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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Semicircle</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="Go to an article in another language. Available in 27 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-27" 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">27 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Semic%C3%ADrculu" title="Semicírculu – Asturian" lang="ast" hreflang="ast" data-title="Semicírculu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%B0%E0%A7%8D%E0%A6%A7%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="অর্ধবৃত্ত – Bangla" lang="bn" hreflang="bn" data-title="অর্ধবৃত্ত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Semicercle" title="Semicercle – Catalan" lang="ca" hreflang="ca" data-title="Semicercle" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Halbkreis" title="Halbkreis – German" lang="de" hreflang="de" data-title="Halbkreis" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%97%CE%BC%CE%B9%CE%BA%CF%8D%CE%BA%CE%BB%CE%B9%CE%BF" title="Ημικύκλιο – Greek" lang="el" hreflang="el" data-title="Ημικύκλιο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" 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/M%C3%AA%C5%9B_ser%C4%87" title="Mêś serć – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Mêś serć" 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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Semic%C3%ADrculo" title="Semicírculo – Spanish" lang="es" hreflang="es" data-title="Semicírculo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zirkuluerdi" title="Zirkuluerdi – Basque" lang="eu" hreflang="eu" data-title="Zirkuluerdi" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%DB%8C%D9%85%E2%80%8C%D8%AF%D8%A7%DB%8C%D8%B1%D9%87" title="نیم‌دایره – Persian" lang="fa" hreflang="fa" data-title="نیم‌دایره" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Leathchiorcal" title="Leathchiorcal – Irish" lang="ga" hreflang="ga" data-title="Leathchiorcal" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%98%EC%9B%90" title="반원 – Korean" lang="ko" hreflang="ko" data-title="반원" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%AB%D5%BD%D5%A1%D5%B7%D6%80%D5%BB%D5%A1%D5%B6" title="Կիսաշրջան – Armenian" lang="hy" hreflang="hy" data-title="Կիսաշրջան" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%B0%E0%A5%8D%E0%A4%A7_%E0%A4%97%E0%A5%8B%E0%A4%B2%E0%A4%BE" title="अर्ध गोला – Hindi" lang="hi" hreflang="hi" data-title="अर्ध गोला" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Polukrug" title="Polukrug – Croatian" lang="hr" hreflang="hr" data-title="Polukrug" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Semicerchio" title="Semicerchio – Italian" lang="it" hreflang="it" data-title="Semicerchio" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A6%D7%99_%D7%9E%D7%A2%D7%92%D7%9C" title="חצי מעגל – Hebrew" lang="he" hreflang="he" data-title="חצי מעגל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8D%8A%E5%86%86" title="半円 – Japanese" lang="ja" hreflang="ja" data-title="半円" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/P%C3%B3%C5%82okr%C4%85g" title="Półokrąg – Polish" lang="pl" hreflang="pl" data-title="Półokrąg" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Semic%C3%ADrculo" title="Semicírculo – Portuguese" lang="pt" hreflang="pt" data-title="Semicírculo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%83%D0%BA%D1%80%D1%83%D0%B3" title="Полукруг – Russian" lang="ru" hreflang="ru" data-title="Полукруг" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Semicircle" title="Semicircle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Semicircle" 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-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Kalahating_bilog" title="Kalahating bilog – Tagalog" lang="tl" hreflang="tl" data-title="Kalahating bilog" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%B0%E0%AF%88%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="அரைவட்டம் – Tamil" lang="ta" hreflang="ta" data-title="அரைவட்டம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B2%D0%BA%D0%BE%D0%BB%D0%BE" title="Півколо – Ukrainian" lang="uk" hreflang="uk" data-title="Півколо" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_b%C3%A1n_nguy%E1%BB%87t" title="Hình bán nguyệt – Vietnamese" lang="vi" hreflang="vi" data-title="Hình bán nguyệt" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8D%8A%E5%9C%93" title="半圓 – Cantonese" lang="yue" hreflang="yue" data-title="半圓" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%8A%E5%9C%93" title="半圓 – Chinese" lang="zh" hreflang="zh" data-title="半圓" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Geometric shape</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the music album, see <a href="/wiki/Semicircle_(album)" title="Semicircle (album)">Semicircle (album)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3;">Semicircle</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Semicircle-curve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Semicircle-curve.svg/220px-Semicircle-curve.svg.png" decoding="async" width="220" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Semicircle-curve.svg/330px-Semicircle-curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Semicircle-curve.svg/440px-Semicircle-curve.svg.png 2x" data-file-width="896" data-file-height="540" /></a></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Area" title="Area">Area</a></th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">πr<sup>2</sup></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Perimeter" title="Perimeter">Perimeter</a></th><td class="infobox-data"><span class="texhtml">(π+2)r</span></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> (and more specifically <a href="/wiki/Geometry" title="Geometry">geometry</a>), a <b>semicircle</b> is a one-dimensional <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of points that forms half of a <a href="/wiki/Circle" title="Circle">circle</a>. It is a <a href="/wiki/Circular_arc" title="Circular arc">circular arc</a> that measures 180° (equivalently, <span class="texhtml mvar" style="font-style:italic;">π</span> <a