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title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E5%9C%86" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-历史" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#历史"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>历史</span> </div> </a> <ul id="toc-历史-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-性质" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#性质"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>性质</span> </div> </a> <button aria-controls="toc-性质-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>开关性质子章节</span> </button> <ul id="toc-性质-sublist" class="vector-toc-list"> <li id="toc-解析几何" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#解析几何"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>解析几何</span> </div> </a> <ul id="toc-解析几何-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-圆心" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#圆心"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>圆心</span> </div> </a> <ul id="toc-圆心-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-弦" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#弦"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>弦</span> </div> </a> <ul id="toc-弦-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-弧" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#弧"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>弧</span> </div> </a> <ul id="toc-弧-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-直径、半径" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#直径、半径"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>直径、半径</span> </div> </a> <ul id="toc-直径、半径-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-切线" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#切线"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>切线</span> </div> </a> <ul id="toc-切线-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-割线" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#割线"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>割线</span> </div> </a> <ul id="toc-割线-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-周长" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#周长"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>周长</span> </div> </a> <ul id="toc-周长-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-面积" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#面积"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.9</span> <span>面积</span> </div> </a> <ul id="toc-面积-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-对称性" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#对称性"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.10</span> <span>对称性</span> </div> </a> <ul id="toc-对称性-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-圓心角、圆周角" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#圓心角、圆周角"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>圓心角、圆周角</span> </div> </a> <button aria-controls="toc-圓心角、圆周角-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>开关圓心角、圆周角子章节</span> </button> <ul id="toc-圓心角、圆周角-sublist" class="vector-toc-list"> <li id="toc-圆心角定理" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#圆心角定理"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>圆心角定理</span> </div> </a> <ul id="toc-圆心角定理-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-圆周角定理" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#圆周角定理"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>圆周角定理</span> </div> </a> <ul id="toc-圆周角定理-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-垂径定理" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#垂径定理"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>垂径定理</span> </div> </a> <button aria-controls="toc-垂径定理-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>开关垂径定理子章节</span> </button> <ul id="toc-垂径定理-sublist" class="vector-toc-list"> <li id="toc-知二推三" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#知二推三"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>知二推三</span> </div> </a> <ul id="toc-知二推三-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-推论" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#推论"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>推论</span> </div> </a> <ul id="toc-推论-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-兩圓位置關係" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#兩圓位置關係"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>兩圓位置關係</span> </div> </a> <button aria-controls="toc-兩圓位置關係-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>开关兩圓位置關係子章节</span> </button> <ul id="toc-兩圓位置關係-sublist" class="vector-toc-list"> <li id="toc-圆系方程" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#圆系方程"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>圆系方程</span> </div> </a> <ul id="toc-圆系方程-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-其他定义" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#其他定义"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>其他定义</span> </div> </a> <ul id="toc-其他定义-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-其它" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#其它"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>其它</span> </div> </a> <button aria-controls="toc-其它-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>开关其它子章节</span> </button> <ul id="toc-其它-sublist" class="vector-toc-list"> <li id="toc-相關的立体图形" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#相關的立体图形"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>相關的立体图形</span> </div> </a> <ul id="toc-相關的立体图形-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-圓和其他平面形狀" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#圓和其他平面形狀"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>圓和其他平面形狀</span> </div> </a> <ul id="toc-圓和其他平面形狀-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-圓的問題" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#圓的問題"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>圓的問題</span> </div> </a> <ul id="toc-圓的問題-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-参考资料" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#参考资料"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>参考资料</span> </div> </a> <button aria-controls="toc-参考资料-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>开关参考资料子章节</span> </button> <ul id="toc-参考资料-sublist" class="vector-toc-list"> <li id="toc-注释" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#注释"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>注释</span> </div> </a> <ul id="toc-注释-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-资料" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#资料"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>资料</span> </div> </a> <ul id="toc-资料-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-扩展阅读" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#扩展阅读"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>扩展阅读</span> </div> </a> <ul id="toc-扩展阅读-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>外部链接</span> </div> </a> <ul id="toc-外部链接-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <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">开关目录</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">圆</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="前往另一种语言写成的文章。145种语言可用" > <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-145" 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">145种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Sirkel" title="Sirkel – 南非荷兰语" lang="af" hreflang="af" data-title="Sirkel" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Kreis_(Geometrie)" title="Kreis (Geometrie) – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Kreis (Geometrie)" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%AD%E1%89%A5" title="ክብ – 阿姆哈拉语" lang="am" hreflang="am" data-title="ክብ" data-language-autonym="አማርኛ" data-language-local-name="阿姆哈拉语" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Circumferencia" title="Circumferencia – 阿拉贡语" lang="an" hreflang="an" data-title="Circumferencia" data-language-autonym="Aragonés" data-language-local-name="阿拉贡语" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar badge-Q17437796 badge-featuredarticle mw-list-item" title="典范条目"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9" title="دائرة – 阿拉伯语" lang="ar" hreflang="ar" data-title="دائرة" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%AF%D8%A7%D9%8A%D8%B1%D9%87" title="دايره – 埃及阿拉伯文" lang="arz" hreflang="arz" data-title="دايره" data-language-autonym="مصرى" data-language-local-name="埃及阿拉伯文" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="বৃত্ত – 阿萨姆语" lang="as" hreflang="as" data-title="বৃত্ত" data-language-autonym="অসমীয়া" data-language-local-name="阿萨姆语" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Circunferencia" title="Circunferencia – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Circunferencia" data-language-autonym="Asturianu" data-language-local-name="阿斯图里亚斯语" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Muyu" title="Muyu – 艾马拉语" lang="ay" hreflang="ay" data-title="Muyu" data-language-autonym="Aymar aru" data-language-local-name="艾马拉语" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87evr%C9%99" title="Çevrə – 阿塞拜疆语" lang="az" hreflang="az" data-title="Çevrə" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87" title="دایره – South Azerbaijani" lang="azb" hreflang="azb" data-title="دایره" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D3%98%D0%B9%D0%BB%D3%99%D0%BD%D3%99" title="Әйләнә – 巴什基尔语" lang="ba" hreflang="ba" data-title="Әйләнә" data-language-autonym="Башҡортса" data-language-local-name="巴什基尔语" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Apskr%C4%97t%C4%97ms" title="Apskrėtėms – 薩莫吉希亞文" lang="sgs" hreflang="sgs" data-title="Apskrėtėms" data-language-autonym="Žemaitėška" data-language-local-name="薩莫吉希亞文" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Bilog" title="Bilog – Central Bikol" lang="bcl" hreflang="bcl" data-title="Bilog" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Акружнасць – 白俄罗斯语" lang="be" hreflang="be" data-title="Акружнасць" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" 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%90%D0%BA%D1%80%D1%83%D0%B6%D1%8B%D0%BD%D0%B0" title="Акружына – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Акружына" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%BA%D1%80%D1%8A%D0%B6%D0%BD%D0%BE%D1%81%D1%82" title="Окръжност – 保加利亚语" lang="bg" hreflang="bg" data-title="Окръжност" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="বৃত্ত – 孟加拉语" lang="bn" hreflang="bn" data-title="বৃত্ত" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A6%E0%BE%92%E0%BD%BC%E0%BD%A2%E0%BC%8B%E0%BD%90%E0%BD%B2%E0%BD%82%E0%BC%8B" title="སྒོར་ཐིག་ – 藏语" lang="bo" hreflang="bo" data-title="སྒོར་ཐིག་" data-language-autonym="བོད་ཡིག" data-language-local-name="藏语" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Kelc%27h" title="Kelc'h – 布列塔尼语" lang="br" hreflang="br" data-title="Kelc'h" data-language-autonym="Brezhoneg" data-language-local-name="布列塔尼语" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Kružnica" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" 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/Circumfer%C3%A8ncia" title="Circumferència – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Circumferència" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%D8%A7%D8%B2%D9%86%DB%95_(%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95)" title="بازنە (ئەندازە) – 中库尔德语" lang="ckb" hreflang="ckb" data-title="بازنە (ئەندازە)" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kru%C5%BEnice" title="Kružnice – 捷克语" lang="cs" hreflang="cs" data-title="Kružnice" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</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%D0%BA%C4%83%D1%88" title="Çавракăш – 楚瓦什语" lang="cv" hreflang="cv" data-title="Çавракăш" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cylch" title="Cylch – 威尔士语" lang="cy" hreflang="cy" data-title="Cylch" data-language-autonym="Cymraeg" data-language-local-name="威尔士语" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Cirkel" title="Cirkel – 丹麦语" lang="da" hreflang="da" data-title="Cirkel" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kreis" title="Kreis – 德语" lang="de" hreflang="de" data-title="Kreis" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Cera_krejza" title="Cera krejza – 下索布语" lang="dsb" hreflang="dsb" data-title="Cera krejza" data-language-autonym="Dolnoserbski" data-language-local-name="下索布语" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%8D%CE%BA%CE%BB%CE%BF%CF%82" title="Κύκλος – 希腊语" lang="el" hreflang="el" data-title="Κύκλος" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Ser%C4%87_(giometr%C3%ACa)" title="Serć (giometrìa) – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Serć (giometrìa)" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Circle" title="Circle – 英语" lang="en" hreflang="en" data-title="Circle" data-language-autonym="English" data-language-local-name="英语" 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/Cirklo" title="Cirklo – 世界语" lang="eo" hreflang="eo" data-title="Cirklo" data-language-autonym="Esperanto" data-language-local-name="世界语" 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/C%C3%ADrculo" title="Círculo – 西班牙语" lang="es" hreflang="es" data-title="Círculo" data-language-autonym="Español" data-language-local-name="西班牙语" 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/Ringjoon" title="Ringjoon – 爱沙尼亚语" lang="et" hreflang="et" data-title="Ringjoon" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zirkulu" title="Zirkulu – 巴斯克语" lang="eu" hreflang="eu" data-title="Zirkulu" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa badge-Q17437798 badge-goodarticle mw-list-item" title="优良条目"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87" title="دایره – 波斯语" lang="fa" hreflang="fa" data-title="دایره" data-language-autonym="فارسی" data-language-local-name="波斯语" 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/Ympyr%C3%A4" title="Ympyrä – 芬兰语" lang="fi" hreflang="fi" data-title="Ympyrä" data-language-autonym="Suomi" data-language-local-name="芬兰语" 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%B5rjuun" title="Tsõõrjuun – 佛羅文" lang="vro" hreflang="vro" data-title="Tsõõrjuun" data-language-autonym="Võro" data-language-local-name="佛羅文" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Iwirini" title="Iwirini – 斐济语" lang="fj" hreflang="fj" data-title="Iwirini" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="斐济语" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Sirkul" title="Sirkul – 法罗语" lang="fo" hreflang="fo" data-title="Sirkul" data-language-autonym="Føroyskt" data-language-local-name="法罗语" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cercle" title="Cercle – 法语" lang="fr" hreflang="fr" data-title="Cercle" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kreis_(Geometrii)" title="Kreis (Geometrii) – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Kreis (Geometrii)" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ciorcal" title="Ciorcal – 爱尔兰语" lang="ga" hreflang="ga" data-title="Ciorcal" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%9C%93%E5%BD%A2" title="圓形 – 赣语" lang="gan" hreflang="gan" data-title="圓形" data-language-autonym="贛語" data-language-local-name="赣语" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Serk" title="Serk – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Serk" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Cearcall" title="Cearcall – 苏格兰盖尔语" lang="gd" hreflang="gd" data-title="Cearcall" data-language-autonym="Gàidhlig" data-language-local-name="苏格兰盖尔语" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Circunferencia" title="Circunferencia – 加利西亚语" lang="gl" hreflang="gl" data-title="Circunferencia" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B5%E0%AA%B0%E0%AB%8D%E0%AA%A4%E0%AB%81%E0%AA%B3" title="વર્તુળ – 古吉拉特语" lang="gu" hreflang="gu" data-title="વર્તુળ" data-language-autonym="ગુજરાતી" data-language-local-name="古吉拉特语" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Kiarkyl" title="Kiarkyl – 马恩语" lang="gv" hreflang="gv" data-title="Kiarkyl" data-language-autonym="Gaelg" data-language-local-name="马恩语" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%92%D7%9C" title="מעגל – 希伯来语" lang="he" hreflang="he" data-title="מעגל" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%83%E0%A4%A4%E0%A5%8D%E0%A4%A4" title="वृत्त – 印地语" lang="hi" hreflang="hi" data-title="वृत्त" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Circle" title="Circle – 斐濟印地文" lang="hif" hreflang="hif" data-title="Circle" data-language-autonym="Fiji Hindi" data-language-local-name="斐濟印地文" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – 克罗地亚语" lang="hr" hreflang="hr" data-title="Kružnica" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – 上索布语" lang="hsb" hreflang="hsb" data-title="Kružnica" data-language-autonym="Hornjoserbsce" data-language-local-name="上索布语" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/S%C3%A8k_(non)" title="Sèk (non) – 海地克里奥尔语" lang="ht" hreflang="ht" data-title="Sèk (non)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="海地克里奥尔语" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%B6r_(geometria)" title="Kör (geometria) – 匈牙利语" lang="hu" hreflang="hu" data-title="Kör (geometria)" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%87%D6%80%D5%BB%D5%A1%D5%B6%D5%A1%D5%A3%D5%AB%D5%AE" title="Շրջանագիծ – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Շրջանագիծ" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Circulo" title="Circulo – 国际语" lang="ia" hreflang="ia" data-title="Circulo" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lingkaran" title="Lingkaran – 印度尼西亚语" lang="id" hreflang="id" data-title="Lingkaran" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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/Cirklo" title="Cirklo – 伊多语" lang="io" hreflang="io" data-title="Cirklo" data-language-autonym="Ido" data-language-local-name="伊多语" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hringur_(r%C3%BAmfr%C3%A6%C3%B0i)" title="Hringur (rúmfræði) – 冰岛语" lang="is" hreflang="is" data-title="Hringur (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Circonferenza" title="Circonferenza – 意大利语" lang="it" hreflang="it" data-title="Circonferenza" data-language-autonym="Italiano" data-language-local-name="意大利语" 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%95%B0%E5%AD%A6)" title="円 (数学) – 日语" lang="ja" hreflang="ja" data-title="円 (数学)" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Soerkl" title="Soerkl – 牙買加克里奧爾英文" lang="jam" hreflang="jam" data-title="Soerkl" data-language-autonym="Patois" data-language-local-name="牙買加克里奧爾英文" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Bunderan" title="Bunderan – 爪哇语" lang="jv" hreflang="jv" data-title="Bunderan" data-language-autonym="Jawa" data-language-local-name="爪哇语" class="interlanguage-link-target"><span>Jawa</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%E1%83%AC%E1%83%98%E1%83%A0%E1%83%98" title="წრეწირი – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="წრეწირი" data-language-autonym="ქართული" data-language-local-name="格鲁吉亚语" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tawinest" title="Tawinest – 卡拜尔语" lang="kab" hreflang="kab" data-title="Tawinest" data-language-autonym="Taqbaylit" data-language-local-name="卡拜尔语" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A8%D0%B5%D2%A3%D0%B1%D0%B5%D1%80" title="Шеңбер – 哈萨克语" lang="kk" hreflang="kk" data-title="Шеңбер" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9A%E1%9E%84%E1%9F%92%E1%9E%9C%E1%9E%84%E1%9F%8B" title="រង្វង់ – 高棉语" lang="km" hreflang="km" data-title="រង្វង់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="高棉语" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B3%83%E0%B2%A4%E0%B3%8D%E0%B2%A4" title="ವೃತ್ತ – 卡纳达语" lang="kn" hreflang="kn" data-title="ವೃತ್ತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="卡纳达语" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90_(%EA%B8%B0%ED%95%98%ED%95%99)" title="원 (기하학) – 韩语" lang="ko" hreflang="ko" data-title="원 (기하학)" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Gilover" title="Gilover – 库尔德语" lang="ku" hreflang="ku" data-title="Gilover" data-language-autonym="Kurdî" data-language-local-name="库尔德语" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Kylgh" title="Kylgh – 康沃尔语" lang="kw" hreflang="kw" data-title="Kylgh" data-language-autonym="Kernowek" data-language-local-name="康沃尔语" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B9%D0%BB%D0%B0%D0%BD%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Айлана (математика) – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Айлана (математика)" data-language-autonym="Кыргызча" data-language-local-name="柯尔克孜语" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Circulus" title="Circulus – 拉丁语" lang="la" hreflang="la" data-title="Circulus" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Krees_(Geometrie)" title="Krees (Geometrie) – 卢森堡语" lang="lb" hreflang="lb" data-title="Krees (Geometrie)" data-language-autonym="Lëtzebuergesch" data-language-local-name="卢森堡语" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Sirculo" title="Sirculo – 新共同語言" lang="lfn" hreflang="lfn" data-title="Sirculo" data-language-autonym="Lingua Franca Nova" data-language-local-name="新共同語言" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Cirkel" title="Cirkel – 林堡语" lang="li" hreflang="li" data-title="Cirkel" data-language-autonym="Limburgs" data-language-local-name="林堡语" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/S%C3%A9rcc" title="Sércc – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Sércc" data-language-autonym="Lombard" data-language-local-name="倫巴底文" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Apskritimas" title="Apskritimas – 立陶宛语" lang="lt" hreflang="lt" data-title="Apskritimas" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" 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%B7a_l%C4%ABnija" title="Riņķa līnija – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Riņķa līnija" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Faribolana" title="Faribolana – 马拉加斯语" lang="mg" hreflang="mg" data-title="Faribolana" data-language-autonym="Malagasy" data-language-local-name="马拉加斯语" 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%9E%D2%A5%D0%B3%D0%BE" title="Оҥго – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Оҥго" 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-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Lingkaran" title="Lingkaran – 米南佳保语" lang="min" hreflang="min" data-title="Lingkaran" data-language-autonym="Minangkabau" data-language-local-name="米南佳保语" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="典范条目"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0" title="Кружница – 马其顿语" lang="mk" hreflang="mk" data-title="Кружница" data-language-autonym="Македонски" data-language-local-name="马其顿语" 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%B5%E0%B5%83%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%82" title="വൃത്തം – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="വൃത്തം" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B9%D1%80%D0%BE%D0%B3" title="Тойрог – 蒙古语" lang="mn" hreflang="mn" data-title="Тойрог" data-language-autonym="Монгол" data-language-local-name="蒙古语" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A5%81%E0%A4%B3" title="वर्तुळ – 马拉地语" lang="mr" hreflang="mr" data-title="वर्तुळ" data-language-autonym="मराठी" data-language-local-name="马拉地语" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bulatan" title="Bulatan – 马来语" lang="ms" hreflang="ms" data-title="Bulatan" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%85%E1%80%80%E1%80%BA%E1%80%9D%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8" title="စက်ဝိုင်း – 缅甸语" lang="my" hreflang="my" data-title="စက်ဝိုင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="缅甸语" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Krink" title="Krink – 低地德语" lang="nds" hreflang="nds" data-title="Krink" data-language-autonym="Plattdüütsch" data-language-local-name="低地德语" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A5%83%E0%A4%A4" title="वृत – 尼泊尔语" lang="ne" hreflang="ne" data-title="वृत" data-language-autonym="नेपाली" data-language-local-name="尼泊尔语" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%9A%E0%A4%BE%E0%A4%95%E0%A4%83" title="चाकः – 尼瓦尔语" lang="new" hreflang="new" data-title="चाकः" data-language-autonym="नेपाल भाषा" data-language-local-name="尼瓦尔语" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Cirkel" title="Cirkel – 荷兰语" lang="nl" hreflang="nl" data-title="Cirkel" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Sirkel" title="Sirkel – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Sirkel" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sirkel" title="Sirkel – 书面挪威语" lang="nb" hreflang="nb" data-title="Sirkel" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Cercle" title="Cercle – 奥克语" lang="oc" hreflang="oc" data-title="Cercle" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Geengoo" title="Geengoo – 奥罗莫语" lang="om" hreflang="om" data-title="Geengoo" data-language-autonym="Oromoo" data-language-local-name="奥罗莫语" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AC%E0%AD%83%E0%AC%A4%E0%AD%8D%E0%AC%A4" title="ବୃତ୍ତ – 奥里亚语" lang="or" hreflang="or" data-title="ବୃତ୍ତ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="奥里亚语" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9A%E0%A9%B1%E0%A8%95%E0%A8%B0" title="ਚੱਕਰ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਚੱਕਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pih mw-list-item"><a href="https://pih.wikipedia.org/wiki/Sirkil" title="Sirkil – Norfuk / Pitkern" lang="pih" hreflang="pih" data-title="Sirkil" data-language-autonym="Norfuk / Pitkern" data-language-local-name="Norfuk / Pitkern" class="interlanguage-link-target"><span>Norfuk / Pitkern</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Okr%C4%85g" title="Okrąg – 波兰语" lang="pl" hreflang="pl" data-title="Okrąg" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%DB%81" title="دائرہ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="دائرہ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%AF%D8%B1%D8%AF%DA%A9%D9%87" title="گردکه – 普什图语" lang="ps" hreflang="ps" data-title="گردکه" data-language-autonym="پښتو" data-language-local-name="普什图语" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Circunfer%C3%AAncia" title="Circunferência – 葡萄牙语" lang="pt" hreflang="pt" data-title="Circunferência" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/P%27allta_muyu" title="P'allta muyu – 克丘亚语" lang="qu" hreflang="qu" data-title="P'allta muyu" data-language-autonym="Runa Simi" data-language-local-name="克丘亚语" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Cerc" title="Cerc – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Cerc" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-roa-rup mw-list-item"><a href="https://roa-rup.wikipedia.org/wiki/%C8%9Aerc%C4%BEiu" title="Țercľiu – 阿罗马尼亚语" lang="rup" hreflang="rup" data-title="Țercľiu" data-language-autonym="Armãneashti" data-language-local-name="阿罗马尼亚语" class="interlanguage-link-target"><span>Armãneashti</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Окружность – 俄语" lang="ru" hreflang="ru" data-title="Окружность" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B3" title="Круг – 盧森尼亞文" lang="rue" hreflang="rue" data-title="Круг" data-language-autonym="Русиньскый" data-language-local-name="盧森尼亞文" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/C%C3%ACrculu_(giometr%C3%ACa)" title="Cìrculu (giometrìa) – 西西里语" lang="scn" hreflang="scn" data-title="Cìrculu (giometrìa)" data-language-autonym="Sicilianu" data-language-local-name="西西里语" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Raing" title="Raing – 苏格兰语" lang="sco" hreflang="sco" data-title="Raing" data-language-autonym="Scots" data-language-local-name="苏格兰语" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AF%D9%88%D9%84" title="گول – 信德语" lang="sd" hreflang="sd" data-title="گول" data-language-autonym="سنڌي" data-language-local-name="信德语" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Kružnica" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Circle" title="Circle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Circle" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Kružnica" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" 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/Kro%C5%BEnica" title="Krožnica – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Krožnica" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Denderedzwa" title="Denderedzwa – 绍纳语" lang="sn" hreflang="sn" data-title="Denderedzwa" data-language-autonym="ChiShona" data-language-local-name="绍纳语" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Goobo" title="Goobo – 索马里语" lang="so" hreflang="so" data-title="Goobo" data-language-autonym="Soomaaliga" data-language-local-name="索马里语" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rrethi" title="Rrethi – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Rrethi" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" 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%B6%D0%BD%D0%B8%D1%86%D0%B0" title="Кружница – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Кружница" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Bunderan_(%C3%A9lmu_ukur)" title="Bunderan (élmu ukur) – 巽他语" lang="su" hreflang="su" data-title="Bunderan (élmu ukur)" data-language-autonym="Sunda" data-language-local-name="巽他语" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Cirkel" title="Cirkel – 瑞典语" lang="sv" hreflang="sv" data-title="Cirkel" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Duara" title="Duara – 斯瓦希里语" lang="sw" hreflang="sw" data-title="Duara" data-language-autonym="Kiswahili" data-language-local-name="斯瓦希里语" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="வட்டம் – 泰米尔语" lang="ta" hreflang="ta" data-title="வட்டம்" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B1%83%E0%B0%A4%E0%B1%8D%E0%B0%A4%E0%B0%AE%E0%B1%81" title="వృత్తము – 泰卢固语" lang="te" hreflang="te" data-title="వృత్తము" data-language-autonym="తెలుగు" data-language-local-name="泰卢固语" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%A7%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1" title="รูปวงกลม – 泰语" lang="th" hreflang="th" data-title="รูปวงกลม" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Bilog" title="Bilog – 他加禄语" lang="tl" hreflang="tl" data-title="Bilog" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87ember" title="Çember – 土耳其语" lang="tr" hreflang="tr" data-title="Çember" data-language-autonym="Türkçe" data-language-local-name="土耳其语" 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/%D3%98%D0%B9%D0%BB%D3%99%D0%BD%D3%99" title="Әйләнә – 鞑靼语" lang="tt" hreflang="tt" data-title="Әйләнә" data-language-autonym="Татарча / tatarça" data-language-local-name="鞑靼语" 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%D0%BE%D0%BB%D0%BE" title="Коло – 乌克兰语" lang="uk" hreflang="uk" data-title="Коло" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%DB%81" title="دائرہ – 乌尔都语" lang="ur" hreflang="ur" data-title="دائرہ" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Aylana" title="Aylana – 乌兹别克语" lang="uz" hreflang="uz" data-title="Aylana" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Sercio" title="Sercio – 威尼斯语" lang="vec" hreflang="vec" data-title="Sercio" data-language-autonym="Vèneto" data-language-local-name="威尼斯语" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_tr%C3%B2n" title="Đường tròn – 越南语" lang="vi" hreflang="vi" data-title="Đường tròn" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Lidong" title="Lidong – 瓦瑞语" lang="war" hreflang="war" data-title="Lidong" data-language-autonym="Winaray" data-language-local-name="瓦瑞语" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9C%86" title="圆 – 吴语" lang="wuu" hreflang="wuu" data-title="圆" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Isazinge" title="Isazinge – 科萨语" lang="xh" hreflang="xh" data-title="Isazinge" data-language-autonym="IsiXhosa" data-language-local-name="科萨语" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%A8%D7%99%D7%99%D7%96" title="קרייז – 意第绪语" lang="yi" hreflang="yi" data-title="קרייז" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%92b%C3%ACr%C3%ADpo" title="Òbìrípo – 约鲁巴语" lang="yo" hreflang="yo" data-title="Òbìrípo" data-language-autonym="Yorùbá" data-language-local-name="约鲁巴语" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Luenz" title="Luenz – 壮语" lang="za" hreflang="za" data-title="Luenz" data-language-autonym="Vahcuengh" data-language-local-name="壮语" class="interlanguage-link-target"><span>Vahcuengh</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%9C%93" title="圓 – 文言文" lang="lzh" hreflang="lzh" data-title="圓" data-language-autonym="文言" data-language-local-name="文言文" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/%C3%8E%E2%81%BF-h%C3%AAng" title="Îⁿ-hêng – 闽南语" lang="nan" hreflang="nan" data-title="Îⁿ-hêng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="闽南语" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9C%93%E5%BD%A2" title="圓形 – 粤语" lang="yue" hreflang="yue" data-title="圓形" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</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/Q17278#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</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="命名空间"> <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"> 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mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E5%9C%86" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E5%9C%86" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E5%9C%86" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E5%9C%86" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E5%9C%86" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E5%9C%86" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E5%9C%86" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E5%9C%86" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E5%9C%86" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <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/%E5%9C%86"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E5%9C%86&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E5%9C%86&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <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="工具" > <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">工具</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">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </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/%E5%9C%86"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E5%9C%86&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E5%9C%86&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E5%9C%86" title="列出所有与本页相链的页面[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E5%9C%86" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E5%9C%86&oldid=84402584" title="此页面该修订版本的固定链接"><span>固定链接</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E5%9C%86&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E5%9C%86&id=84402584&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E5%259C%2586"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E5%259C%2586"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E5%9C%86&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印页面</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"> 在其他项目中 </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:Circle_area" hreflang="en"><span>维基共享资源</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://zh.wikiquote.org/wiki/%E5%9C%93" hreflang="zh"><span>维基语录</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q17278" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r83732972">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-night .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-Merge_from plainlinks metadata ambox ambox-move" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Mergefrom.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Mergefrom.svg/50px-Mergefrom.svg.png" decoding="async" width="50" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Mergefrom.svg/75px-Mergefrom.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Mergefrom.svg/100px-Mergefrom.svg.png 2x" data-file-width="50" data-file-height="20" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">建議将<b><a href="/wiki/%E5%9C%86%E7%B3%BB%E6%96%B9%E7%A8%8B" title="圆系方程">圆系方程</a></b><a href="/wiki/Wikipedia:%E5%90%88%E5%B9%B6" class="mw-redirect" title="Wikipedia:合并">併入</a>此條目或章節。（<a href="/wiki/Talk:%E5%9C%86" title="Talk:圆">討論</a>）<span class="hide-when-compact"></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><a href="/wiki/Wikipedia:%E6%B6%88%E6%AD%A7%E4%B9%89" title="Wikipedia:消歧义"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/25px-Disambig_gray.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/38px-Disambig_gray.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/50px-Disambig_gray.svg.png 2x" data-file-width="220" data-file-height="168" /></a></span><style data-mw-deduplicate="TemplateStyles:r74069148">body:not(.skin-minerva) .mw-parser-output .ifmobile>.mobile{display:none}body.skin-minerva .mw-parser-output .ifmobile>.nomobile{display:inherit;display:initial}</style><span class="ifmobile"><span class="nomobile">  </span><span class="mobile"></span></span>关于其他意义，请见「<b><a href="/wiki/%E5%9C%93_(%E6%B6%88%E6%AD%A7%E7%BE%A9)" class="mw-disambig" title="圓 (消歧義)">圆 (消歧义)</a></b>」。</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><a href="/wiki/Wikipedia:%E6%B6%88%E6%AD%A7%E4%B9%89" title="Wikipedia:消歧义"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/24px-Confusion_grey.svg.png" decoding="async" width="24" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/36px-Confusion_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/48px-Confusion_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r74069148"><span class="ifmobile"><span class="nomobile">  </span><span class="mobile"></span></span><b>提示</b>：此条目页的主题不是<b><a href="/wiki/%E9%9B%B6%E9%82%8A%E5%BD%A2" class="mw-redirect" title="零邊形">零邊形</a></b>或<b><a href="/wiki/%E7%84%A1%E9%99%90%E9%82%8A%E5%BD%A2" title="無限邊形">無限邊形</a></b>。</div> <style data-mw-deduplicate="TemplateStyles:r83732082">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}html.skin-theme-clientpref-night .mw-parser-output .infobox td div:not(.notheme)[style]{background:transparent!important;color:var(--color-base,#202122)}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox td div:not(.notheme)[style]{background:transparent!important;color:var(--color-base,#202122)}}html.skin-theme-clientpref-night .mw-parser-output .infobox td div.NavHead:not(.notheme)[style]{background:transparent!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox td div.NavHead:not(.notheme)[style]{background:transparent!important}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3;">圓</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File: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></span><div class="infobox-caption"><div class="legend-line"><span style="display:inline-block; vertical-align:middle; width:1.67em; height:0px; border-style:none; border-top:2px dotted #000; border-top:black solid 3px;"> </span> 圆周<i>C</i></div> <div class="legend-line"><span style="display:inline-block; vertical-align:middle; width:1.67em; height:0px; border-style:none; border-top:2px dotted #000; border-top:blue solid 2px;"> </span> 直徑<i>D</i></div> <div class="legend-line"><span style="display:inline-block; vertical-align:middle; width:1.67em; height:0px; border-style:none; border-top:2px dotted #000; border-top:red solid 2px;"> </span> 半徑<i>R</i></div> <div class="legend-line"><span style="display:inline-block; vertical-align:middle; width:1.67em; height:0px; border-style:none; border-top:2px dotted #000; border-top:green solid 2px;"> </span> 原點<i>O</i></div></div></td></tr><tr><th scope="row" class="infobox-label" style="background:#e7dcc3;;">類型</th><td class="infobox-data" style=""><a href="/wiki/%E5%9C%86%E9%94%A5%E6%9B%B2%E7%BA%BF" title="圆锥曲线">圓錐曲線</a></td></tr><tr><th scope="row" class="infobox-label" style="background:#e7dcc3;;"><span class="ilh-all" data-orig-title="鮑爾斯縮寫" data-lang-code="verse-and-dimensions" data-lang-name="verse-and-dimensions的wikia" data-foreign-title="The_Fifty-Nine_Icosahedra#Notes_on_the_list"><span class="ilh-page"><a href="/w/index.php?title=%E9%AE%91%E7%88%BE%E6%96%AF%E7%B8%AE%E5%AF%AB&action=edit&redlink=1" class="new" title="鮑爾斯縮寫（页面不存在）">鮑爾斯縮寫<br /></a></span><span class="noprint ilh-comment">（<span class="ilh-lang">verse-and-dimensions的wikia</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://community.fandom.com/wiki/w:c:verse-and-dimensions:Bowers_acronym" class="extiw" title="wikia:verse-and-dimensions:Bowers acronym"><span lang="en" dir="auto">Bowers acronym</span></a></span>）</span></span></th><td class="infobox-data" style="">circ<span style="float:right"><span class="mw-valign-text-top" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q17278?uselang=zh#P9997" title="在维基数据编辑"><img alt="在维基数据编辑" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="background:#e7dcc3;;"><a href="/wiki/%E7%A9%BA%E9%96%93%E5%B0%8D%E7%A8%B1%E7%BE%A4" title="空間對稱群">對稱群</a></th><td class="infobox-data" style=""><a href="/wiki/%E6%AD%A3%E4%BA%A4%E7%BE%A4" title="正交群"><style data-mw-deduplicate="TemplateStyles:r58896141">'"`UNIQ--templatestyles-0000000A-QINU`"'</style><span class="serif"><span class="texhtml">O(2)</span></span></a></td></tr><tr><th scope="row" class="infobox-label" style="background:#e7dcc3;;"><a href="/wiki/%E9%9D%A2%E7%A9%8D" class="mw-redirect" title="面積">面積</a></th><td class="infobox-data" style=""><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml">πR<sup>2</sup></span></span></td></tr><tr><th scope="row" class="infobox-label" style="background:#e7dcc3;;"><a href="/wiki/%E5%91%A8%E9%95%BF" title="周长">周长</a></th><td class="infobox-data" style=""><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml">C = 2πR</span></span></td></tr><tr><td colspan="2" class="infobox-navbar"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist-pipe dd::after,.mw-parser-output .hlist-pipe li::after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd::after,.mw-parser-output .hlist-hyphen li::after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd::after,.mw-parser-output .hlist-comma li::after{content:"、";font-weight:normal}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:"（"counter(listitem)"\a0 "}.mw-parser-output ul.cslist,.mw-parser-output ul.sslist{margin:0;padding:0;display:inline-block;list-style:none}.mw-parser-output .cslist li,.mw-parser-output .sslist li{margin:0;display:inline-block}.mw-parser-output .cslist li::after{content:"，"}.mw-parser-output .sslist li::after{content:"；"}.mw-parser-output .cslist li:last-child::after,.mw-parser-output .sslist li:last-child::after{content:none}</style><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Infobox_polygon" title="Template:Infobox polygon"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Infobox_polygon" title="Template talk:Infobox polygon"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:Infobox_polygon" title="Special:编辑页面/Template:Infobox polygon"><abbr title="编辑该模板">编</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>圆</b> （英語：<span lang="en">circle</span>）的第一个定义是：根據<a href="/wiki/%E6%AD%90%E5%B9%BE%E9%87%8C%E5%BE%97" class="mw-redirect" title="歐幾里得">歐幾里得</a>的《<a href="/wiki/%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC" title="几何原本">几何原本</a>》，在同一<a href="/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学)">平面</a>内到定点 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> 的距离等于定长 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 的点的集合<sup id="cite_ref-几何原本_1-0" class="reference"><a href="#cite_note-几何原本-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。