AAncia" title="Circunferência">circunferência</a>. Um círculo é considerado fechado se contiver a <a href="/wiki/Circunfer%C3%AAncia" title="Circunferência">circunferência</a> que constitui seu limite, e aberto se não contiver.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Um círculo de raio <span class="texhtml mvar" style="font-style:italic;">r</span> e centro <span class="texhtml mvar" style="font-style:italic;">O</span> é geralmente denotado como <span class="texhtml"><i>C</i>(<i>O</i>; <i>r</i>)</span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etimologia">Etimologia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit&section=1" title="Editar secção: Etimologia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%ADrculo&action=edit&section=1" title="Editar código-fonte da secção: Etimologia"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A palavra círculo deriva do grego κίρκος/κύκλος (<i>kirkos</i>/<i>kuklos</i>), que é uma metátese do grego homérico κρίκος (<i>krikos</i>), que significa "aro" ou "anel".<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> As origens das palavras <a href="/wiki/Circo" title="Circo">circo</a> e <a href="https://en.wiktionary.org/wiki/circuito" class="extiw" title="wiktionary:circuito">circuito</a> estão intimamente relacionadas. </p> <div class="mw-heading mw-heading2"><h2 id="Fórmulas"><span id="F.C3.B3rmulas"></span>Fórmulas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit&section=2" title="Editar secção: Fórmulas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%ADrculo&action=edit&section=2" title="Editar código-fonte da secção: Fórmulas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Em coordenadas cartesianas, o círculo aberto de centro <span class="texhtml"><i>O</i>(a, b)</span> e o raio <span class="texhtml mvar" style="font-style:italic;">R</span> é dado pela fórmula:<sup id="cite_ref-odm_11-0" class="reference"><a href="#cite_note-odm-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(O;r)=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}<R^{2}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>O</mi> <mo>;</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <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> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(O;r)=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}<R^{2}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d616eab620b3652e8b33acffcfaebd38da6001b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.571ex; height:3.176ex;" alt="{\displaystyle C(O;r)=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}<R^{2}\}}" /></span> </p><p>enquanto o círculo fechado com o mesmo centro e raio é dado por: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {C}}(O;r)=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}\leq R^{2}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>O</mi> <mo>;</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <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> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {C}}(O;r)=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}\leq R^{2}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50a852370e84915729aa27dcc9df24c2a707762e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.401ex; height:3.509ex;" alt="{\displaystyle {\overline {C}}(O;r)=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}\leq R^{2}\}.}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Propriedades">Propriedades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit&section=3" title="Editar secção: Propriedades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%ADrculo&action=edit&section=3" title="Editar código-fonte da secção: Propriedades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O círculo tem <a href="/wiki/Simetria_circular_e_esf%C3%A9rica" title="Simetria circular e esférica">simetria circular</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>O círculo aberto e o fechado não são topologicamente equivalentes (ou seja, não são <a href="/wiki/Homeomorfismo" title="Homeomorfismo">homeomórficos</a>), pois têm propriedades topológicas diferentes um do outro. Por exemplo, todo círculo fechado é <a href="/wiki/Espa%C3%A7o_compacto" title="Espaço compacto">compacto</a>, ao passo que todo círculo aberto não é compacto.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Entretanto, do ponto de vista da topologia algébrica, eles compartilham muitas propriedades: ambos são <a href="/wiki/Espa%C3%A7o_contr%C3%A1ctil" title="Espaço contráctil">contraíveis</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> e, portanto, são <a href="/wiki/Homotopia#Equivalência_homotópica" title="Homotopia">homotopicamente equivalentes</a> a um único ponto. Isso implica que seus <a href="/wiki/Grupo_fundamental" title="Grupo fundamental">grupos fundamentais</a> são triviais e todos os grupos de <a href="/wiki/Homologia_(matem%C3%A1tica)" title="Homologia (matemática)">homologia</a> são triviais, exceto o 0, que é isomórfico a <span class="texhtml"><b>Z</b></span>. A <a href="/wiki/Caracter%C3%ADstica_de_Euler" title="Característica de Euler">característica de Euler</a> de um ponto (e, portanto, também a de um círculo fechado ou aberto) é 1.