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Available in 21 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-21" 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">21 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B6%D9%84%D8%B9_%D9%85%D9%82%D8%B9%D8%B1" title="مضلع مقعر – Arabic" lang="ar" hreflang="ar" data-title="مضلع مقعر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Konkavni_mnogougao" title="Konkavni mnogougao – Bosnian" lang="bs" hreflang="bs" data-title="Konkavni mnogougao" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Polygon_ceugrwm" title="Polygon ceugrwm – Welsh" lang="cy" hreflang="cy" data-title="Polygon ceugrwm" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%BF%CE%AF%CE%BB%CE%BF_%CF%80%CE%BF%CE%BB%CF%8D%CE%B3%CF%89%CE%BD%CE%BF" title="Κοίλο πολύγωνο – Greek" lang="el" hreflang="el" data-title="Κοίλο πολύγωνο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Pol%C3%ADgono_c%C3%B3ncavo" title="Polígono cóncavo – Spanish" lang="es" hreflang="es" data-title="Polígono cóncavo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%86%D8%AF%D8%B6%D9%84%D8%B9%DB%8C_%D9%85%D9%82%D8%B9%D8%B1" title="چندضلعی مقعر – Persian" lang="fa" hreflang="fa" data-title="چندضلعی مقعر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Polygone_non_convexe" title="Polygone non convexe – French" lang="fr" hreflang="fr" data-title="Polygone non convexe" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%EB%AA%A9_%EB%8B%A4%EA%B0%81%ED%98%95" title="오목 다각형 – Korean" lang="ko" hreflang="ko" data-title="오목 다각형" 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%B3%D5%B8%D5%A3%D5%A1%D5%BE%D5%B8%D6%80_%D5%A2%D5%A1%D5%A6%D5%B4%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6" title="Գոգավոր բազմանկյուն – Armenian" lang="hy" hreflang="hy" data-title="Գոգավոր բազմանկյուն" 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%B5%E0%A4%A4%E0%A4%B2_%E0%A4%AC%E0%A4%B9%E0%A5%81%E0%A4%AD%E0%A5%81%E0%A4%9C" title="अवतल बहुभुज – Hindi" lang="hi" hreflang="hi" data-title="अवतल बहुभुज" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-he badge-Q70894304 mw-list-item" title=""><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A6%D7%95%D7%9C%D7%A2_%D7%A7%D7%A2%D7%95%D7%A8" title="מצולע קעור – Hebrew" lang="he" hreflang="he" data-title="מצולע קעור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Konk%C3%A1v_soksz%C3%B6g" title="Konkáv sokszög – Hungarian" lang="hu" hreflang="hu" data-title="Konkáv sokszög" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B4%D0%BB%D0%B0%D0%B1%D0%BD%D0%B0%D1%82_%D0%BC%D0%BD%D0%BE%D0%B3%D1%83%D0%B0%D0%B3%D0%BE%D0%BB%D0%BD%D0%B8%D0%BA" title="Вдлабнат многуаголник – Macedonian" lang="mk" hreflang="mk" data-title="Вдлабнат многуаголник" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%87%B9%E5%A4%9A%E8%A7%92%E5%BD%A2" title="凹多角形 – Japanese" lang="ja" hreflang="ja" data-title="凹多角形" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Pol%C3%ADgono_c%C3%B4ncavo" title="Polígono côncavo – Portuguese" lang="pt" hreflang="pt" data-title="Polígono côncavo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Poligon_concav" title="Poligon concav – Romanian" lang="ro" hreflang="ro" data-title="Poligon concav" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%B1%DB%95%DA%AF%DB%86%D8%B4%DB%95%DB%8C_%D9%82%D9%88%D9%BE%D8%A7%D9%88" title="فرەگۆشەی قوپاو – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فرەگۆشەی قوپاو" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%81%E0%AE%B4%E0%AE%BF%E0%AE%B5%E0%AF%81%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%B2%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%AE%E0%AF%8D" title="குழிவுப் பல்கோணம் – Tamil" lang="ta" hreflang="ta" data-title="குழிவுப் பல்கோணம்" 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%A3%D0%B2%D1%96%D0%B3%D0%BD%D1%83%D1%82%D0%B8%D0%B9_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA" title="Увігнутий многокутник – Ukrainian" lang="uk" hreflang="uk" data-title="Увігнутий многокутник" 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/%C4%90a_gi%C3%A1c_l%C3%B5m" title="Đa giác lõm – Vietnamese" lang="vi" hreflang="vi" data-title="Đa giác lõm" 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 mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%87%B9%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="凹多边形 – Chinese" lang="zh" hreflang="zh" data-title="凹多边形" 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 href="https://www.wikidata.org/wiki/Special:EntityPage/Q12243537#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <div id="p-associated-pages" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Simple_polygon.svg/200px-Simple_polygon.svg.png" decoding="async" width="200" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Simple_polygon.svg/300px-Simple_polygon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Simple_polygon.svg/400px-Simple_polygon.svg.png 2x" data-file-width="588" data-file-height="521" /></a><figcaption>An example of a concave polygon.