此定点 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> 称为圆心（center of a circle），此定长 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 称为半径（radius）。 </p><p>圆的第二个定义是：平面内一动点到两定点的距离的比，等于一个不为1的常数，则此动点的轨迹是圆<sup id="cite_ref-高中数学必修1_2-0" class="reference"><a href="#cite_note-高中数学必修1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>；此圆属于一种<span class="ilh-all ilh-blue" data-orig-title="阿波罗尼奥斯圆" data-lang-code="en" data-lang-name="英语" data-foreign-title="Circles of Apollonius"><span class="ilh-page"><a href="/wiki/%E9%98%BF%E6%B3%A2%E7%BD%97%E5%B0%BC%E5%A5%A5%E6%96%AF%E5%9C%86" class="mw-redirect" title="阿波罗尼奥斯圆">阿波罗尼奥斯圆</a></span></span>（circles of Apollonius）。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="历史"><span id=".E5.8E.86.E5.8F.B2"></span>历史</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=1" title="编辑章节：历史"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">更多信息：<a href="/wiki/%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC" title="几何原本">几何原本</a>和<a href="/wiki/%E5%A2%A8%E5%AD%90" title="墨子">墨子</a></div> <p>古代人最早是从<a href="/wiki/%E5%A4%AA%E9%98%B3" title="太阳">太阳</a>、阴历十五的<a href="/wiki/%E6%9C%88%E4%BA%AE" class="mw-redirect" title="月亮">月亮</a>得到圆的概念的。在一万八千年前的<a href="/wiki/%E5%B1%B1%E9%A1%B6%E6%B4%9E%E4%BA%BA" title="山顶洞人">山顶洞人</a>曾经在<a href="/wiki/%E7%89%99%E9%BD%92" title="牙齒">兽牙</a>、<a href="/wiki/%E7%A0%BE%E7%9F%B3" title="砾石">砾石</a>和石珠上钻孔，那些孔有的就很像圆。<sup id="cite_ref-历史_3-0" class="reference"><a href="#cite_note-历史-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>到了<a href="/w/index.php?title=%E9%99%B6%E5%99%A8%E6%97%B6%E4%BB%A3&action=edit&redlink=1" class="new" title="陶器时代（页面不存在）">陶器时代</a>，许多陶器都是圆的。圆的陶器是将泥土放在一个转盘上制成的。<sup id="cite_ref-圆的历史_4-0" class="reference"><a href="#cite_note-圆的历史-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>当人们开始纺线，又制出了圆形的<a href="/w/index.php?title=%E7%9F%B3%E7%BA%BA%E9%94%A4&action=edit&redlink=1" class="new" title="石纺锤（页面不存在）">石纺锤</a>或<a href="/w/index.php?title=%E9%99%B6%E7%BA%BA%E9%94%A4&action=edit&redlink=1" class="new" title="陶纺锤（页面不存在）">陶纺锤</a>。古代人还发现搬运圆的木头时滚着走比较省劲。后来他们在搬运重物的时候，就把几段圆木垫在大树、大石头下面滚着走。<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>约在6000年前，<a href="/wiki/%E7%BE%8E%E7%B4%A2%E4%B8%8D%E8%BE%BE%E7%B1%B3%E4%BA%9A" title="美索不达米亚">美索不达米亚</a>人，做出了世界上第一个轮子——圆型的木盘。<sup id="cite_ref-圆的历史_4-1" class="reference"><a href="#cite_note-圆的历史-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>大约在4000多年前，人们将圆的木盘固定在木架下，这就成了最初的车子。古代<a href="/wiki/%E5%9F%83%E5%8F%8A" title="埃及">埃及</a>人认为：圆，是神赐给人的神圣图形。一直到两千多年前中国的<a href="/wiki/%E5%A2%A8%E5%AD%90" title="墨子">墨子</a>给圆下了一个定义：圆，一中同长也。意思是说：圆有一个<a href="/wiki/%E5%9C%93%E5%BF%83" class="mw-redirect" title="圓心">圆心</a>，圆心到<a href="/wiki/%E5%9C%86%E5%91%A8" class="mw-redirect" title="圆周">圆周</a>上各点的距离（即<a href="/wiki/%E5%8D%8A%E5%BE%84" title="半径">半径</a>）都相等。<sup id="cite_ref-圆的历史_4-2" class="reference"><a href="#cite_note-圆的历史-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="性质"><span id=".E6.80.A7.E8.B4.A8"></span>性质</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=2" title="编辑章节：性质"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="解析几何"><span id=".E8.A7.A3.E6.9E.90.E5.87.A0.E4.BD.95"></span>解析几何</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=3" title="编辑章节：解析几何"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E7%B3%BB" class="mw-redirect" title="直角坐标系">直角坐标系</a>中的定义：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-x_{m})^{2}+(y-y_{m})^{2}=r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <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> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-x_{m})^{2}+(y-y_{m})^{2}=r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34a3cca0cd77612b2f8eba3cb6e8eccce520afe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.754ex; height:3.176ex;" alt="{\displaystyle (x-x_{m})^{2}+(y-y_{m})^{2}=r^{2}}"></span>，其中r是半径，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{m},y_{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{m},y_{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad1904bbe759ab59a4fe514a28dfbd10fff49c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.662ex; height:2.843ex;" alt="{\displaystyle (x_{m},y_{m})}"></span>是圆心坐标。</li> <li><a href="/wiki/%E5%8F%83%E6%95%B8%E6%96%B9%E7%A8%8B" title="參數方程">参数方程</a>的定义：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{m}+a\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{m}+a\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfdbc3b9d93567547dc58c28c0c0136fde3aafe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.479ex; height:2.509ex;" alt="{\displaystyle x=x_{m}+a\cos \theta }"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y_{m}+a\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=y_{m}+a\sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88ced878a975d821e6cf3f117c7014e3c59a6e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.859ex; height:2.509ex;" alt="{\displaystyle y=y_{m}+a\sin \theta }"></span>。</li> <li><a href="/wiki/%E6%9E%81%E5%9D%90%E6%A0%87" class="mw-redirect" title="极坐标">极坐标</a><a href="/wiki/%E6%96%B9%E7%A8%8B" title="方程">方程</a>的定义（圆心在原点）：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e193cdce556664593cd0a8347617f5284d0364c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.377ex; height:1.676ex;" alt="{\displaystyle r=a}"></span>。</li></ul> <div class="mw-heading mw-heading3"><h3 id="圆心"><span id=".E5.9C.86.E5.BF.83"></span>圆心</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=4" title="编辑章节：圆心"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>圆是在同一平面内到定点的距离等于定长的点的集合，这个定点叫做圆的圆心（通常用<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>表示）。<sup id="cite_ref-数学_6-0" class="reference"><a href="#cite_note-数学-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="弦"><span id=".E5.BC.A6"></span>弦</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=5" title="编辑章节：弦"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>圆周上任何两点相连的<a href="/wiki/%E7%BA%BF%E6%AE%B5" title="线段">线段</a>称为圆的<a href="/wiki/%E5%BC%A6_(%E5%B9%BE%E4%BD%95)" title="弦 (幾何)">弦</a>（英語：<span lang="en">chord</span>）。如图2，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>分别为圆上任意两点，那么<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f91c39c977790f3cc4e768d5aad89bb1696110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.622ex; height:3.009ex;" alt="{\displaystyle {\overline {AB}}}"></span>就是圆的<a href="/wiki/%E5%BC%A6_(%E5%B9%BE%E4%BD%95)" title="弦 (幾何)">弦</a>。 </p> <div class="mw-heading mw-heading3"><h3 id="弧"><span id=".E5.BC.A7"></span>弧</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=6" title="编辑章节：弧"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>圆周上任意两<a href="/wiki/%E7%82%B9" title="点">点</a>间的部分叫做<a href="/wiki/%E5%BC%A7" title="弧">弧</a>（英語：<span lang="en">arc</span>），通常用符号<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \frown }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⌢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \frown }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/904123c3e03f3f99c5acbecf3cdf08b8f8438b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.281ex; margin-bottom: -0.452ex; width:2.324ex; height:1.343ex;" alt="{\displaystyle \frown }"></span>表示。弧分为半圆、优弧、劣弧三种。<sup id="cite_ref-数学_6-1" class="reference"><a href="#cite_note-数学-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="直径、半径"><span id=".E7.9B.B4.E5.BE.84.E3.80.81.E5.8D.8A.E5.BE.84"></span>直径、半径</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=7" title="编辑章节：直径、半径"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>直径（英語：<span lang="en">diameter</span>）：经过圆心的<a href="/wiki/%E5%BC%A6_(%E5%B9%BE%E4%BD%95)" title="弦 (幾何)">弦</a>稱作直径（用<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>表示）。<sup id="cite_ref-高中数学必修1_2-1" class="reference"><a href="#cite_note-高中数学必修1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>半径（英語：<span lang="en">radius</span>）：在圆中，连接圆心和圆上任意一点的线段叫做圆的半径，半径用字母<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>表示。</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\{X\in E\mid {}{\overline {MX}}<=r\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo>∈</mo> <mi>E</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>X</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo><=</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\{X\in E\mid {}{\overline {MX}}<=r\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb3869920767dbb074dd8bac490dee10a4dff6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.697ex; height:3.509ex;" alt="{\displaystyle k=\{X\in E\mid {}{\overline {MX}}<=r\}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="切线"><span id=".E5.88.87.E7.BA.BF"></span>切线</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=8" title="编辑章节：切线"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%88%87%E7%BA%BF" title="切线">切线</a></div> <p>假如一条直线与圆相交僅有一个交点，那么称这条直线是这个圆的<a href="/wiki/%E5%88%87%E7%BA%BF" title="切线">切线</a>，与圆相交的<a href="/wiki/%E7%82%B9" title="点">点</a>叫做切点。<sup id="cite_ref-高中数学必修1_2-2" class="reference"><a href="#cite_note-高中数学必修1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>如下图，<a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {QP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>Q</mi> <mi>P</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {QP}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67acd6b6d988d81e65adf6a7f59a7445c0e4a1b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.863ex; height:3.343ex;" alt="{\displaystyle {\overline {QP}}}"></span>与圆只有一个交点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>，那么<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {QP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>Q</mi> <mi>P</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {QP}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67acd6b6d988d81e65adf6a7f59a7445c0e4a1b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.863ex; height:3.343ex;" alt="{\displaystyle {\overline {QP}}}"></span>就是圆的<a href="/wiki/%E5%88%87%E7%BA%BF" title="切线">切线</a>。过圆上一点的切线：设该点为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{o},y_{o})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x_{o},y_{o})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb893997bf6289af9b20454df2de7ffc1196c3b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.117ex; height:2.843ex;" alt="{\displaystyle P(x_{o},y_{o})}"></span>，圆的方程为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8888bb9e8d52fd12fbc05a98715f64c105a884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.162ex; height:3.176ex;" alt="{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}"></span>，则圆在该点的切线方程为：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{o}-a)(x-a)+(y_{o}-b)(y-b)=r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{o}-a)(x-a)+(y_{o}-b)(y-b)=r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df73ccfcdfcd15cfdafe41f84284c7b9b46e30dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.109ex; height:3.176ex;" alt="{\displaystyle (x_{o}-a)(x-a)+(y_{o}-b)(y-b)=r^{2}}"></span> </p> <ul><li>性质定理：圆的切线垂直于经过切点的半径。</li> <li>推论1：经过圆心且垂直于<a href="/wiki/%E5%88%87%E7%BA%BF" title="切线">切线</a>的<a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a>必经过切点。</li> <li>推论2：经过切点且垂直于切线的直线必经过圆心。</li></ul> <div class="mw-heading mw-heading3"><h3 id="割线"><span id=".E5.89.B2.E7.BA.BF"></span>割线</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=9" title="编辑章节：割线"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一条<a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a>与一条弧线有两个公共点，这条直线是这条曲线的割线（英語：<span lang="en">Secant Theorem</span>）。<sup id="cite_ref-高中数学必修1_2-3" class="reference"><a href="#cite_note-高中数学必修1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>如图，直线<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {QO}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>Q</mi> <mi>O</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {QO}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c173f6b7ec73400ae97142c46b536b17a3758301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.727ex; height:3.343ex;" alt="{\displaystyle {\overline {QO}}}"></span>与圆有两个公共点，那么<a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {QO}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>Q</mi> <mi>O</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {QO}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c173f6b7ec73400ae97142c46b536b17a3758301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.727ex; height:3.343ex;" alt="{\displaystyle {\overline {QO}}}"></span>就是圆的割线。 </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Secant-unit-circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Secant-unit-circle.svg/300px-Secant-unit-circle.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Secant-unit-circle.svg/450px-Secant-unit-circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Secant-unit-circle.svg/600px-Secant-unit-circle.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>θ 的正割是从O到Q的距离。</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="周长"><span id=".E5.91.A8.E9.95.BF"></span>周长</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=10" title="编辑章节：周长"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>圆的一周的长度称为圆的<a href="/wiki/%E5%91%A8%E9%95%BF" title="周长">周长</a>（记作<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>）。圆的周长与半径的关系是： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\pi d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>π</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\pi d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96831e95e6c0012cf052dc01e823f623d8e811ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.413ex; height:2.176ex;" alt="{\displaystyle C=\pi d}"></span> 或 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=2\pi r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=2\pi r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398305eb631c365e21449ec4e9c9d7dca3f8c788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.408ex; height:2.176ex;" alt="{\displaystyle C=2\pi r}"></span>，</dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>是<a href="/wiki/%E5%9C%86%E5%91%A8%E7%8E%87" class="mw-redirect" title="圆周率">圆周率</a>。 </p> <div class="mw-heading mw-heading3"><h3 id="面积"><span id=".E9.9D.A2.E7.A7.AF"></span>面积</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=11" title="编辑章节：面积"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%9C%86%E7%9A%84%E9%9D%A2%E7%A7%AF" title="圆的面积">圆的面积</a>与半径的关系是：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f7b7f93f93e7ba7bebb97efbe88e181ce332e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.276ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}}"></span>。 </p> <div class="mw-heading mw-heading3"><h3 id="对称性"><span id=".E5.AF.B9.E7.A7.B0.E6.80.A7"></span>对称性</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=12" title="编辑章节：对称性"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>圆既是<a href="/wiki/%E8%BD%B4%E5%AF%B9%E7%A7%B0%E5%9B%BE%E5%BD%A2" class="mw-redirect" title="轴对称图形">轴对称图形</a>又是<a href="/wiki/%E4%B8%AD%E5%BF%83%E5%AF%B9%E7%A7%B0%E5%9B%BE%E5%BD%A2" title="中心对称图形">中心对称图形</a>，圆的对称轴为经过圆心<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>的任意<a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a>，圆的对称中心为圆心<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>。<sup id="cite_ref-数学_6-2" class="reference"><a href="#cite_note-数学-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="圓心角、圆周角"><span id=".E5.9C.93.E5.BF.83.E8.A7.92.E3.80.81.E5.9C.86.E5.91.A8.E8.A7.92"></span>圓心角、圆周角</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=13" title="编辑章节：圓心角、圆周角"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">更多信息：<a href="/wiki/%E6%89%87%E5%BD%A2" title="扇形">扇形</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sehnentangentenwinkel.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Sehnentangentenwinkel.png/250px-Sehnentangentenwinkel.png" decoding="async" width="250" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Sehnentangentenwinkel.png/375px-Sehnentangentenwinkel.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Sehnentangentenwinkel.png/500px-Sehnentangentenwinkel.png 2x" data-file-width="2312" data-file-height="2354" /></a><figcaption>图2：弦、圆周角、圆心角</figcaption></figure> <ul><li>圆心角：<a href="/wiki/%E9%A0%82%E9%BB%9E_(%E5%B9%BE%E4%BD%95)" title="頂點 (幾何)">顶点</a>在圆心的<a href="/wiki/%E8%A7%92" title="角">角</a>叫圆心角，圆心角的度数等于它所对的弧的度数，公式表示为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {L}{2\pi r}}\cdot 2\pi ={\frac {L}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mn>2</mn> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {L}{2\pi r}}\cdot 2\pi ={\frac {L}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b21d366afa0b3105da26d25019661032e538533" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.259ex; height:5.176ex;" alt="{\displaystyle \theta ={\frac {L}{2\pi r}}\cdot 2\pi ={\frac {L}{r}}}"></span>。<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-高中数学必修1_2-4" class="reference"><a href="#cite_note-高中数学必修1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>如右图，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>为圆的圆心，那么<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle AMB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>M</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle AMB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4501803baa0a10cc7347603028ab987fc1c9d889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.627ex; height:2.176ex;" alt="{\displaystyle \angle AMB}"></span>为圆心角。</li> <li>圆周角：<a href="/wiki/%E9%A0%82%E9%BB%9E_(%E5%B9%BE%E4%BD%95)" title="頂點 (幾何)">顶点</a>在圆周上，<a href="/wiki/%E8%A7%92" title="角">角</a>两边和圆相交的角叫圆周角。如右图，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle ACB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ACB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cb3ffaff70a9b25c9564d8ef5d9f801222754c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.952ex; height:2.176ex;" alt="{\displaystyle \angle ACB}"></span>的顶点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>在圆周上，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle ACB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ACB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cb3ffaff70a9b25c9564d8ef5d9f801222754c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.952ex; height:2.176ex;" alt="{\displaystyle \angle ACB}"></span>的两边<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea69bf8ccc972402f794c261e00b222f396bcce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.692ex; height:3.009ex;" alt="{\displaystyle {\overline {AC}}}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650fedad268814c3e14377171e917b73a0698ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.713ex; height:3.009ex;" alt="{\displaystyle {\overline {BC}}}"></span>分别交在圆周上，那么<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle ACB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ACB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cb3ffaff70a9b25c9564d8ef5d9f801222754c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.952ex; height:2.176ex;" alt="{\displaystyle \angle ACB}"></span>就是圆周角。</li></ul> <div class="mw-heading mw-heading3"><h3 id="圆心角定理"><span id=".E5.9C.86.E5.BF.83.E8.A7.92.E5.AE.9A.E7.90.86"></span>圆心角定理</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=14" title="编辑章节：圆心角定理"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>同圆或等圆中，相等的圆心角所对的<a href="/wiki/%E5%BC%A6_(%E5%B9%BE%E4%BD%95)" title="弦 (幾何)">弦</a>相等，所对的<a href="/wiki/%E5%BC%A7" title="弧">弧</a>相等，弦心距<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup>相等，此定理也称“一推三定理”。<sup id="cite_ref-数学_6-3" class="reference"><a href="#cite_note-数学-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="圆周角定理"><span id=".E5.9C.86.E5.91.A8.E8.A7.92.E5.AE.9A.E7.90.86"></span>圆周角定理</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=15" title="编辑章节：圆周角定理"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>圆周角定理：同弧所对的圆周角等于它所对的圆心的<a href="/wiki/%E8%A7%92" title="角">角</a>的一半。<sup id="cite_ref-数学_6-4" class="reference"><a href="#cite_note-数学-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <br />如上图，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>为圆心，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}"></span>分别为圆周上的<a href="/wiki/%E7%82%B9" title="点">点</a>，那麼：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle AMB=2\;\angle ACB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>M</mi> <mi>B</mi> <mo>=</mo> <mn>2</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle AMB=2\;\angle ACB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2683c6b7072b934976086c7898ab817bdce7d2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.485ex; height:2.176ex;" alt="{\displaystyle \angle AMB=2\;\angle ACB}"></span> </p> <dl><dd>证明：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \because BM=CM,AM=CM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∵</mo> <mi>B</mi> <mi>M</mi> <mo>=</mo> <mi>C</mi> <mi>M</mi> <mo>,</mo> <mi>A</mi> <mi>M</mi> <mo>=</mo> <mi>C</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \because BM=CM,AM=CM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a96b5d213cc06c78e857b898751581609823884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.235ex; height:2.509ex;" alt="{\displaystyle \because BM=CM,AM=CM}"></span> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \because \angle BCM=\angle CBM,\angle ACM=\angle CAM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∵</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>M</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>M</mi> <mo>,</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>M</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>A</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \because \angle BCM=\angle CBM,\angle ACM=\angle CAM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a14fdeda08f798fa1deb24dc653469e8c768c5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.987ex; height:2.509ex;" alt="{\displaystyle \because \angle BCM=\angle CBM,\angle ACM=\angle CAM}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \therefore \angle BMS=\angle BCM+\angle CBM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∴</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>M</mi> <mi>S</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>M</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \therefore \angle BMS=\angle BCM+\angle CBM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b8cdc6d9ffac4ee664d1c2a845f50b7f59effa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.819ex; height:2.343ex;" alt="{\displaystyle \therefore \angle BMS=\angle BCM+\angle CBM}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \because \angle AMS=\angle ACM+\angle CAM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∵</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>M</mi> <mi>S</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>M</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>A</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \because \angle AMS=\angle ACM+\angle CAM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a553e5b5fa1c21c5d0ffb6cc9fd7b6f40a082d07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.756ex; height:2.343ex;" alt="{\displaystyle \because \angle AMS=\angle ACM+\angle CAM}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \therefore \angle BMS+\angle AMS=2(\angle BCM+\angle ACM)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∴</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>M</mi> <mi>S</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>M</mi> <mi>S</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>M</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \therefore \angle BMS+\angle AMS=2(\angle BCM+\angle ACM)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/725a82851334c06b635661c89df8548ea2766976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.973ex; height:2.843ex;" alt="{\displaystyle \therefore \angle BMS+\angle AMS=2(\angle BCM+\angle ACM)}"></span></dd> <dd>即：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle AMB=2\;\angle ACB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>M</mi> <mi>B</mi> <mo>=</mo> <mn>2</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle AMB=2\;\angle ACB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2683c6b7072b934976086c7898ab817bdce7d2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.485ex; height:2.176ex;" alt="{\displaystyle \angle AMB=2\;\angle ACB}"></span></dd></dl></dd></dl></dd></dl> <p>圆周角定理的推论： </p> <ol><li>同弧或等<a href="/wiki/%E5%BC%A7" title="弧">弧</a>所对的圆周角相等；同圆或等圆中，相等的圆周<a href="/wiki/%E8%A7%92" title="角">角</a>所对的弧是等弧。</li> <li>半圆或直径所对的圆周角是<a href="/wiki/%E7%9B%B4%E8%A7%92" title="直角">直角</a>；圆周角是<a href="/wiki/%E7%9B%B4%E8%A7%92" title="直角">直角</a>所对的弧的半圆，所对的弦是直径。</li> <li>若<a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形">三角形</a>一边上的<a href="/wiki/%E4%B8%AD%E7%BA%BF" class="mw-redirect" title="中线">中线</a>等于这边的一半，那么这个三角形是<a href="/wiki/%E7%9B%B4%E8%A7%92%E4%B8%89%E8%A7%92%E5%BD%A2" title="直角三角形">直角三角形</a>。</li></ol> <div class="mw-heading mw-heading2"><h2 id="垂径定理"><span id=".E5.9E.82.E5.BE.84.E5.AE.9A.E7.90.86"></span>垂径定理</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=16" title="编辑章节：垂径定理"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%9E%82%E5%BE%84%E5%AE%9A%E7%90%86" title="垂径定理">垂径定理</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:%E5%9E%82%E5%BE%84%E5%AE%9A%E7%90%86.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/%E5%9E%82%E5%BE%84%E5%AE%9A%E7%90%86.jpg/300px-%E5%9E%82%E5%BE%84%E5%AE%9A%E7%90%86.jpg" decoding="async" width="300" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/%E5%9E%82%E5%BE%84%E5%AE%9A%E7%90%86.jpg/450px-%E5%9E%82%E5%BE%84%E5%AE%9A%E7%90%86.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/b/bd/%E5%9E%82%E5%BE%84%E5%AE%9A%E7%90%86.jpg 2x" data-file-width="479" data-file-height="317" /></a><figcaption>垂径定理示意图</figcaption></figure> <p>垂径定理是一种常用的<b><a href="/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6" title="几何学">几何学</a></b>的<a href="/wiki/%E5%AE%9A%E7%90%86" title="定理">定理</a>。 </p><p>定理定义：垂直于弦的<a href="/wiki/%E7%9B%B4%E5%BE%84" title="直径">直径</a>平分这条弦，并且平分弦所对的两条<a href="/wiki/%E5%BC%A7" title="弧">弧</a>。<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="知二推三"><span id=".E7.9F.A5.E4.BA.8C.E6.8E.A8.E4.B8.89"></span>知二推三</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=17" title="编辑章节：知二推三"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一条直线，在下列5条中只要具备其中任意两条作为条件，就可以推出其他三条结论。称为“知二推三”。 </p> <ul><li>平分弦所对的优弧</li> <li>平分弦所对的劣弧（前两条合起来就是平分弦所对的两条弧）</li> <li>平分弦（不是直径）</li> <li>垂直于弦</li> <li>经过圆心</li></ul> <div class="mw-heading mw-heading3"><h3 id="推论"><span id=".E6.8E.A8.E8.AE.BA"></span>推论</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=18" title="编辑章节：推论"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>BE过<a href="/wiki/%E5%9C%93%E5%BF%83" class="mw-redirect" title="圓心">圆心</a>O，AD=DC，则BE垂直AC并平分AC、AEC两条弧。即“平分<b>非直径</b>的弦的直径垂直于弦并平分弦所对的两弧。”</li> <li>AD=DC且BE垂直AC，则BE过圆心O且平分AC、AEC两条弧。即“弦的垂直平分线过圆心且平分弦所对的两弧。”</li> <li>BE是<a href="/wiki/%E7%9B%B4%E5%BE%84" title="直径">直径</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\frown }{AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4608fbf7084ad30e15487e824daf74260640f810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{AB}}}"></span>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\frown }{AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{AE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981431a501cfd72ead5457ab534a6614b2c385ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.519ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{AE}}}"></span>）＝<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\frown }{BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d5cffe92c89482266f356039a49ba3d4333277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{BC}}}"></span>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\frown }{CE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>E</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{CE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a370485c5a32c42ba4217cd8cf705afc83b20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.542ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{CE}}}"></span>），则BE过圆心O，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\frown }{AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{AE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981431a501cfd72ead5457ab534a6614b2c385ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.519ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{AE}}}"></span>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\frown }{AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4608fbf7084ad30e15487e824daf74260640f810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{AB}}}"></span>）＝<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\frown }{CE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>E</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{CE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a370485c5a32c42ba4217cd8cf705afc83b20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.542ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{CE}}}"></span>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {\frown }{BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d5cffe92c89482266f356039a49ba3d4333277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{BC}}}"></span>）。