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>nota 2<span class="cite-bracket">]</span></a></sup> </p><p>Todo mapa contínuo do círculo fechado para ele mesmo tem pelo menos um <a href="/wiki/Ponto_fixo" title="Ponto fixo">ponto fixo</a> (não é necessário que o mapa seja <a href="/wiki/Fun%C3%A7%C3%A3o_bijectiva" title="Função bijectiva">bijetivo</a> ou mesmo <a href="/wiki/Fun%C3%A7%C3%A3o_sobrejectiva" title="Função sobrejectiva">sobrejetivo</a>); esse é o caso <span class="texhtml"><i>n</i> = 2</span> do <a href="/wiki/Teorema_do_ponto_fixo_de_Brouwer" title="Teorema do ponto fixo de Brouwer">teorema do ponto fixo de Brouwer</a>.<sup id="cite_ref-FOOTNOTEArnold2013132_17-0" class="reference"><a href="#cite_note-FOOTNOTEArnold2013132-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> A afirmação é falsa para o círculo aberto:<sup id="cite_ref-FOOTNOTEArnold2013135ex._1_18-0" class="reference"><a href="#cite_note-FOOTNOTEArnold2013135ex._1-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> considere, por exemplo, a função </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=\left({\frac {x+{\sqrt {1-y^{2}}}}{2}},y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=\left({\frac {x+{\sqrt {1-y^{2}}}}{2}},y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb2756fce1c7ad332a679333ff9f66fcec1087e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.124ex; height:7.509ex;" alt="{\displaystyle f(x,y)=\left({\frac {x+{\sqrt {1-y^{2}}}}{2}},y\right)}" /></span> </p><p>que mapeia cada ponto do <a href="/wiki/C%C3%ADrculo_unit%C3%A1rio" title="Círculo unitário">círculo unitário</a> aberto para outro ponto no circulo unitário aberto à direita do ponto dado. Mas para o círculo unitário fechado, ele fixa cada ponto no semicírculo <span class="texhtml">x<sup>2</sup> + y<sup>2</sup> = 1, x > 0</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Como_uma_distribuição_estatística"><span id="Como_uma_distribui.C3.A7.C3.A3o_estat.C3.ADstica"></span>Como uma distribuição estatística</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit&section=4" title="Editar secção: Como uma distribuição estatística" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%ADrculo&action=edit&section=4" title="Editar código-fonte da secção: Como uma distribuição estatística"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Discdist.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Discdist.svg/250px-Discdist.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Discdist.svg/330px-Discdist.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Discdist.svg/440px-Discdist.svg.png 2x" data-file-width="250" data-file-height="250" /></a><figcaption>A distância média de um local a partir de pontos em um círculo</figcaption></figure> <p>Uma distribuição uniforme em um círculo unitário é ocasionalmente encontrada em <a href="/wiki/Estat%C3%ADstica" title="Estatística">estatística</a>. Ela ocorre mais comumente em <a href="/wiki/Investiga%C3%A7%C3%A3o_operacional" title="Investigação operacional">investigação operacional</a> na matemática do planejamento urbano, onde pode ser usada para modelar uma população dentro de uma cidade. Outros usos podem tirar proveito do fato de ser uma distribuição para a qual é fácil calcular a probabilidade de que um determinado conjunto de desigualdades lineares seja satisfeito. (As <a href="/wiki/Distribui%C3%A7%C3%A3o_normal" title="Distribuição normal">distribuições gaussianas</a> no plano exigem <a href="/wiki/Integra%C3%A7%C3%A3o_num%C3%A9rica" title="Integração numérica">quadratura numérica</a>). </p><p>"Um argumento engenhoso por meio de funções elementares" mostra que a <a href="/wiki/Dist%C3%A2ncia_euclidiana" title="Distância euclidiana">distância euclidiana</a> média entre dois pontos no círculo é <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r69565977">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">128</span><span class="sr-only">/</span><span class="den">45<span class="texhtml mvar" style="font-style:italic;">π</span></span></span>⁠</span> ≈ 0.90541</span>,<sup id="cite_ref-lew_19-0" class="reference"><a href="#cite_note-lew-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> enquanto a integração direta em <a href="/wiki/Coordenadas_polares" title="Coordenadas polares">coordenadas polares</a> mostra que a distância média ao quadrado é 1. </p><p>Se