</figcaption></figure> <p>A <a href="/wiki/Simple_polygon" title="Simple polygon">simple polygon</a> that is not <a href="/wiki/Convex_polygon" title="Convex polygon">convex</a> is called <b>concave</b>,<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> <b>non-convex</b><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> or <b>reentrant</b>.<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> A concave polygon will always have at least one <a href="/wiki/Reflex_angle" class="mw-redirect" title="Reflex angle">reflex interior angle</a>—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive.<sup id="cite_ref-MOR_4-0" class="reference"><a href="#cite_note-MOR-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Polygon">Polygon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Concave_polygon&action=edit&section=1" title="Edit section: Polygon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some lines containing interior points of a concave polygon intersect its boundary at more than two points.<sup id="cite_ref-MOR_4-1" class="reference"><a href="#cite_note-MOR-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Some <a href="/wiki/Diagonal#Polygons" title="Diagonal">diagonals</a> of a concave polygon lie partly or wholly outside the polygon.<sup id="cite_ref-MOR_4-2" class="reference"><a href="#cite_note-MOR-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Some <a href="/wiki/Extended_side" title="Extended side">sidelines</a> of a concave polygon fail to divide the plane into two half-planes one of which entirely contains the polygon. None of these three statements holds for a convex polygon. </p><p>As with any simple polygon, the sum of the <a href="/wiki/Internal_angle" class="mw-redirect" title="Internal angle">internal angles</a> of a concave polygon is <span class="texhtml mvar" style="font-style:italic;">π</span>×(<i>n</i> − 2) <a href="/wiki/Radian" title="Radian">radians</a>, equivalently 180×(<i>n</i> − 2) degrees (°), where <i>n</i> is the number of sides. </p><p>It is always possible to <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> a concave polygon into a set of convex polygons. A <a href="/wiki/Polynomial-time" class="mw-redirect" title="Polynomial-time">polynomial-time algorithm</a> for finding a decomposition into as few convex polygons as possible is described by <a href="#CITEREFChazelleDobkin1985">Chazelle & Dobkin (1985)</a>.<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>A <a href="/wiki/Triangle" title="Triangle">triangle</a> can never be concave, but there exist concave polygons with <i>n</i> sides for any <i>n</i> > 3. An example of a concave <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> is the <a href="/wiki/Dart_(geometry)" class="mw-redirect" title="Dart (geometry)">dart</a>. </p><p>At least one interior angle does not contain all other vertices in its edges and interior. </p><p>The <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of the concave polygon's vertices, and that of its edges, contains points that are exterior to the polygon. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Concave_polygon&action=edit&section=2" title="Edit section: Notes"><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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMcConnell2006" class="citation cs2">McConnell, Jeffrey J. (2006), <a rel="nofollow" class="external text" href="https://archive.org/details/computergraphics0000mcco/page/130"><i>Computer Graphics: Theory Into Practice</i></a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/computergraphics0000mcco/page/130">130</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7637-2250-2" title="Special:BookSources/0-7637-2250-2"><bdi>0-7637-2250-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Graphics%3A+Theory+Into+Practice&rft.pages=130&rft.date=2006&rft.isbn=0-7637-2250-2&rft.aulast=McConnell&rft.aufirst=Jeffrey+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcomputergraphics0000mcco%2Fpage%2F130&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConcave+polygon" class="Z3988"></span>.