即“平分弦所对的一条弧的直径垂直平分弦且平分弦所对的另一条弧。”</li></ol> <div class="mw-heading mw-heading2"><h2 id="兩圓位置關係"><span id=".E5.85.A9.E5.9C.93.E4.BD.8D.E7.BD.AE.E9.97.9C.E4.BF.82"></span>兩圓位置關係</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=19" title="编辑章节：兩圓位置關係"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Two_circles.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/Two_circles.png" decoding="async" width="268" height="263" class="mw-file-element" data-file-width="268" data-file-height="263" /></a><figcaption></figcaption></figure> <p>兩個不同大小的圓（半徑分別為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>及<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>，圓心距為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>，其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r<R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda3f5af44b388094e37b9ddeb9524a3433ef30c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle r<R}"></span>）之間的關係如下：<sup id="cite_ref-高中数学必修1_2-5" class="reference"><a href="#cite_note-高中数学必修1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87f7389ad2498c0f93551ec4fc92a882548484f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=0}"></span>：兩圓不相交（內含），互為<a href="/wiki/%E5%90%8C%E5%BF%83%E5%9C%93" class="mw-redirect" title="同心圓">同心圓</a>。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<d<R-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>d</mi> <mo><</mo> <mi>R</mi> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<d<R-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29b9e57c16dc2b0d266de45fbd147f83e755d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.228ex; height:2.343ex;" alt="{\displaystyle 0<d<R-r}"></span>：兩圓不相交（內含，亦稱「內離」）。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=R-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>R</mi> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=R-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47dbff89e2666bb9e3dd6423ec276d04b6b25fd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.967ex; height:2.343ex;" alt="{\displaystyle d=R-r}"></span>：兩圓相交於一點（內切），有1條共同切線。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=R+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=R+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3976c94c0b3757e834461e0ce7a80a3080ba3dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.967ex; height:2.343ex;" alt="{\displaystyle d=R+r}"></span>：兩圓相交於一點（外切），有3條共同切線。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R-r<d<R+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>−</mo> <mi>r</mi> <mo><</mo> <mi>d</mi> <mo><</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R-r<d<R+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/910fa39bab4755ade44805de6f6744e39eef6f04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.719ex; height:2.343ex;" alt="{\displaystyle R-r<d<R+r}"></span>：兩圓相交於兩點，有2條共同切線。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d>R+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>></mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d>R+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598e8f4026358b0d63a245b4d16fd09c5dca4702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.967ex; height:2.343ex;" alt="{\displaystyle d>R+r}"></span>：兩圓不相交（外離），有4條共同切線。</li></ol> <div class="mw-heading mw-heading3"><h3 id="圆系方程"><span id=".E5.9C.86.E7.B3.BB.E6.96.B9.E7.A8.8B"></span>圆系方程</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=20" title="编辑章节：圆系方程"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%9C%86%E7%B3%BB%E6%96%B9%E7%A8%8B" title="圆系方程">圆系方程</a></div> <p>在解析几何中，符合特定条件的某些圆构成一个<a href="/w/index.php?title=%E5%9C%86%E7%B3%BB&action=edit&redlink=1" class="new" title="圆系（页面不存在）">圆系</a>，一个圆系所具有的共同形式的方程称为圆系方程。例如求半径到直线距离的方程就可以叫圆系方程。<sup id="cite_ref-高中数学必修1_2-6" class="reference"><a href="#cite_note-高中数学必修1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <br />在方程<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8888bb9e8d52fd12fbc05a98715f64c105a884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.162ex; height:3.176ex;" alt="{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}"></span>中，若圆心<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span>为定点，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>为参变数，则它表示<a href="/wiki/%E5%90%8C%E5%BF%83%E5%9C%86" class="mw-redirect" title="同心圆">同心圆</a>的圆系<a href="/wiki/%E6%96%B9%E7%A8%8B" title="方程">方程</a>。若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>是常量，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>（或<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>）为参变数，则它表示半径相同，圆心在同一直线上（平行于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>轴或<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>轴）的圆系方程。 </p> <ul><li>过两圆<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c19064464bbbe4eedb0bb5033566dd8b789635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.003ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}"></span>与<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e973da6f9c964cf717300f2592b3cb73460e202a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.003ex; height:3.176ex;" alt="{\displaystyle x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2}=0}"></span>交点的圆系方程为： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}+\lambda (x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2})=0\quad (\lambda \neq -1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}+\lambda (x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2})=0\quad (\lambda \neq -1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe6a1d7dfa1e6818785f1bc922ec59afe3c2d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:75.953ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}+\lambda (x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2})=0\quad (\lambda \neq -1).}"></span></dd></dl></li> <li>过直线<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax+By+C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax+By+C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e84e63a5c044f46835839af53ccbe7a695187da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.7ex; height:2.509ex;" alt="{\displaystyle Ax+By+C=0}"></span>与圆<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c19064464bbbe4eedb0bb5033566dd8b789635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.003ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}"></span>交点的圆系方程为： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}+\lambda (Ax+By+C)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}+\lambda (Ax+By+C)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc9efa2f59252566a70bed310adaa63daa7de75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.094ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}+\lambda (Ax+By+C)=0.}"></span></dd></dl></li> <li>过两圆<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c19064464bbbe4eedb0bb5033566dd8b789635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.003ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}=0}"></span>与<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e973da6f9c964cf717300f2592b3cb73460e202a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.003ex; height:3.176ex;" alt="{\displaystyle x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2}=0}"></span>交点的直线方程为： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}-(x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2})=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}-(x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2})=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1933bd7f124502c8c1287d5c470c75115eacc79e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.042ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+y_{1}^{2}+D_{1}x+E_{1}y+F_{1}-(x_{2}^{2}+y_{2}^{2}+D_{2}x+E_{2}y+F_{2})=0.}"></span></dd></dl></li></ul> <div class="mw-heading mw-heading2"><h2 id="其他定义"><span id=".E5.85.B6.E4.BB.96.E5.AE.9A.E4.B9.89"></span>其他定义</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=21" title="编辑章节：其他定义"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">更多信息：<a href="/wiki/%E6%A4%AD%E5%9C%86" title="椭圆">椭圆</a>和<a href="/wiki/%E7%A6%BB%E5%BF%83%E7%8E%87" class="mw-redirect" title="离心率">离心率</a></div> <ul><li><a href="/wiki/%E6%A4%AD%E5%9C%86" title="椭圆">椭圆</a>是<a href="/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学)">平面</a>上到两个固定点的距离之和为<a href="/wiki/%E5%B8%B8%E6%95%B0" title="常数">常数</a>的点之轨迹，椭圆的形状可以用<a href="/wiki/%E7%A6%BB%E5%BF%83%E7%8E%87" class="mw-redirect" title="离心率">离心率</a>来表示；圆可以看作是一种特殊的椭圆，即当椭圆的两个<a href="/wiki/%E7%84%A6%E9%BB%9E" title="焦點">焦点</a>重合，<a href="/wiki/%E7%A6%BB%E5%BF%83%E7%8E%87" class="mw-redirect" title="离心率">离心率</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbe653d369abd71fc0be98efbf59e9edc1bdfb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon =0}"></span>的情况。</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">更多信息：<a href="/wiki/%E7%AB%8B%E4%BD%93%E5%87%A0%E4%BD%95" title="立体几何">立体几何</a>和<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97" title="欧几里得">欧几里得</a></div> <ul><li>在<a href="/wiki/%E7%AB%8B%E9%AB%94%E5%B9%BE%E4%BD%95" class="mw-redirect" title="立體幾何">三維空間</a>，球面被設定為是在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab2376f6a3a3b77ddc94ade4f6fbc96e85ca29b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{3}}"></span>空間中與一個定點距離為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>的所有<a href="/wiki/%E9%BB%9E" class="mw-redirect" title="點">點</a>的集合，此處r是一個正的<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a>，稱為半徑，固定的點稱為球心或中心，並且不屬於球面的範圍。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}"></span>是球的特例，稱為單位球。</li> <li>在測度<a href="/wiki/%E7%A9%BA%E9%96%93" title="空間">空間</a>中，圓的定義仍舊指距離一定點等距（在該測度下）的點的<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>。</li></ul> <div class="mw-heading mw-heading2"><h2 id="其它"><span id=".E5.85.B6.E5.AE.83"></span>其它</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=22" title="编辑章节：其它"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="相關的立体图形"><span id=".E7.9B.B8.E9.97.9C.E7.9A.84.E7.AB.8B.E4.BD.93.E5.9B.BE.E5.BD.A2"></span>相關的立体图形</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=23" title="编辑章节：相關的立体图形"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E6%88%AA%E9%9D%A2_(%E5%B9%BE%E4%BD%95)" title="截面 (幾何)">截面</a>為圓的<a href="/wiki/%E4%B8%89%E7%B6%AD%E7%A9%BA%E9%96%93" title="三維空間">三維</a><a href="/wiki/%E5%BD%A2%E7%8B%80" title="形狀">形狀</a>有： </p> <ul><li><a href="/wiki/%E7%90%83%E9%AB%94" class="mw-redirect" title="球體">球體</a></li> <li><a href="/wiki/%E6%89%81%E7%90%83%E9%AB%94" class="mw-redirect" title="扁球體">扁球體</a></li> <li><a href="/wiki/%E5%9C%86%E9%94%A5%E4%BD%93" class="mw-redirect" title="圆锥体">圓錐體</a></li> <li><a href="/wiki/%E5%9C%86%E6%9F%B1%E4%BD%93" title="圆柱体">圓柱體</a></li> <li><a href="/wiki/%E5%9C%86%E5%8F%B0" title="圆台">圆台</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="圓和其他平面形狀"><span id=".E5.9C.93.E5.92.8C.E5.85.B6.E4.BB.96.E5.B9.B3.E9.9D.A2.E5.BD.A2.E7.8B.80"></span>圓和其他平面形狀</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=24" title="编辑章节：圓和其他平面形狀"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%A4%96%E6%8E%A5%E5%9C%93" title="外接圓">外接圓</a></li> <li><a href="/wiki/%E5%85%A7%E5%88%87%E5%9C%93" class="mw-redirect" title="內切圓">內切圓</a></li> <li><a href="/wiki/%E6%97%81%E5%88%87%E5%9C%93" title="旁切圓">旁切圓</a></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E7%AD%89%E5%91%A8%E5%AE%9A%E7%90%86" title="等周定理">等周定理</a></div> <ul><li>當多邊形的每條邊固定，以有外接圓的圖形<a href="/wiki/%E9%9D%A2%E7%A7%AF" title="面积">面积</a>最大。<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="圓的問題"><span id=".E5.9C.93.E7.9A.84.E5.95.8F.E9.A1.8C"></span>圓的問題</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=25" title="编辑章节：圓的問題"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%8C%96%E5%9C%93%E7%82%BA%E6%96%B9" title="化圓為方">化圓為方</a>是指用<a href="/wiki/%E5%B0%BA%E8%A7%84%E4%BD%9C%E5%9B%BE" title="尺规作图">尺规作图</a>的方法畫出一個和已知圓面積相同的正方形。已经证明这是不可能的。<sup id="cite_ref-clj_11-0" class="reference"><a href="#cite_note-clj-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/%E5%A1%94%E6%96%AF%E5%9F%BA%E5%88%86%E5%89%B2%E5%9C%93%E5%95%8F%E9%A1%8C" title="塔斯基分割圓問題">塔斯基分割圓問題</a>要求用分割的方法來使已知圓變成正方形。</li> <li><a href="/wiki/%E9%AB%98%E6%96%AF%E5%9C%93%E5%95%8F%E9%A1%8C" title="高斯圓問題">高斯圆问题</a>。</li></ul> <div class="mw-heading mw-heading2"><h2 id="参考资料"><span id=".E5.8F.82.E8.80.83.E8.B5.84.E6.96.99"></span>参考资料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=26" title="编辑章节：参考资料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="注释"><span id=".E6.B3.A8.E9.87.8A"></span>注释</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=27" title="编辑章节：注释"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="notelist" style="list-style-type: lower-alpha;"> <ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">L为<a href="/wiki/%E6%89%87%E5%BD%A2" title="扇形">扇形</a><a href="/wiki/%E5%BC%A7" title="弧">弧</a>长，变形公式<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=r\cdot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=r\cdot \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567df00fda8040b74a4907b66e134e82d585d7c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.5ex; height:2.176ex;" alt="{\displaystyle L=r\cdot \theta }"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">弦心距指的是<a href="/wiki/%E5%9C%93%E5%BF%83" class="mw-redirect" title="圓心">圆心</a>到<a href="/wiki/%E5%BC%A6_(%E5%B9%BE%E4%BD%95)" title="弦 (幾何)">弦</a>的距离</span> </li> </ol> </div> <div class="mw-heading mw-heading3"><h3 id="资料"><span id=".E8.B5.84.E6.96.99"></span>资料</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=28" title="编辑章节：资料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-几何原本-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-几何原本_1-0">^</a></b></span> <span class="reference-text"><cite class="citation book">欧几里得[原著]/燕晓东（译）. 