for dado um local arbitrário a uma distância <span class="texhtml mvar" style="font-style:italic;">q</span> do centro do círculo, também é interessante determinar a distância média <span class="texhtml"><i>b</i>(<i>q</i>)</span> dos pontos na distribuição até esse local e o quadrado médio dessas distâncias. O último valor pode ser calculado diretamente como <span class="texhtml">q<sup>2</sup> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r69565977" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Distância_média_até_um_ponto_interno_arbitrário"><span id="Dist.C3.A2ncia_m.C3.A9dia_at.C3.A9_um_ponto_interno_arbitr.C3.A1rio"></span>Distância média até um ponto interno arbitrário</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit&section=5" title="Editar secção: Distância média até um ponto interno arbitrário" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%ADrculo&action=edit&section=5" title="Editar código-fonte da secção: Distância média até um ponto interno arbitrário"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Cjcdiscin.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Cjcdiscin.svg/250px-Cjcdiscin.svg.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Cjcdiscin.svg/330px-Cjcdiscin.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Cjcdiscin.svg/340px-Cjcdiscin.svg.png 2x" data-file-width="340" data-file-height="340" /></a><figcaption>A distância média de um disco até um ponto interno</figcaption></figure> <p>Para encontrar a distância <span class="texhtml"><i>b</i>(<i>q</i>)</span>, precisamos analisar separadamente os casos em que a localização é interna ou externa, ou seja, em que <span class="texhtml"><i>q</i> ≶ 1</span>, e descobrimos que em ambos os casos o resultado só pode ser expresso em termos de <a href="/wiki/Integral_el%C3%ADptica" title="Integral elíptica">integrais elípticas</a> completas. </p><p>Se considerarmos uma localização interna, nosso objetivo (olhando para o diagrama) é calcular o valor esperado de <span class="texhtml mvar" style="font-style:italic;">r</span> sob uma distribuição cuja densidade é <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r69565977" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">π</span></span>⁠</span></span> para <span class="texhtml">0 ≤ <i>r</i> ≤ <i>s</i>(θ)</span>, integrando em coordenadas polares centradas no local fixo para o qual a área de uma célula é <span class="texhtml"><i>r</i> d<i>r</i> dθ</span> ; portanto </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(q)={\frac {1}{\pi }}\int _{0}^{2\pi }{\textrm {d}}\theta \int _{0}^{s(\theta )}r^{2}{\textrm {d}}r={\frac {1}{3\pi }}\int _{0}^{2\pi }s(\theta )^{3}{\textrm {d}}\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> </mrow> <mi>θ</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> </mrow> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> </mrow> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(q)={\frac {1}{\pi }}\int _{0}^{2\pi }{\textrm {d}}\theta \int _{0}^{s(\theta )}r^{2}{\textrm {d}}r={\frac {1}{3\pi }}\int _{0}^{2\pi }s(\theta )^{3}{\textrm {d}}\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54965d915c6e3f4151cb1b4cee689d31d6befd7a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.234ex; height:6.343ex;" alt="{\displaystyle b(q)={\frac {1}{\pi }}\int _{0}^{2\pi }{\textrm {d}}\theta \int _{0}^{s(\theta )}r^{2}{\textrm {d}}r={\frac {1}{3\pi }}\int _{0}^{2\pi }s(\theta )^{3}{\textrm {d}}\theta .}" /></span> </p><p>Aqui <span class="texhtml"><i>s</i>(θ)</span> pode ser encontrado em termos de <span class="texhtml mvar" style="font-style:italic;">q</span> e <span class="texhtml">θ</span> usando a <a href="/wiki/Lei_dos_cossenos" title="Lei dos cossenos">lei dos cossenos</a>. As etapas necessárias para avaliar a integral, juntamente com várias referências, podem ser encontradas no artigo de Lew <i>et al</i>.;<sup id="cite_ref-lew_19-1" class="reference"><a href="#cite_note-lew-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> o resultado é que </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(q)={\frac {4}{9\pi }}{\biggl \{}4(q^{2}-1)K(q^{2})+(q^{2}+7)E(q^{2}){\biggr \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>9</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mn>4</mn> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(q)={\frac {4}{9\pi }}{\biggl \{}4(q^{2}-1)K(q^{2})+(q^{2}+7)E(q^{2}){\biggr \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6593a61449804b8cbe8bc202ee31c02cb6662fff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.414ex; height:6.176ex;" alt="{\displaystyle b(q)={\frac {4}{9\pi }}{\biggl \{}4(q^{2}-1)K(q^{2})+(q^{2}+7)E(q^{2}){\biggr \}}}" /></span> </p><p>onde <span class="texhtml mvar" style="font-style:italic;">K</span> e <span class="texhtml mvar" style="font-style:italic;">E</span> são integrais elípticas completas do primeiro e segundo tipos.