href="/wiki/Radians" class="mw-redirect" title="Radians">radians</a>, or a <a href="/wiki/Turn_(geometry)#Subdivision_of_turns" class="mw-redirect" title="Turn (geometry)">half-turn</a>). It only has one line of symmetry (<a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a>). </p><p>In non-technical usage, the term "semicircle" is sometimes used to refer to either a <a href="/wiki/Closed_curve" class="mw-redirect" title="Closed curve">closed curve</a> that also includes the diameter segment from one end of the arc to the other or to the half-<a href="/wiki/Disc_(geometry)" class="mw-redirect" title="Disc (geometry)">disk</a>, which is a two-dimensional <a href="/wiki/Region_(mathematics)" class="mw-redirect" title="Region (mathematics)">geometric region</a> that further includes all the interior points. </p><p>By <a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a>, any <a href="/wiki/Triangle" title="Triangle">triangle</a> <a href="/wiki/Inscribed_figure" title="Inscribed figure">inscribed</a> in a semicircle with a <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a> at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a>, with a <a href="/wiki/Right_angle" title="Right angle">right angle</a> at the third vertex. </p><p>All lines intersecting the semicircle <a href="/wiki/Perpendicular" title="Perpendicular">perpendicularly</a> are <a href="/wiki/Concurrent_lines" title="Concurrent lines">concurrent</a> at the center of the circle containing the given semicircle. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Arithmetic_and_geometric_means">Arithmetic and geometric means</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semicircle&action=edit&section=1" title="Edit section: Arithmetic and geometric means"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Quadrature_of_rectangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Quadrature_of_rectangle.svg/170px-Quadrature_of_rectangle.svg.png" decoding="async" width="170" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Quadrature_of_rectangle.svg/255px-Quadrature_of_rectangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Quadrature_of_rectangle.svg/340px-Quadrature_of_rectangle.svg.png 2x" data-file-width="512" data-file-height="768" /></a><figcaption>Construction of a square with the same area as a given oblong</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:AM_GM_inequality_visual_proof.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/AM_GM_inequality_visual_proof.svg/220px-AM_GM_inequality_visual_proof.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/AM_GM_inequality_visual_proof.svg/330px-AM_GM_inequality_visual_proof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/AM_GM_inequality_visual_proof.svg/440px-AM_GM_inequality_visual_proof.svg.png 2x" data-file-width="512" data-file-height="341" /></a><figcaption><a href="/wiki/Proof_without_words" title="Proof without words">Proof without words</a> of the <span class="nowrap"><a href="/wiki/AM%E2%80%93GM_inequality" title="AM–GM inequality">AM–GM inequality</a>:</span><br style="margin-bottom:1ex;" />PR is the diameter of a circle centered on O; its radius AO is the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> of <i>a</i> and <i>b</i>. Using the <a href="/wiki/Geometric_mean_theorem" title="Geometric mean theorem">geometric mean theorem</a>, triangle PGR's <a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">altitude</a> GQ is the <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a>. For any ratio <span class="nowrap"><i>a</i>:<i>b</i>,</span> <span class="nowrap">AO ≥ GQ.</span></figcaption></figure> <p>A semicircle can be used to <a href="/wiki/Compass_and_straightedge_constructions" class="mw-redirect" title="Compass and straightedge constructions">construct</a> the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic</a> and <a href="/wiki/Geometric_mean" title="Geometric mean">geometric</a> means of two lengths using straight-edge and compass. For a semicircle with a diameter of <i>a</i> + <i>b</i>, the length of its <a href="/wiki/Radius" title="Radius">radius</a> is the arithmetic mean of <i>a</i> and <i>b</i> (since the radius is half of the diameter). </p><p>The <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> can be found by dividing the diameter into two segments of lengths <i>a</i> and <i>b</i>, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean. This can be proven by applying the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> to three similar right triangles, each having as vertices the point where the perpendicular touches the semicircle and two of the three endpoints of the segments of lengths <i>a</i> and <i>b</i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p> The construction of the geometric mean can be used to transform any rectangle into a square of the same area, a problem called the <a href="/wiki/Quadrature_(mathematics)" class="mw-redirect" title="Quadrature (mathematics)">quadrature</a> of a rectangle. The side length of the square is the geometric mean of the side lengths of the rectangle. More generally, it is used as a <a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">lemma</a> in a general method for transforming any polygonal shape into a similar copy of itself with the area of any other given polygonal shape.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></p><div style="clear:left;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Farey_diagram">Farey diagram</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semicircle&action=edit&section=2" title="Edit section: Farey diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="//upload.wikimedia.org/wikipedia/commons/0/0f/Comparison_Ford_circles_Farey_diagram.svg"><img resource="/wiki/File:Comparison_Ford_circles_Farey_diagram.