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeff2008" class="citation cs2">Leff, Lawrence (2008), <i>Let's Review: Geometry</i>, Hauppauge, NY: Barron's Educational Series, p. 66, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7641-4069-3" title="Special:BookSources/978-0-7641-4069-3"><bdi>978-0-7641-4069-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Let%27s+Review%3A+Geometry&rft.place=Hauppauge%2C+NY&rft.pages=66&rft.pub=Barron%27s+Educational+Series&rft.date=2008&rft.isbn=978-0-7641-4069-3&rft.aulast=Leff&rft.aufirst=Lawrence&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConcave+polygon" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMason1946" class="citation cs2">Mason, J.I. (1946), "On the angles of a polygon", <i>The Mathematical Gazette</i>, <b>30</b> (291), The Mathematical Association: 237–238, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3611229">10.2307/3611229</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3611229">3611229</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=On+the+angles+of+a+polygon&rft.volume=30&rft.issue=291&rft.pages=237-238&rft.date=1946&rft_id=info%3Adoi%2F10.2307%2F3611229&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3611229%23id-name%3DJSTOR&rft.aulast=Mason&rft.aufirst=J.I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConcave+polygon" class="Z3988"></span>.</span> </li> <li id="cite_note-MOR-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-MOR_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MOR_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-MOR_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathopenref.com/polygonconcave.html">"Definition and properties of concave polygons with interactive animation"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Definition+and+properties+of+concave+polygons+with+interactive+animation.&rft_id=http%3A%2F%2Fwww.mathopenref.com%2Fpolygonconcave.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConcave+polygon" class="Z3988"></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChazelleDobkin1985" class="citation cs2"><a href="/wiki/Bernard_Chazelle" title="Bernard Chazelle">Chazelle, Bernard</a>; <a href="/wiki/David_P._Dobkin" title="David P. Dobkin">Dobkin, David P.</a> (1985), "Optimal convex decompositions", in Toussaint, G.T. (ed.), <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/~chazelle/pubs/OptimalConvexDecomp.pdf"><i>Computational Geometry</i></a> <span class="cs1-format">(PDF)</span>, Elsevier, pp. 63–133</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Optimal+convex+decompositions&rft.btitle=Computational+Geometry&rft.pages=63-133&rft.pub=Elsevier&rft.date=1985&rft.aulast=Chazelle&rft.aufirst=Bernard&rft.au=Dobkin%2C+David+P.&rft_id=http%3A%2F%2Fwww.cs.princeton.edu%2F~chazelle%2Fpubs%2FOptimalConvexDecomp.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConcave+polygon" class="Z3988"></span>.</span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 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td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Polygons_(List)" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;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 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href="/wiki/Equilateral_triangle" title="Equilateral triangle">Equilateral</a></li> <li><a href="/wiki/Ideal_triangle" title="Ideal triangle">Ideal</a></li> <li><a href="/wiki/Isosceles_triangle" title="Isosceles triangle">Isosceles</a></li> <li><a href="/wiki/Kepler_triangle" title="Kepler triangle">Kepler</a></li> <li><a href="/wiki/Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Obtuse</a></li> <li><a href="/wiki/Right_triangle" title="Right triangle">Right</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilaterals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a></li> <li><a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">Bicentric</a></li> <li><a href="/wiki/Crossed_quadrilateral" class="mw-redirect" title="Crossed quadrilateral">Crossed</a></li> <li><a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">Cyclic</a></li> <li><a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">Equidiagonal</a></li> <li><a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">Ex-tangential</a></li> <li><a href="/wiki/Harmonic_quadrilateral" title="Harmonic quadrilateral">Harmonic</a></li> <li><a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">Isosceles trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li> <li><a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">Orthodiagonal</a></li> <li><a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Right_kite" title="Right kite">Right kite</a></li> <li><a href="/wiki/Right_trapezoid" class="mw-redirect" title="Right trapezoid">Right trapezoid</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">Tangential</a></li> <li><a href="/wiki/Tangential_trapezoid" title="Tangential trapezoid">Tangential trapezoid</a></li> <li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By number <br />of sides</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">1–10 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Monogon" title="Monogon">Monogon (1)</a></li> <li><a href="/wiki/Digon" title="Digon">Digon (2)</a></li> <li><a href="/wiki/Triangle" title="Triangle">Triangle (3)</a></li> <li><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral (4)</a></li> <li><a href="/wiki/Pentagon" title="Pentagon">Pentagon (5)</a></li> <li><a href="/wiki/Hexagon" title="Hexagon">Hexagon (6)</a></li> <li><a