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <span class="texhtml"><i>b</i>(0) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r69565977" /><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span></span>; <span class="texhtml"><i>b</i>(1) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r69565977" /><span class="sfrac">⁠<span class="tion"><span class="num">32</span><span class="sr-only">/</span><span class="den">9π</span></span>⁠</span> ≈ 1.13177</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Distância_média_até_um_ponto_externo_arbitrário"><span id="Dist.C3.A2ncia_m.C3.A9dia_at.C3.A9_um_ponto_externo_arbitr.C3.A1rio"></span>Distância média até um ponto externo arbitrário</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit&section=6" title="Editar secção: Distância média até um ponto externo arbitrário" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%ADrculo&action=edit&section=6" title="Editar código-fonte da secção: Distância média até um ponto externo arbitrário"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Cjcdiscex.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Cjcdiscex.svg/220px-Cjcdiscex.svg.png" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Cjcdiscex.svg/330px-Cjcdiscex.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Cjcdiscex.svg/440px-Cjcdiscex.svg.png 2x" data-file-width="490" data-file-height="340" /></a><figcaption>A distância média de um círculo até um ponto externo</figcaption></figure> <p>Voltando a um local externo, podemos configurar a integral de maneira semelhante, desta vez obtendo </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(q)={\frac {2}{3\pi }}\int _{0}^{{\textrm {sin}}^{-1}{\tfrac {1}{q}}}{\biggl \{}s_{+}(\theta )^{3}-s_{-}(\theta )^{3}{\biggr \}}{\textrm {d}}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sin</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mstyle> </mrow> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> </mrow> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(q)={\frac {2}{3\pi }}\int _{0}^{{\textrm {sin}}^{-1}{\tfrac {1}{q}}}{\biggl \{}s_{+}(\theta )^{3}-s_{-}(\theta )^{3}{\biggr \}}{\textrm {d}}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7e855be04c6dc5a3a4faf7a01d3ece4782cf2e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.143ex; height:7.343ex;" alt="{\displaystyle b(q)={\frac {2}{3\pi }}\int _{0}^{{\textrm {sin}}^{-1}{\tfrac {1}{q}}}{\biggl \{}s_{+}(\theta )^{3}-s_{-}(\theta )^{3}{\biggr \}}{\textrm {d}}\theta }" /></span> </p><p>onde a lei dos cossenos nos diz que <span class="texhtml"><i>s</i><sub>+</sub>(θ)</span> e <span class="texhtml"><i>s</i><sub>–</sub>(θ)</span> são as raízes para <span class="texhtml mvar" style="font-style:italic;">s</span> da equação </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>q</mi> <mi>s</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>cos</mtext> </mrow> </mrow> <mi>θ</mi> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace"></mspace> <mo>−</mo> <mspace width="negativethinmathspace"></mspace> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83e8796578f5b46c7aa0b2ab68485a7f1e25e38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.007ex; height:3.009ex;" alt="{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}" /></span> </p><p>Portanto </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>q</mi> <mi>s</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>cos</mtext> </mrow> </mrow> <mi>θ</mi> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace"></mspace> <mo>−</mo> <mspace width="negativethinmathspace"></mspace> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83e8796578f5b46c7aa0b2ab68485a7f1e25e38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.007ex; height:3.009ex;" alt="{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}" /></span> </p><p>Podemos substituir <span class="texhtml"><i>u</i> = <i>q</i> sinθ </span> para obter </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>q</mi> <mi>s</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>cos</mtext> </mrow> </mrow> <mi>θ</mi> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace"></mspace> <mo>−</mo> <mspace width="negativethinmathspace"></mspace> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83e8796578f5b46c7aa0b2ab68485a7f1e25e38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.007ex; height:3.009ex;" alt="{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}" /></span> </p><p>usando integrais padrão.