几何原本. 南京: 江苏人民出版社. 2014. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9787214067593" title="Special:网络书源/9787214067593"><span title="国际标准书号">ISBN</span> 9787214067593</a>. <q>圆是一个在同一平面内到定点的距离等于定长的点的集合，这个定点就是圆心。</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.au=%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%5B%E5%8E%9F%E8%91%97%5D%2F%E7%87%95%E6%99%93%E4%B8%9C%EF%BC%88%E8%AF%91%EF%BC%89&rft.btitle=%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC&rft.date=2014&rft.genre=book&rft.isbn=9787214067593&rft.place=%E5%8D%97%E4%BA%AC&rft.pub=%E6%B1%9F%E8%8B%8F%E4%BA%BA%E6%B0%91%E5%87%BA%E7%89%88%E7%A4%BE&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-高中数学必修1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-高中数学必修1_2-0"><sup><b>2.0</b></sup></a> <a href="#cite_ref-高中数学必修1_2-1"><sup><b>2.1</b></sup></a> <a href="#cite_ref-高中数学必修1_2-2"><sup><b>2.2</b></sup></a> <a href="#cite_ref-高中数学必修1_2-3"><sup><b>2.3</b></sup></a> <a href="#cite_ref-高中数学必修1_2-4"><sup><b>2.4</b></sup></a> <a href="#cite_ref-高中数学必修1_2-5"><sup><b>2.5</b></sup></a> <a href="#cite_ref-高中数学必修1_2-6"><sup><b>2.6</b></sup></a></span> <span class="reference-text"><cite class="citation book"><a rel="nofollow" class="external text" href="http://www.pep.com.cn">高中数学必修1</a>. 北京: 人民教育出版社. 2014 <span class="reference-accessdate"> [<span class="nowrap">2020-10-04</span>]</span>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9787107177057" title="Special:网络书源/9787107177057"><span title="国际标准书号">ISBN</span> 9787107177057</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20170613074807/http://www.pep.com.cn/">存档</a>于2017-06-13）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.btitle=%E9%AB%98%E4%B8%AD%E6%95%B0%E5%AD%A6%E5%BF%85%E4%BF%AE1&rft.date=2014&rft.genre=book&rft.isbn=9787107177057&rft.place=%E5%8C%97%E4%BA%AC&rft.pub=%E4%BA%BA%E6%B0%91%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=http%3A%2F%2Fwww.pep.com.cn&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-历史-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-历史_3-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a rel="nofollow" class="external text" href="http://www.pep.com.cn">历史</a>. 北京: 人民教育出版社. 2014 <span class="reference-accessdate"> [<span class="nowrap">2020-10-04</span>]</span>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9787107155598" title="Special:网络书源/9787107155598"><span title="国际标准书号">ISBN</span> 9787107155598</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20170613074807/http://www.pep.com.cn/">存档</a>于2017-06-13）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.btitle=%E5%8E%86%E5%8F%B2&rft.date=2014&rft.genre=book&rft.isbn=9787107155598&rft.place=%E5%8C%97%E4%BA%AC&rft.pub=%E4%BA%BA%E6%B0%91%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=http%3A%2F%2Fwww.pep.com.cn&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-圆的历史-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-圆的历史_4-0"><sup><b>4.0</b></sup></a> <a href="#cite_ref-圆的历史_4-1"><sup><b>4.1</b></sup></a> <a href="#cite_ref-圆的历史_4-2"><sup><b>4.2</b></sup></a></span> <span class="reference-text"><cite class="citation book"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20211121223914/https://wenku.baidu.com/link?url=uA8UNKJKwViV0tguKcf3EuIR-rF48sNP0d1m9bJORQr6Sk7erhfn2IAO2xhgp16nyaeAjxIAt-7IK9EWbsAj5t45IF6J6g9U6bxz0hDJeUy">圆的历史</a>. <span class="reference-accessdate"> [<span class="nowrap">2015-08-25</span>]</span>. （<a rel="nofollow" class="external text" href="http://wenku.baidu.com/link?url=uA8UNKJKwViV0tguKcf3EuIR-rF48sNP0d1m9bJORQr6Sk7erhfn2IAO2xhgp16nyaeAjxIAt-7IK9EWbsAj5t45IF6J6g9U6bxz0hDJeUy">原始内容</a>存档于2021-11-21）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.btitle=%E5%9C%86%E7%9A%84%E5%8E%86%E5%8F%B2&rft.genre=book&rft_id=http%3A%2F%2Fwenku.baidu.com%2Flink%3Furl%3DuA8UNKJKwViV0tguKcf3EuIR-rF48sNP0d1m9bJORQr6Sk7erhfn2IAO2xhgp16nyaeAjxIAt-7IK9EWbsAj5t45IF6J6g9U6bxz0hDJeUy&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"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304142718/http://www.cnposts.com/Translation/6404.aspx">古代人是如何搬运重物的？</a>. <span class="reference-accessdate"> [<span class="nowrap">2015-08-25</span>]</span>. （<a rel="nofollow" class="external text" href="http://www.cnposts.com/Translation/6404.aspx">原始内容</a>存档于2016-03-04）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.btitle=%E5%8F%A4%E4%BB%A3%E4%BA%BA%E6%98%AF%E5%A6%82%E4%BD%95%E6%90%AC%E8%BF%90%E9%87%8D%E7%89%A9%E7%9A%84%EF%BC%9F&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.cnposts.com%2FTranslation%2F6404.aspx&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-数学-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-数学_6-0"><sup><b>6.0</b></sup></a> <a href="#cite_ref-数学_6-1"><sup><b>6.1</b></sup></a> <a href="#cite_ref-数学_6-2"><sup><b>6.2</b></sup></a> <a href="#cite_ref-数学_6-3"><sup><b>6.3</b></sup></a> <a href="#cite_ref-数学_6-4"><sup><b>6.4</b></sup></a></span> <span class="reference-text"><cite class="citation book"><a rel="nofollow" class="external text" href="http://www.pep.com.cn">数学</a>. 北京: 北京师范大学出版社. 2014 <span class="reference-accessdate"> [<span class="nowrap">2020-10-04</span>]</span>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9787303136933" title="Special:网络书源/9787303136933"><span title="国际标准书号">ISBN</span> 9787303136933</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20170613074807/http://www.pep.com.cn/">存档</a>于2017-06-13）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.btitle=%E6%95%B0%E5%AD%A6&rft.date=2014&rft.genre=book&rft.isbn=9787303136933&rft.place=%E5%8C%97%E4%BA%AC&rft.pub=%E5%8C%97%E4%BA%AC%E5%B8%88%E8%8C%83%E5%A4%A7%E5%AD%A6%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=http%3A%2F%2Fwww.pep.com.cn&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"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97" title="欧几里得">欧几里得</a>. 第I卷第12个命题. 几何原本.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.atitle=%E7%AC%ACI%E5%8D%B7%E7%AC%AC12%E4%B8%AA%E5%91%BD%E9%A2%98&rft.au=%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97&rft.btitle=%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC&rft.genre=bookitem&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"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">J. Steiner, <i>Einfacher Beweis der isoperimetrischen Hauptsätze</i>, J. reine angew Math. <b>18</b>, (1838), pp. 281–296; and Gesammelte Werke Vol. 2, pp. 77–91, Reimer, Berlin, (1882).</span> </li> <li id="cite_note-clj-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-clj_11-0">^</a></b></span> <span class="reference-text"><cite class="citation web">曹亮吉. <a rel="nofollow" class="external text" href="http://episte.math.ntu.edu.tw/articles/sm/sm_09_04_1/index.html">《三等分任意角可能吗？》</a>. 原載於科學月刊第九卷第四期. <span class="reference-accessdate"> [<span class="nowrap">2015-08-26</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20140623211803/http://episte.math.ntu.edu.tw/articles/sm/sm_09_04_1/index.html">存档</a>于2014-06-23）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.atitle=%E3%80%8A%E4%B8%89%E7%AD%89%E5%88%86%E4%BB%BB%E6%84%8F%E8%A7%92%E5%8F%AF%E8%83%BD%E5%90%97%EF%BC%9F%E3%80%8B&rft.au=%E6%9B%B9%E4%BA%AE%E5%90%89&rft.genre=unknown&rft.jtitle=%E5%8E%9F%E8%BC%89%E6%96%BC%E7%A7%91%E5%AD%B8%E6%9C%88%E5%88%8A%E7%AC%AC%E4%B9%9D%E5%8D%B7%E7%AC%AC%E5%9B%9B%E6%9C%9F&rft_id=http%3A%2F%2Fepiste.math.ntu.edu.tw%2Farticles%2Fsm%2Fsm_09_04_1%2Findex.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=29" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="navigation" aria-label="Portals" class="noprint portal plainlist tright" style="margin:0.5em 0 0.5em 1em;border:solid #aaa 1px"> <ul style="display:table;box-sizing:border-box;padding:0.1em;max-width:175px;background:var(--background-color-base,#f9f9f9);font-size:85%;line-height:110%;font-weight:bold"> <li style="display:table-row"><span style="display:table-cell;padding:0.2em;vertical-align:middle;text-align:center"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span style="display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle"><a href="/wiki/Portal:%E6%95%B0%E5%AD%A6" title="Portal:数学">数学主题</a></span></li></ul></div> <ul><li><a href="/wiki/%E5%9C%86%E9%94%A5%E6%9B%B2%E7%BA%BF" title="圆锥曲线">圆锥曲线</a></li> <li><a href="/wiki/%E7%90%83_(%E6%95%B0%E5%AD%A6)" title="球 (数学)">球 (数学)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="扩展阅读"><span id=".E6.89.A9.E5.B1.95.E9.98.85.E8.AF.BB"></span>扩展阅读</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=30" title="编辑章节：扩展阅读"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book">Pedoe, Dan. <a rel="nofollow" class="external text" href="https://archive.org/details/geometrycomprehe0000pedo">Geometry: a comprehensive course</a>. Dover. 1988.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.au=Pedoe%2C+Dan&rft.btitle=Geometry%3A+a+comprehensive+course&rft.date=1988&rft.genre=book&rft.pub=Dover&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometrycomprehe0000pedo&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/Curves/Circle.html">"Circle" in The MacTutor History of Mathematics archive</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20190407191943/http://www-history.mcs.st-andrews.ac.uk/history/Curves/Circle.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li></ul> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%9C%86&action=edit&section=31" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r82655521">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/%E7%BB%B4%E5%9F%BA%E6%96%87%E5%BA%93" title="维基文库">英文维基文库</a>中的《<a href="/wiki/%E5%A4%A7%E8%8B%B1%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8%E7%AC%AC%E5%8D%81%E4%B8%80%E7%89%88" title="大英百科全書第十一版">1911年版大英百科全書</a>》條目：<b><a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Circle" class="extiw" title="wikisource:1911 Encyclopædia Britannica/Circle">Circle</a></b></div></div> </div> <ul><li><cite id="CITEREFHazewinkel2001" class="citation">Hazewinkel, Michiel (编), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=p/c022260">Circle</a>, <a href="/wiki/%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6" title="数学百科全书">数学百科全书</a>, <a href="/wiki/Springer_Science%2BBusiness_Media" class="mw-redirect" title="Springer Science+Business Media">Springer</a>, 2001, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-1-55608-010-4" title="Special:网络书源/978-1-55608-010-4"><span title="国际标准书号">ISBN</span> 978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.atitle=Circle&rft.aufirst=Michiel&rft.aulast=Hazewinkel&rft.btitle=%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6&rft.date=2001&rft.genre=bookitem&rft.isbn=978-1-55608-010-4&rft.pub=Springer&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dp%2Fc022260&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a href="https://planetmath.org/alphabetical.html" class="extiw" title="planetmath:4236">Circle (PlanetMath.org website)</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Circle"><cite class="citation web"><a href="/wiki/%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0" title="埃里克·韦斯坦因">埃里克·韦斯坦因</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Circle.html">Circle</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%9C%86&rft.atitle=Circle&rft.au=%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0&rft.genre=unknown&rft.jtitle=MathWorld&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FCircle.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/tocs/circlestoc.html">Interactive Java applets</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20170606141014/http://www.mathopenref.com/tocs/circlestoc.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） for the properties of and elementary constructions involving circles.</li> <li><a rel="nofollow" class="external text" href="http://www.mathwarehouse.com/geometry/circle/interactive-circle-equation.php">Interactive Standard Form Equation of Circle</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20161215140155/http://www.mathwarehouse.com/geometry/circle/interactive-circle-equation.php">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） Click and drag points to see standard form equation in action</li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/pythagoras/Munching/circle.shtml">Munching on Circles</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20190317024803/http://www.cut-the-knot.org/pythagoras/Munching/circle.shtml">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li></ul> <div style="clear: both; height: 1em"></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84261037">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output 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scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E7%82%B9" title="点">点</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%A0%82%E9%BB%9E_(%E5%B9%BE%E4%BD%95)" title="頂點 (幾何)">頂點</a></li> <li><a href="/wiki/%E4%BA%A4%E9%BB%9E" title="交點">交點</a></li> <li><a href="/wiki/%E4%B8%AD%E9%BB%9E" title="中點">中點</a></li> <li><a href="/wiki/%E8%A7%92" title="角">角</a> <ul><li><a href="/wiki/%E5%90%8C%E7%95%8C%E8%A7%92" title="同界角">同界角</a></li></ul></li> <li><a href="/wiki/%E6%9E%81%E5%80%BC%E7%82%B9" title="极值点">極值點</a></li> <li><a href="/w/index.php?title=%E6%9C%80%E5%80%BC%E9%BB%9E&action=edit&redlink=1" class="new" title="最值點（页面不存在）">最值點</a></li> <li><a href="/wiki/%E4%B8%B4%E7%95%8C%E7%82%B9_(%E6%95%B0%E5%AD%A6)" title="临界点 (数学)">臨界點</a></li> <li><a href="/wiki/%E9%A9%BB%E7%82%B9" title="驻点">驻点</a></li> <li><a href="/wiki/%E9%9E%8D%E9%BB%9E" title="鞍點">鞍點</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a>和<a href="/wiki/%E6%9B%B2%E7%BA%BF" title="曲线">曲线</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BA%BF%E6%AE%B5" title="线段">线段</a></li> <li><a href="/wiki/%E5%B0%84%E7%BA%BF" title="射线">射线</a></li> <li><a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a></li> <li><a href="/wiki/%E5%88%87%E7%BA%BF" title="切线">切线</a></li> <li><a href="/wiki/%E6%B3%95%E7%BA%BF" title="法线">（主）法线</a></li> <li><a href="/w/index.php?title=%E5%89%AF%E6%B3%95%E7%B7%9A&action=edit&redlink=1" class="new" title="副法線（页面不存在）">副法線</a></li> <li><a href="/wiki/%E6%9B%B2%E7%BA%BF" title="曲线">曲线</a></li> <li><a href="/wiki/%E5%9C%86%E9%94%A5%E6%9B%B2%E7%BA%BF" title="圆锥曲线">圆锥曲线</a></li> <li><a href="/wiki/%E5%8F%8C%E6%9B%B2%E7%BA%BF" title="双曲线">双曲线</a></li> <li><a