href="/wiki/Heptagon" title="Heptagon">Heptagon (7)</a></li> <li><a href="/wiki/Octagon" title="Octagon">Octagon (8)</a></li> <li><a href="/wiki/Nonagon" title="Nonagon">Nonagon/Enneagon (9)</a></li> <li><a href="/wiki/Decagon" title="Decagon">Decagon (10)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11–20 sides</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hendecagon" title="Hendecagon">Hendecagon (11)</a></li> <li><a href="/wiki/Dodecagon" title="Dodecagon">Dodecagon (12)</a></li> <li><a href="/wiki/Tridecagon" title="Tridecagon">Tridecagon (13)</a></li> <li><a href="/wiki/Tetradecagon" title="Tetradecagon">Tetradecagon (14)</a></li> <li><a href="/wiki/Pentadecagon" title="Pentadecagon">Pentadecagon (15)</a></li> <li><a href="/wiki/Hexadecagon" title="Hexadecagon">Hexadecagon (16)</a></li> <li><a href="/wiki/Heptadecagon" title="Heptadecagon">Heptadecagon (17)</a></li> <li><a href="/wiki/Octadecagon" title="Octadecagon">Octadecagon (18)</a></li> <li><a href="/wiki/Icosagon" title="Icosagon">Icosagon (20)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">>20 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Icositrigon" title="Icositrigon">Icositrigon (23)</a></li> <li><a href="/wiki/Icositetragon" title="Icositetragon">Icositetragon (24)</a></li> <li><a href="/wiki/Triacontagon" title="Triacontagon">Triacontagon (30)</a></li> <li><a href="/wiki/257-gon" title="257-gon">257-gon</a></li> <li><a href="/wiki/Chiliagon" title="Chiliagon">Chiliagon (1000)</a></li> <li><a href="/wiki/Myriagon" title="Myriagon">Myriagon (10,000)</a></li> <li><a href="/wiki/65537-gon" title="65537-gon">65537-gon</a></li> <li><a href="/wiki/Megagon" title="Megagon">Megagon (1,000,000)</a></li> <li><a href="/wiki/Apeirogon" title="Apeirogon">Apeirogon (∞)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Star_polygon" title="Star polygon">Star polygons</a><br /></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentagram" title="Pentagram">Pentagram</a></li> <li><a href="/wiki/Hexagram" title="Hexagram">Hexagram</a></li> <li><a href="/wiki/Heptagram" title="Heptagram">Heptagram</a></li> <li><a href="/wiki/Octagram" title="Octagram">Octagram</a></li> <li><a href="/wiki/Enneagram_(geometry)" title="Enneagram (geometry)">Enneagram</a></li> <li><a href="/wiki/Decagram_(geometry)" title="Decagram (geometry)">Decagram</a></li> <li><a href="/wiki/Hendecagram" title="Hendecagram">Hendecagram</a></li> <li><a href="/wiki/Dodecagram" title="Dodecagram">Dodecagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Concave</a></li> <li><a href="/wiki/Convex_polygon" title="Convex polygon">Convex</a></li> <li><a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">Cyclic</a></li> <li><a href="/wiki/Equiangular_polygon" title="Equiangular polygon">Equiangular</a></li> <li><a href="/wiki/Equilateral_polygon" title="Equilateral polygon">Equilateral</a></li> <li><a href="/wiki/Infinite_skew_polygon" title="Infinite skew polygon">Infinite skew</a></li> <li><a href="/wiki/Isogonal_figure" title="Isogonal figure">Isogonal</a></li> <li><a href="/wiki/Isotoxal_figure" title="Isotoxal figure">Isotoxal</a></li> <li><a href="/wiki/Magic_polygon" title="Magic polygon">Magic</a></li> <li><a href="/wiki/Pseudotriangle" title="Pseudotriangle">Pseudotriangle</a></li> <li><a href="/wiki/Rectilinear_polygon" title="Rectilinear polygon">Rectilinear</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular</a></li> <li><a href="/wiki/Reinhardt_polygon" title="Reinhardt polygon">Reinhardt</a></li> <li><a href="/wiki/Simple_polygon" title="Simple polygon">Simple</a></li> <li><a href="/wiki/Skew_polygon" title="Skew polygon">Skew</a></li> <li><a href="/wiki/Star-shaped_polygon" title="Star-shaped polygon">Star-shaped</a></li> <li><a href="/wiki/Tangential_polygon" title="Tangential polygon">Tangential</a></li> <li><a href="/wiki/Weakly_simple_polygon" title="Weakly simple polygon">Weakly simple</a></li></ul> </div></td></tr></tbody></table></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=Concave_polygon&action=edit&section=3" 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-Concave_polygon"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ConcavePolygon.html">"Concave polygon"</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=Concave+polygon&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FConcavePolygon.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConcave+polygon" 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=Concave_polygon&oldid=1176982298">https://en.wikipedia.org/w/index.php?title=Concave_polygon&oldid=1176982298</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:Types_of_polygons" title="Category:Types of polygons">Types of polygons</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 25 September 2023, at 05:36<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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