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Portanto, novamente <span class="texhtml"><i>b</i>(1) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r69565977" /><span class="sfrac">⁠<span class="tion"><span class="num">32</span><span class="sr-only">/</span><span class="den">9π</span></span>⁠</span></span> e também<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{q\to \infty }b(q)=q+{\tfrac {1}{8q}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>b</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>q</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{q\to \infty }b(q)=q+{\tfrac {1}{8q}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e539fd050e99d0bd77587f1fad6bad12f520d62" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.376ex; height:4.509ex;" alt="{\displaystyle \lim _{q\to \infty }b(q)=q+{\tfrac {1}{8q}}.}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%ADrculo&veaction=edit&section=7" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%ADrculo&action=edit&section=7" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Coroa_circular" title="Coroa circular">Coroa circular</a></li> <li><a href="/wiki/Bola_(matem%C3%A1tica)" title="Bola (matemática)">Bola (matemática)</a></li> <li><a href="/wiki/Segmento_circular" title="Segmento circular">Segmento circular</a></li></ul> <h2 id="Notas_e_referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Notas_e_refer.C3.AAncias"></span>Notas e referências</h2><h3 id="Notas" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso.">Notas</h3> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Em inglês, o termos utilizado é <i><span lang="en">disk</span></i>, de forma que em alguns livros, especialmente aqueles direcionados para o ensino superior, utilizem a palavra disco para se referir à figura, como pode ser observado em Geometria euclidiana plana, de <a href="/wiki/Jo%C3%A3o_Lucas_Marques_Barbosa" title="João Lucas Marques Barbosa">Barbosa</a>,<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> ou em Curso de análise, de <a href="/wiki/Elon_Lages_Lima" title="Elon Lages Lima">Lima</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> No entanto, no ensino básico, a palavra utilizada é círculo, como pode ser observado nos livros didáticos das editoras <a href="/wiki/FTD" title="FTD">FTD</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Editora do Brasil,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Edições SM,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> e <a href="/wiki/Moderna" title="Moderna">Moderna</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><a href="#cite_ref-16">↑</a></span> <span class="reference-text">Em dimensões maiores, a característica de Euler de uma esfera fechada permanece igual a +1, mas a característica de Euler de uma esfera aberta é +1 para esferas de dimensão par e -1 para esferas de dimensão ímpar.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="reflist" style="list-style-type: lower-alpha;"></div> <div class="reflist" style="list-style-type: decimal;"></div> <div class="reflist" style="list-style-type: lower-alpha;"></div> <div class="reflist" style="list-style-type: decimal;"></div> <div class="reflist" style="list-style-type: lower-alpha;"></div> <h3 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h3> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation book">Barbosa, João Lucas Marques (Agosto de 1997) [1995]. <a rel="nofollow" class="external text" href="https://archive.org/details/geometriaeuclidianaplanajoaolucasempdfescrito/page/n19/mode/2up"><i>Geometria Euclidiana Plana</i></a>. Rio de Janeiro, RJ: Sociedade Brasileira de Matemática. p. 12. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-85-85818-02-9" title="Especial:Fontes de livros/978-85-85818-02-9">978-85-85818-02-9</a>. <q>Todo ponto C que satisfaz a desigualdade <i><span style="text-decoration:overline;">AC</span></i> < <i>r</i> é dito estar dentro do círculo. Se, ao invés, <i><span style="text-decoration:overline;">AC</span></i> > <i>r</i>, então C é dito estar fora do círculo. O conjunto dos pontos que estão dentro do círculo é chamado de <i>disco</i> de raio <i>r</i> e centro A.</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.aufirst=Jo%C3%A3o+Lucas+Marques&rft.aulast=Barbosa&rft.btitle=Geometria+Euclidiana+Plana&rft.date=1997-08&rft.genre=book&rft.isbn=978-85-85818-02-9&rft.pages=12&rft.place=Rio+de+Janeiro%2C+RJ&rft.pub=Sociedade+Brasileira+de+Matem%C3%A1tica&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometriaeuclidianaplanajoaolucasempdfescrito%2Fpage%2Fn19%2Fmode%2F2up&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book">Lima, Elon Lages (2014). <i>Curso de análise</i>. Rio de Janeiro, RJ: IMPA. p. 19. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-85-244-0375-0" title="Especial:Fontes de livros/978-85-244-0375-0">978-85-244-0375-0</a>. <q>Agora consideremos o disco D de centro na origem e raio 1. Temos D = {(x, y) ∈ <b>R</b><sup>2</sup>; x<sup>2</sup> + y<sup>2</sup> ≤ 1}.