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Comparison_Ford_circles_Farey_diagram.svg/300px-Comparison_Ford_circles_Farey_diagram.svg.png" decoding="async" width="300" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Comparison_Ford_circles_Farey_diagram.svg/450px-Comparison_Ford_circles_Farey_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Comparison_Ford_circles_Farey_diagram.svg/600px-Comparison_Ford_circles_Farey_diagram.svg.png 2x" data-file-width="512" data-file-height="288" /></a><figcaption>Comparison of Ford circles and a Farey diagram with semicircles for <i>n</i> from 1 to 9. Each semicircle intersects its corresponding circles at right angles. <span class="nowrap">In <a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/0/0f/Comparison_Ford_circles_Farey_diagram.svg">the SVG image,</a></span> hover over a circle or curve to highlight it and its terms.</figcaption></figure> <p>The <a href="/wiki/Farey_sequence" title="Farey sequence">Farey sequence</a> of order <i>n</i> is the <a href="/wiki/Sequence" title="Sequence">sequence</a> of completely reduced <a href="/wiki/Fraction" title="Fraction">fractions</a> which when <a href="/wiki/In_lowest_terms" class="mw-redirect" title="In lowest terms">in lowest terms</a> have <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominators</a> less than or equal to <i>n</i>, arranged in order of increasing size. With a restricted definition, each Farey sequence starts with the value 0, denoted by the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">0</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span>, and ends with the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span>. <a href="/wiki/Ford_circle" title="Ford circle">Ford circles</a> can be constructed <a href="/wiki/Tangent" title="Tangent">tangent</a> to their neighbours, and to the x-axis at these points. Semicircles joining adjacent points on the x-axis pass through the points of contact at right angles.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Equation">Equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semicircle&action=edit&section=3" title="Edit section: Equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The equation of a semicircle with midpoint <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_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}"></span> on the diameter between its endpoints and which is entirely concave from below is </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 y=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}}}"> <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"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a00e632e96a4375501949dc489cf2d26ecc37b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:25.972ex; height:4.843ex;" alt="{\displaystyle y=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}}}"></span></dd></dl> <p>If it is entirely concave from above, the equation is </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 y=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}}}"> <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"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606540d1b6b1f7f0b2d8bfa86cddb5446ff4b2fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:25.972ex; height:4.843ex;" alt="{\displaystyle y=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Arbelos">Arbelos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semicircle&action=edit&section=4" title="Edit section: Arbelos"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Arbelos.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Arbelos.svg/300px-Arbelos.svg.png" decoding="async" width="300" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Arbelos.svg/450px-Arbelos.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Arbelos.svg/600px-Arbelos.svg.png 2x" data-file-width="512" data-file-height="257" /></a><figcaption>An <b>arbelos</b> (grey region)</figcaption></figure> <p>An <a href="/wiki/Arbelos" title="Arbelos">arbelos</a> is a region in the <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> bounded by three semicircles connected at their endpoints, all on the same side of a <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a> (the <i>baseline</i>) that contains their <a href="/wiki/Diameter" title="Diameter">diameters</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semicircle&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Amphitheater" class="mw-redirect" title="Amphitheater">Amphitheater</a></li> <li><a href="/wiki/Archimedes%27_twin_circles" class="mw-redirect" title="Archimedes' twin circles">Archimedes' twin circles</a></li> <li><a href="/wiki/Archimedes%27_quadruplets" title="Archimedes' quadruplets">Archimedes' quadruplets</a></li> <li><a href="/wiki/Great_semicircle" class="mw-redirect" title="Great semicircle">Great semicircle</a></li> <li><a href="/wiki/Salinon" title="Salinon">Salinon</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle distribution</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semicircle&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI13.html">Euclid's Elements, Book VI, Proposition 13</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/propVI25.html">Euclid's Elements, Book VI, Proposition 25</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output 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.cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/FordCircle.html">"Ford Circle"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Ford+Circle&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FFordCircle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemicircle" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semicircle&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Semicircle"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Semicircle.html">"Semicircle"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Semicircle&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSemicircle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemicircle" class="Z3988"></span></span></li></ul>    </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="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Semicircle&oldid=1257099423">https://en.wikipedia.org/w/index.php?title=Semicircle&oldid=1257099423</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Elementary_geometry" title="Category:Elementary geometry">Elementary geometry</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 November 2024, at 06:32<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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