href="/wiki/%E6%8A%9B%E7%89%A9%E7%BA%BF" title="抛物线">抛物线</a></li> <li><a href="/wiki/%E6%AD%A3%E5%BC%A6%E6%9B%B2%E7%B7%9A" title="正弦曲線">正弦曲線</a></li> <li><a href="/wiki/%E8%9A%8C%E7%BA%BF" title="蚌线">蚌线</a></li> <li><a href="/wiki/%E8%9C%97%E7%BA%BF" class="mw-redirect" title="蜗线">蜗线</a></li> <li><a href="/wiki/%E8%9E%BA%E7%BA%BF" title="螺线">螺线</a>（<a href="/wiki/%E9%98%BF%E5%9F%BA%E7%B1%B3%E5%BE%B7%E8%9E%BA%E7%BA%BF" title="阿基米德螺线">阿基米德螺线</a>、<a href="/wiki/%E7%AD%89%E8%A7%92%E8%9E%BA%E7%BA%BF" title="等角螺线">等角螺线</a>……）</li> <li><a href="/wiki/%E6%91%86%E7%BA%BF" title="摆线">摆线</a>（<a href="/wiki/%E6%9C%80%E9%80%9F%E9%99%8D%E7%B7%9A%E5%95%8F%E9%A1%8C" title="最速降線問題">最速降線問題</a>）</li> <li><a href="/wiki/%E6%82%AC%E9%93%BE%E7%BA%BF" title="悬链线">悬链线</a></li> <li><a href="/wiki/%E6%9B%B3%E7%89%A9%E7%BA%BF" title="曳物线">曳物线</a></li> <li><a href="/wiki/%E6%BC%B8%E4%BC%B8%E7%B7%9A" title="漸伸線">漸開線</a></li> <li><a href="/wiki/%E6%B8%90%E5%B1%88%E7%BA%BF" title="渐屈线">渐屈线</a></li> <li><a href="/wiki/%E6%B8%90%E8%BF%91%E7%BA%BF" title="渐近线">渐近线</a></li> <li><a href="/wiki/%E6%B5%8B%E5%9C%B0%E7%BA%BF" title="测地线">测地线</a></li> <li><a href="/wiki/%E9%82%8A_(%E5%B9%BE%E4%BD%95)" title="邊 (幾何)">邊</a></li> <li><a href="/wiki/%E5%91%A8%E9%95%BF" title="周长">周界</a></li> <li><a href="/wiki/%E5%BC%A6_(%E5%B9%BE%E4%BD%95)" title="弦 (幾何)">弦</a></li> <li><a href="/wiki/%E5%BC%A7" title="弧">弧</a></li> <li><a href="/wiki/%E7%9F%A2_(%E5%B9%BE%E4%BD%95)" title="矢 (幾何)">矢</a></li> <li><a href="/wiki/%E5%9E%82%E7%9B%B4%E5%B9%B3%E5%88%86%E7%B7%9A" title="垂直平分線">垂直平分線</a></li> <li><a href="/wiki/%E4%BB%A3%E6%95%B8%E6%9B%B2%E7%B7%9A" title="代數曲線">代數曲線</a></li> <li><a href="/wiki/%E6%A4%AD%E5%9C%86%E6%9B%B2%E7%BA%BF" title="椭圆曲线">椭圆曲线</a></li> <li><a href="/wiki/%E8%B6%85%E6%A9%A2%E5%9C%93" title="超橢圓">超橢圓</a></li> <li><a href="/wiki/%E6%98%9F%E5%BD%A2%E7%BA%BF" title="星形线">星形线</a></li> <li><a href="/wiki/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" title="三尖瓣线">三尖瓣线</a></li> <li><a href="/wiki/%E6%96%B9%E5%9C%93%E5%BD%A2" title="方圓形">方圓形</a></li> <li><a href="/wiki/%E5%8B%92%E6%B4%9B%E4%B8%89%E8%A7%92%E5%BD%A2" title="勒洛三角形">勒洛三角形</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学)">平面</a>圖形</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a class="mw-selflink selflink">圆</a>（<a href="/wiki/%E5%B9%BF%E4%B9%89%E5%9C%86" title="广义圆">广义圆</a>）</li> <li><a href="/wiki/%E6%A4%AD%E5%9C%86" title="椭圆">椭圆</a></li> <li><a href="/wiki/%E6%89%87%E5%BD%A2" title="扇形">扇形</a></li> <li><a href="/wiki/%E5%BC%93%E5%BD%A2" title="弓形">弓形</a></li> <li><a href="/wiki/%E7%8E%AF%E5%BD%A2" title="环形">环形</a></li> <li><a href="/wiki/%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="多边形">多边形</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形">三角形</a></li> <li><a href="/wiki/%E5%9B%9B%E9%82%8A%E5%BD%A2" title="四邊形">四邊形</a></li> <li><a href="/wiki/%E4%BA%94%E8%BE%B9%E5%BD%A2" title="五边形">五边形</a></li> <li><a href="/wiki/%E5%85%AD%E8%BE%B9%E5%BD%A2" title="六边形">六边形</a></li> <li><a href="/wiki/%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="多边形">多边形</a></li> <li><a href="/wiki/%E6%AD%A3%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="正多边形">正多边形</a></li> <li><a href="/wiki/%E6%A2%AF%E5%BD%A2" title="梯形">梯形</a></li> <li><a href="/wiki/%E5%B9%B3%E8%A1%8C%E5%9B%9B%E8%BE%B9%E5%BD%A2" title="平行四边形">平行四边形</a></li> <li><a href="/wiki/%E8%8F%B1%E5%BD%A2" title="菱形">菱形</a></li> <li><a href="/wiki/%E7%9F%A9%E5%BD%A2" title="矩形">矩形</a></li> <li><a href="/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形">正方形</a></li> <li><a href="/wiki/%E9%B7%82%E5%BD%A2" title="鷂形">鷂形</a></li> <li><a href="/wiki/%E5%8D%B5%E5%BD%A2%E7%BA%BF" title="卵形线">卵形线</a></li> <li><a href="/wiki/%E7%B4%A1%E9%8C%98%E5%BD%A2" title="紡錘形">梭形</a></li> <li><a href="/wiki/%E6%98%9F%E5%BD%A2%E5%A4%9A%E9%82%8A%E5%BD%A2" title="星形多邊形">星形</a></li> <li><a href="/wiki/%E4%BA%94%E8%A7%92%E6%98%9F" title="五角星">五角星</a></li> <li><a href="/wiki/%E5%85%AD%E8%A7%92%E6%98%9F" title="六角星">六角星</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E4%B8%89%E7%B6%AD%E7%A9%BA%E9%96%93" title="三維空間">立體</a>圖形</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%A4%9A%E9%9D%A2%E4%BD%93" title="多面体">多面体</a></li> <li><a href="/wiki/%E6%9F%8F%E6%8B%89%E5%9C%96%E7%AB%8B%E9%AB%94" title="柏拉圖立體">正多面體</a></li> <li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94" title="四面體">四面體</a></li> <li><a href="/wiki/%E9%95%B7%E6%96%B9%E9%AB%94" title="長方體">長方體</a></li> <li><a href="/wiki/%E7%AB%8B%E6%96%B9%E9%AB%94" title="立方體">立方體</a></li> <li><a href="/wiki/%E5%B9%B3%E8%A1%8C%E5%85%AD%E9%9D%A2%E4%BD%93" title="平行六面体">平行六面体</a></li> <li><a href="/wiki/%E6%A3%B1%E6%9F%B1" title="棱柱">棱柱</a></li> <li><a href="/wiki/%E5%8F%8D%E6%A3%B1%E6%9F%B1" title="反棱柱">反棱柱</a></li> <li><a href="/wiki/%E6%A3%B1%E9%94%A5" title="棱锥">棱锥</a></li> <li><a href="/wiki/%E9%94%A5%E5%8F%B0" title="锥台">棱台</a></li> <li><a href="/wiki/%E5%9C%86%E6%9F%B1%E4%BD%93" title="圆柱体">圆柱体</a></li> <li><a href="/wiki/%E5%9C%86%E9%94%A5" title="圆锥">圆锥</a></li> <li><a href="/wiki/%E5%9C%86%E5%8F%B0" title="圆台">圆台</a></li> <li><a href="/wiki/%E6%A4%AD%E7%90%83" title="椭球">椭球</a>（<a href="/wiki/%E9%A1%9E%E7%90%83%E9%9D%A2" title="類球面">長球體</a>、<a href="/wiki/%E9%A1%9E%E7%90%83%E9%9D%A2" title="類球面">扁球體</a>）</li> <li><a href="/wiki/%E7%90%83_(%E6%95%B0%E5%AD%A6)" title="球 (数学)">球體</a></li> <li><a href="/wiki/%E7%90%83%E7%BC%BA" title="球缺">球缺</a></li> <li><a href="/wiki/%E7%90%83%E5%86%A0" title="球冠">球冠</a></li> <li><a href="/wiki/%E7%90%83%E5%8F%B0" title="球台">球台</a></li> <li><a href="/wiki/%E9%9B%A2%E5%BF%83%E7%8E%87" title="離心率">準線</a></li> <li><a href="/w/index.php?title=%E6%AF%8D%E7%BA%BF_(%E5%87%A0%E4%BD%95)&action=edit&redlink=1" class="new" title="母线 (几何)（页面不存在）">母線</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E6%9B%B2%E9%9D%A2" title="曲面">曲面</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BA%8C%E6%AC%A1%E6%9B%B2%E9%9D%A2" title="二次曲面">二次曲面</a></li> <li><a href="/wiki/%E6%97%8B%E8%BD%89%E6%9B%B2%E9%9D%A2" title="旋轉曲面">旋轉曲面</a></li> <li><a href="/wiki/%E6%8A%9B%E7%89%A9%E9%9D%A2" title="抛物面">抛物面</a></li> <li><a href="/wiki/%E9%9B%99%E6%9B%B2%E9%9D%A2" title="雙曲面">雙曲面</a></li> <li><a href="/w/index.php?title=%E9%A9%AC%E9%9E%8D%E9%9D%A2&action=edit&redlink=1" class="new" title="马鞍面（页面不存在）">马鞍面</a></li> <li><a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a></li> <li><a href="/wiki/%E6%A4%AD%E7%90%83" title="椭球">橢球面</a></li> <li><a href="/wiki/%E9%A1%9E%E7%90%83%E9%9D%A2" title="類球面">類球面</a></li> <li><a href="/wiki/%E7%8E%AF%E9%9D%A2" title="环面">环面</a></li> <li><a href="/wiki/%E8%8E%AB%E6%AF%94%E4%B9%8C%E6%96%AF%E5%B8%A6" title="莫比乌斯带">莫比乌斯带</a></li> <li><a href="/wiki/%E6%B5%81%E5%BD%A2" title="流形">流形</a></li> <li><a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%9B%B2%E9%9D%A2" title="黎曼曲面">黎曼曲面</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">高維空間</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%B6%85%E5%B9%B3%E9%9D%A2" title="超平面">超平面</a></li> <li><a href="/wiki/%E7%B6%AD%E9%9D%A2" title="維面">超面</a></li> <li><a href="/wiki/%E8%B6%85%E6%9B%B2%E9%9D%A2" title="超曲面">超曲面</a></li> <li><a href="/wiki/%E8%83%9E_(%E7%B5%90%E6%A7%8B)" title="胞 (結構)">胞</a></li> <li><a href="/wiki/%E5%A4%9A%E8%83%9E%E5%BD%A2" title="多胞形">多胞形</a></li> <li><a href="/wiki/%E8%B6%85%E7%90%83%E9%9D%A2" title="超球面">超球體</a></li> <li><a href="/wiki/%E8%B6%85%E6%96%B9%E5%BD%A2" title="超方形">超方形</a></li> <li><a href="/wiki/%E5%9B%9B%E7%B6%AD%E8%B6%85%E6%AD%A3%E6%96%B9%E9%AB%94" title="四維超正方體">超立方體</a></li> <li><a href="/wiki/%E5%85%8B%E8%8E%B1%E5%9B%A0%E7%93%B6" title="克莱因瓶">克莱因瓶</a></li> <li><a href="/wiki/%E5%9B%9B%E7%B6%AD%E6%9F%B1%E9%AB%94%E6%9F%B1" title="四維柱體柱">四維柱體柱</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">圖形關係</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9B%B8%E4%BC%BC_(%E5%B9%BE%E4%BD%95)" title="相似 (幾何)">相似</a></li> <li><a href="/wiki/%E5%85%A8%E7%AD%89" title="全等">全等</a></li> <li><a href="/wiki/%E5%B0%8D%E7%A8%B1" title="對稱">對稱</a></li> <li><a href="/wiki/%E5%B9%B3%E8%A1%8C" title="平行">平行</a></li> <li><a href="/wiki/%E5%9E%82%E7%9B%B4" title="垂直">垂直</a></li> <li><a href="/wiki/%E7%9B%B8%E4%BA%A4" title="相交">相交</a></li> <li><a href="/wiki/%E5%88%87%E7%BA%BF" title="切线">相切</a></li> <li><a href="/w/index.php?title=%E7%9B%B8%E9%9B%A2&action=edit&redlink=1" class="new" title="相離（页面不存在）">相離</a></li> <li><a href="/wiki/%E9%95%9C%E5%83%8F_(%E5%87%A0%E4%BD%95)" title="镜像 (几何)">镜像</a></li> <li><a href="/wiki/%E6%97%8B%E8%BD%AC" title="旋转">旋转</a></li> <li><a href="/wiki/%E5%8F%8D%E6%BC%94" title="反演">反演</a></li> <li><a href="/wiki/%E6%88%AA%E9%9D%A2_(%E5%B9%BE%E4%BD%95)" title="截面 (幾何)">截面</a></li> <li><a href="/wiki/%E7%BC%A9%E6%94%BE" title="缩放">缩放</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形">三角形</a>關係</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9B%B8%E4%BC%BC%E4%B8%89%E8%A7%92%E5%BD%A2" title="相似三角形">相似三角形</a></li> <li><a href="/wiki/%E5%85%A8%E7%AD%89%E4%B8%89%E8%A7%92%E5%BD%A2" title="全等三角形">全等三角形</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="量 (数学)">量</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%B7%9D%E7%A6%BB" title="距离">距离</a></li> <li><a href="/wiki/%E9%95%BF%E5%BA%A6" title="长度">长度</a></li> <li><a href="/wiki/%E5%91%A8%E9%95%BF" title="周长">周长</a></li> <li><a href="/wiki/%E5%BC%A7%E9%95%BF" title="弧长">弧长</a></li> <li><a href="/wiki/%E9%AB%98%E5%BA%A6" title="高度">高度</a></li> <li><a href="/wiki/%E9%9D%A2%E7%A7%AF" title="面积">面积</a></li> <li><a href="/wiki/%E8%A1%A8%E9%9D%A2%E7%A9%8D" title="表面積">表面積</a></li> <li><a href="/wiki/%E4%BD%93%E7%A7%AF" title="体积">体积</a></li> <li><a href="/wiki/%E5%AE%B9%E7%A9%8D" class="mw-redirect" title="容積">容積</a></li> <li><a href="/wiki/%E5%BA%A6_(%E8%A7%92)" title="度 (角)">角度</a></li> <li><a href="/wiki/%E6%9B%B2%E7%8E%87" title="曲率">曲率</a></li> <li><a href="/wiki/%E6%9B%B2%E7%BA%BF%E7%9A%84%E6%8C%A0%E7%8E%87" title="曲线的挠率">撓率</a></li> <li><a href="/wiki/%E9%9B%A2%E5%BF%83%E7%8E%87" title="離心率">離心率</a></li> <li><a href="/wiki/%E5%87%B9%E5%87%B8%E6%80%A7_(%E5%B9%BE%E4%BD%95)" title="凹凸性 (幾何)">凹凸性</a></li> <li><a href="/w/index.php?title=%E6%9C%89%E5%90%91%E6%9B%B2%E9%9D%A2&action=edit&redlink=1" class="new" title="有向曲面（页面不存在）">有向曲面</a></li> <li><a href="/wiki/%E5%8F%AF%E5%B1%95%E6%9B%B2%E9%9D%A2" title="可展曲面">可展曲面</a></li> <li><a href="/wiki/%E7%9B%B4%E7%B4%8B%E6%9B%B2%E9%9D%A2" title="直紋曲面">直紋曲面</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">作圖</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%B0%BA" title="尺">尺</a> <ul><li><a href="/wiki/%E7%9B%B4%E5%B0%BA" title="直尺">直尺</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%B0%BA" title="三角尺">三角尺</a></li></ul></li> <li><a href="/wiki/%E5%9C%86%E8%A7%84" title="圆规">圆规</a></li> <li><a href="/wiki/%E5%B0%BA%E8%A7%84%E4%BD%9C%E5%9B%BE" title="尺规作图">尺规作图</a></li> <li><a href="/wiki/%E4%BA%8C%E5%88%BB%E5%B0%BA%E4%BD%9C%E5%9C%96" title="二刻尺作圖">二刻尺作圖</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">分支</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何">平面幾何</a></li> <li><a href="/wiki/%E7%AB%8B%E4%BD%93%E5%87%A0%E4%BD%95" title="立体几何">立体几何</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%AD%A6" title="三角学">三角学</a></li> <li><a href="/wiki/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95" title="解析几何">解析几何</a></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95" title="微分几何">微分几何</a></li> <li><a href="/wiki/%E6%8B%93%E6%89%91%E5%AD%A6" title="拓扑学">拓扑学</a></li> <li><a href="/wiki/%E5%9B%BE%E8%AE%BA" title="图论">图论</a></li> <li><a href="/wiki/%E6%91%BA%E7%B4%99%E6%95%B8%E5%AD%B8" title="摺紙數學">摺紙數學</a></li> <li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何">欧几里得几何</a></li> <li><a href="/wiki/%E9%9D%9E%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="非欧几里得几何">非欧几里得几何</a>（<a href="/wiki/%E5%8F%8C%E6%9B%B2%E5%87%A0%E4%BD%95" title="双曲几何">双曲几何</a>、<a href="/wiki/%E7%90%83%E9%9D%A2%E5%B9%BE%E4%BD%95%E5%AD%B8" title="球面幾何學">球面幾何</a>……）</li> <li><a href="/wiki/%E5%88%86%E5%BD%A2" title="分形">分形</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">理論</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%AE%9A%E7%90%86" title="定理">定理</a></li> <li><a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a></li> <li><a href="/wiki/%E5%AE%9A%E4%B9%89" title="定义">定义</a></li> <li><a href="/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="數學證明">數學證明</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span typeof="mw:File"><span title="分类"><img alt="分类" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span><b><a href="/wiki/Category:%E5%87%A0%E4%BD%95%E5%AD%A6" title="Category:几何学">分类</a></b></li> <li><span typeof="mw:File"><span title="主题"><img alt="主题" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span><b><a href="/wiki/Portal:%E5%87%A0%E4%BD%95%E5%AD%A6" title="Portal:几何学">主题</a></b></li> <li><span typeof="mw:File"><span title="共享资源页面"><img alt="共享资源页面" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, 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href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"><style data-mw-deduplicate="TemplateStyles:r79005747">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r79005747"></div><div role="navigation" class="navbox authority-control" aria-labelledby="-&#123;zh-cn:规范控制;zh-tw:權威控制;&#125;--&#123;zh-cn:数据库;zh-tw:資料庫&#125;-_frameless&#124;text-top&#124;10px&#124;alt=編輯維基數據鏈接&#124;link=https&#58;//www.wikidata.org/wiki/Q17278#identifiers&#124;class=noprint&#124;編輯維基數據鏈接" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2"><div id="-&#123;zh-cn:规范控制;zh-tw:權威控制;&#125;--&#123;zh-cn:数据库;zh-tw:資料庫&#125;-_frameless&#124;text-top&#124;10px&#124;alt=編輯維基數據鏈接&#124;link=https&#58;//www.wikidata.org/wiki/Q17278#identifiers&#124;class=noprint&#124;編輯維基數據鏈接" style="font-size:110%;margin:0 5em"><a href="/wiki/Help:%E8%A7%84%E8%8C%83%E6%8E%A7%E5%88%B6" title="Help:规范控制">规范控制数据库</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q17278#identifiers" title="編輯維基數據鏈接"><img alt="編輯維基數據鏈接" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, 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href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000321923&P_CON_LNG=ENG">拉脱维亚</a></span></li> <li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="kružnice"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph121971&CON_LNG=ENG">捷克</a></span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">学术</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://vocab.getty.edu/page/aat/300055627">AAT</a></span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a 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