</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.aufirst=Elon+Lages&rft.aulast=Lima&rft.btitle=Curso+de+an%C3%A1lise&rft.date=2014&rft.genre=book&rft.isbn=978-85-244-0375-0&rft.pages=19&rft.place=Rio+de+Janeiro%2C+RJ&rft.pub=IMPA&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite class="citation book">Giovannni Jr., José Ruy; Castrucci, Benedicto (2009). <i>A conquista da Matemática</i>. <b>8</b>. São Paulo, SP: FTD. p. 330. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-85-322-7013-9" title="Especial:Fontes de livros/978-85-322-7013-9">978-85-322-7013-9</a>. <q>A região da circunferência com a sua região interna denomina-se <b>círculo</b>.</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.au=Castrucci%2C+Benedicto&rft.aufirst=Jos%C3%A9+Ruy&rft.aulast=Giovannni+Jr.&rft.btitle=A+conquista+da+Matem%C3%A1tica&rft.date=2009&rft.genre=book&rft.isbn=978-85-322-7013-9&rft.pages=330&rft.place=S%C3%A3o+Paulo%2C+SP&rft.pub=FTD&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation book">Andrini, Álvaro; Vasconcellos, Maria José (2015). <i>Praticando Matemática</i>. <b>9</b>. São Paulo, SP: Editora do Brasil. p. 225. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-85-10-05898-8" title="Especial:Fontes de livros/978-85-10-05898-8">978-85-10-05898-8</a>. <q>Juntando à circunferência os pontos do seu interior, obtemos um círculo.</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.au=Vasconcellos%2C+Maria+Jos%C3%A9&rft.aufirst=%C3%81lvaro&rft.aulast=Andrini&rft.btitle=Praticando+Matem%C3%A1tica&rft.date=2015&rft.genre=book&rft.isbn=978-85-10-05898-8&rft.pages=225&rft.place=S%C3%A3o+Paulo%2C+SP&rft.pub=Editora+do+Brasil&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><cite class="citation book">Chavante, Eduardo Rodrigues (2015). <i>Convergências: Matemática</i>. <b>9</b>. São Paulo, SP: Edições SM. p. 182. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-85-418-0967-2" title="Especial:Fontes de livros/978-85-418-0967-2">978-85-418-0967-2</a>. <q><b>Círculo</b> é uma figura geométrica plana correspondente à união de uma circunferência com todos os pontos do seu interior.</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.aufirst=Eduardo+Rodrigues&rft.aulast=Chavante&rft.btitle=Converg%C3%AAncias%3A+Matem%C3%A1tica&rft.date=2015&rft.genre=book&rft.isbn=978-85-418-0967-2&rft.pages=182&rft.place=S%C3%A3o+Paulo%2C+SP&rft.pub=Edi%C3%A7%C3%B5es+SM&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><cite class="citation book"><i>Projeto Araribá: Matemática</i>. <b>8</b>. São Paulo, SP: Círculo é a região do plano formada por uma circunferência e sua região interna. 2007. p. 227. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-85-16-05518-9" title="Especial:Fontes de livros/978-85-16-05518-9">978-85-16-05518-9</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.btitle=Projeto+Ararib%C3%A1%3A+Matem%C3%A1tica&rft.date=2007&rft.genre=book&rft.isbn=978-85-16-05518-9&rft.pages=227&rft.place=S%C3%A3o+Paulo%2C+SP&rft.pub=C%C3rculo+%C3%A9+a+regi%C3%A3o+do+plano+formada+por+uma+circunfer%C3%AAncia+e+sua+regi%C3%A3o+interna.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><cite class="citation book">Bianchini, Edwaldo (2018). <i>Matemática — Bianchini</i>. <b>9</b>. São Paulo, SP: Moderna. p. 287. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-85-16-11381-0" title="Especial:Fontes de livros/978-85-16-11381-0">978-85-16-11381-0</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.aufirst=Edwaldo&rft.aulast=Bianchini&rft.btitle=Matem%C3%A1tica+%94+Bianchini&rft.date=2018&rft.genre=book&rft.isbn=978-85-16-11381-0&rft.pages=287&rft.place=S%C3%A3o+Paulo%2C+SP&rft.pub=Moderna&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><cite id="CITEREFArnold2013" class="citation">Arnold, B. H. (2013), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TsbDAgAAQBAJ&pg=PA58"><i>Intuitive Concepts in Elementary Topology</i></a>, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/9780486275765" title="Especial:Fontes de livros/9780486275765">9780486275765</a>, Dover Books on Mathematics, Courier Dover Publications, p. 58</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.aufirst=B.+H.&rft.aulast=Arnold&rft.btitle=Intuitive+Concepts+in+Elementary+Topology&rft.date=2013&rft.genre=book&rft.isbn=9780486275765&rft.pages=58&rft.pub=Courier+Dover+Publications&rft.series=Dover+Books+on+Mathematics&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTsbDAgAAQBAJ%26pg%3DPA58&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><cite class="citation web">Liddell, Henry George; Scott, Robert. <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dkri%2Fkos">«krikos»</a>. <i>Perseus</i>. A Greek-English Lexicon. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131106164504/http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dkri%2Fkos">Cópia arquivada em 6 de novembro de 2013</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.atitle=krikos&rft.au=Scott%2C+Robert&rft.aufirst=Henry+George&rft.aulast=Liddell&rft.genre=unknown&rft.jtitle=Perseus&rft_id=https%3A%2F%2Fwww.perseus.tufts.edu%2Fhopper%2Ftext%3Fdoc%3DPerseus%253Atext%253A1999.04.0057%253Aentry%253Dkri%252Fkos&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-odm-11"><span class="mw-cite-backlink"><a href="#cite_ref-odm_11-0">↑</a></span> <span class="reference-text"><cite id="CITEREFClaphamNicholson2014" class="citation">Clapham, Christopher; Nicholson, James (2014), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c69GBAAAQBAJ&pg=PA138"><i>The Concise Oxford Dictionary of Mathematics</i></a>, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/9780199679591" title="Especial:Fontes de livros/9780199679591">9780199679591</a>, Oxford University Press, p. 138</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.au=Nicholson%2C+James&rft.aufirst=Christopher&rft.aulast=Clapham&rft.btitle=The+Concise+Oxford+Dictionary+of+Mathematics&rft.date=2014&rft.genre=book&rft.isbn=9780199679591&rft.pages=138&rft.pub=Oxford+University+Press&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dc69GBAAAQBAJ%26pg%3DPA138&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text"><cite class="citation book">Altmann, Simon L. (1992). <a rel="nofollow" class="external text" href="https://archive.org/details/iconssymmetries0000altm"><i>Icons and Symmetries</i></a> (em inglês). [S.l.]: Oxford University Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/9780198555995" title="Especial:Fontes de livros/9780198555995">9780198555995</a>. <q>disc circular symmetry.</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.aufirst=Simon+L.&rft.aulast=Altmann&rft.btitle=Icons+and+Symmetries&rft.date=1992&rft.genre=book&rft.isbn=9780198555995&rft.pub=Oxford+University+Press&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ficonssymmetries0000altm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text"><cite class="citation book">Maudlin, Tim (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kEbbAgAAQBAJ&pg=PA339"><i>New Foundations for Physical Geometry: The Theory of Linear Structures</i></a>. [S.l.]: Oxford University Press. p. 339. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/9780191004551" title="Especial:Fontes de livros/9780191004551">9780191004551</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.aufirst=Tim&rft.aulast=Maudlin&rft.btitle=New+Foundations+for+Physical+Geometry%3A+The+Theory+of+Linear+Structures&rft.date=2014&rft.genre=book&rft.isbn=9780191004551&rft.pages=339&rft.pub=Oxford+University+Press&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkEbbAgAAQBAJ%26pg%3DPA339&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><cite class="citation book">Cohen, Daniel E. (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=STc4AAAAIAAJ&pg=PA79"><i>Combinatorial Group Theory: A Topological Approach</i></a>. Col: London Mathematical Society Student Texts. <b>14</b>. [S.l.]: Cambridge University Press. p. 79. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/9780521349369" title="Especial:Fontes de livros/9780521349369">9780521349369</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.aufirst=Daniel+E.&rft.aulast=Cohen&rft.btitle=Combinatorial+Group+Theory%3A+A+Topological+Approach&rft.date=1989&rft.genre=book&rft.isbn=9780521349369&rft.pages=79&rft.pub=Cambridge+University+Press&rft.series=London+Mathematical+Society+Student+Texts&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSTc4AAAAIAAJ%26pg%3DPA79&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><cite class="citation book">Klain, Daniel A.; <a href="/wiki/Gian-Carlo_Rota" title="Gian-Carlo Rota">Rota, Gian-Carlo</a> (1997). <i>Introduction to Geometric Probability</i>. Col: Lezioni Lincee. [S.l.]: Cambridge University Press. pp. 46–50</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.au=Rota%2C+Gian-Carlo&rft.aufirst=Daniel+A.&rft.aulast=Klain&rft.btitle=Introduction+to+Geometric+Probability&rft.date=1997&rft.genre=book&rft.pages=46-50&rft.pub=Cambridge+University+Press&rft.series=Lezioni+Lincee&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-FOOTNOTEArnold2013132-17"><span class="mw-cite-backlink"><a href="#cite_ref-FOOTNOTEArnold2013132_17-0">↑</a></span> <span class="reference-text"><a href="#CITEREFArnold2013">Arnold 2013</a>, p. 132.</span> </li> <li id="cite_note-FOOTNOTEArnold2013135ex._1-18"><span class="mw-cite-backlink"><a href="#cite_ref-FOOTNOTEArnold2013135ex._1_18-0">↑</a></span> <span class="reference-text"><a href="#CITEREFArnold2013">Arnold 2013</a>, p. 135, ex. 1.</span> </li> <li id="cite_note-lew-19"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-lew_19-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-lew_19-1">b</a></b></i></sup></span> <span class="reference-text"><cite class="citation journal">Lew, John S.; Frauenthal, James C.; Keyfitz, Nathan (1978). <a rel="nofollow" class="external text" href="http://www.jstor.org/stable/2030357">«On the Average Distances in a Circular Disc»</a>. Society for Industrial and Applied Mathematics. <i>SIAM Review</i>. <b>20</b> (3): 584–592. <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0036-1445">0036-1445</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AC%C3rculo&rft.atitle=On+the+Average+Distances+in+a+Circular+Disc&rft.au=Frauenthal%2C+James+C.&rft.au=Keyfitz%2C+Nathan&rft.aufirst=John+S.&rft.aulast=Lew&rft.date=1978&rft.genre=article&rft.issn=00361445&rft.issue=3&rft.jtitle=SIAM+Review&rft.pages=584-592&rft.volume=20&rft_id=http%3A%2F%2Fwww.jstor.org%2Fstable%2F2030357&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><a href="#cite_ref-20">↑</a></span> <span class="reference-text"><i><a href="/wiki/Handbook_of_Mathematical_Functions" title="Handbook of Mathematical Functions">Handbook of Mathematical Functions</a></i>, 17.3.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><a href="#cite_ref-21">↑</a></span> <span class="reference-text">Gradshteyn e Ryzhik 3.155.7 e 3.169.9, levando em conta a diferença de notação de <i>Handbook of Mathematical Functions</i>. (Compare A&S 17.3.11 com G&R 8.113.) Este artigo segue a notação da A&S.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><a href="#cite_ref-22">↑</a></span> <span class="reference-text"><i>Handbook of Mathematical Functions</i>, 17.3.11 et seq.</span> </li> </ol></div></div> <ul class="noprint navigation-box" style="border-top: solid silver 1px; border-right: solid silver 1px; border-bottom:1px solid silver; border-left: solid silver 1px; padding:3px; background-color: #F9F9F9; text-align: center; margin-top:10px; margin-left: 0; clear: both;"><li style="display: inline;"><span style="white-space: nowrap; margin: auto 1.5em"><span style="margin-right: 0.5em"><span typeof="mw:File"><a href="/wiki/Ficheiro:Nuvola_apps_edu_mathematics-p.svg" title="Portal da matemática"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/25px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/38px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/50px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span style="font-weight:bold"><a href="/wiki/Portal:Matem%C3%A1tica" title="Portal:Matemática">Portal da matemática</a></span></span></li> <li style="display: inline;"><span style="white-space: nowrap; margin: auto 1.5em"><span style="margin-right: 0.5em"><span typeof="mw:File"><a href="/wiki/Ficheiro:Circle-sector.svg" title="Portal da geometria"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Circle-sector.svg/25px-Circle-sector.svg.png" decoding="async" width="25" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Circle-sector.svg/38px-Circle-sector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Circle-sector.svg/50px-Circle-sector.svg.png 2x" data-file-width="360" data-file-height="320" /></a></span></span> <span style="font-weight:bold"><a href="/wiki/Portal:Geometria" title="Portal:Geometria">Portal da geometria</a></span></span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Círculo&oldid=69057706">https://pt.wikipedia.org/w/index.php?title=Círculo&oldid=69057706</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias</a>: <ul><li><a href="/wiki/Categoria:C%C3%ADrculos" title="Categoria:Círculos">Círculos</a></li><li><a href="/wiki/Categoria:Desenho_geom%C3%A9trico" title="Categoria:Desenho geométrico">Desenho geométrico</a></li><li><a href="/wiki/Categoria:Geometria_euclidiana_plana" title="Categoria:Geometria euclidiana plana">Geometria euclidiana plana</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria oculta: <ul><li><a href="/wiki/Categoria:!CS1_ingl%C3%AAs-fontes_em_l%C3%ADngua_(en)" title="Categoria:!CS1 inglês-fontes em língua (en)">!CS1 inglês-fontes em língua (en)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 00h03min de 25 de novembro de 2024.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>60 línguas</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Adicionar tópico</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-585dc88b6-cjggw","wgBackendResponseTime":134,"wgPageParseReport":{"limitreport":{"cputime":"0.273","walltime":"0.690","ppvisitednodes":{"value":4136,"limit":1000000},"postexpandincludesize":{"value":46686,"limit":2097152},"templateargumentsize":{"value":5417,"limit":2097152},"expansiondepth":{"value":11,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":26779,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 264.544 1 -total"," 45.20% 119.578 1 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