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<title>Concave polygon</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Concave polygon</span></span>

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<p>A <a href="Simple_polygon" title="Simple polygon">simple polygon</a> that is not <a href="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="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>

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<div class="mw-heading mw-heading2"><h2 id="Polygon">Polygon</h2></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="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="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="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="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="Partition_of_a_set" title="Partition of a set">partition</a> a concave polygon into a set of convex polygons. A <a href="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>According to Euclidean geometry, a <a href="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="Quadrilateral" title="Quadrilateral">quadrilateral</a> is the <a href="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="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.

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<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</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="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7637-2250-2</bdi></cite>.</span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7641-4069-3</bdi></cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><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: <span class="nowrap">237–</span>238, <a href="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="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>

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<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"><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>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFChazelleDobkin1985" class="citation cs2"><a href="Bernard_Chazelle" title="Bernard Chazelle">Chazelle, Bernard</a>; <a href="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. <span class="nowrap">63–</span>133</cite>.</span>

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</style><div id="Polygons_(List)123" style="font-size:114%;margin:0 4em"><a href="Polygon" title="Polygon">Polygons</a> (<a href="List_of_polygons" title="List of polygons">List</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Triangle" title="Triangle">Triangles</a></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 href="Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Acute</a></li>

<li><a href="Equilateral_triangle" title="Equilateral triangle">Equilateral</a></li>

<li><a href="Ideal_triangle" title="Ideal triangle">Ideal</a></li>

<li><a href="Isosceles_triangle" title="Isosceles triangle">Isosceles</a></li>

<li><a href="Kepler_triangle" title="Kepler triangle">Kepler</a></li>

<li><a href="Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Obtuse</a></li>

<li><a href="Right_triangle" title="Right triangle">Right</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a></li>

<li><a href="Apollonius_quadrilateral" title="Apollonius quadrilateral">Apollonius</a></li>

<li><a href="Bicentric_quadrilateral" title="Bicentric quadrilateral">Bicentric</a></li>

<li><a href="Crossed_quadrilateral" class="mw-redirect" title="Crossed quadrilateral">Crossed</a></li>

<li><a href="Cyclic_quadrilateral" title="Cyclic quadrilateral">Cyclic</a></li>

<li><a href="Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">Equidiagonal</a></li>

<li><a href="Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">Ex-tangential</a></li>

<li><a href="Harmonic_quadrilateral" title="Harmonic quadrilateral">Harmonic</a></li>

<li><a href="Isosceles_trapezoid" title="Isosceles trapezoid">Isosceles trapezoid</a></li>

<li><a href="Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">Orthodiagonal</a></li>

<li><a href="Parallelogram" title="Parallelogram">Parallelogram</a></li>

<li><a href="Rectangle" title="Rectangle">Rectangle</a></li>

<li><a href="Right_kite" title="Right kite">Right kite</a></li>

<li><a href="Right_trapezoid" class="mw-redirect" title="Right trapezoid">Right trapezoid</a></li>

<li><a href="Rhomboid" title="Rhomboid">Rhomboid</a></li>

<li><a href="Rhombus" title="Rhombus">Rhombus</a></li>

<li><a href="Square" title="Square">Square</a></li>

<li><a href="Tangential_quadrilateral" title="Tangential quadrilateral">Tangential</a></li>

<li><a href="Tangential_trapezoid" title="Tangential trapezoid">Tangential trapezoid</a></li>

<li><a href="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="Monogon" title="Monogon">Monogon (1)</a></li>

<li><a href="Digon" title="Digon">Digon (2)</a></li>

<li><a href="Triangle" title="Triangle">Triangle (3)</a></li>

<li><a href="Quadrilateral" title="Quadrilateral">Quadrilateral (4)</a></li>

<li><a href="Pentagon" title="Pentagon">Pentagon (5)</a></li>

<li><a href="Hexagon" title="Hexagon">Hexagon (6)</a></li>

<li><a href="Heptagon" title="Heptagon">Heptagon (7)</a></li>

<li><a href="Octagon" title="Octagon">Octagon (8)</a></li>

<li><a href="Nonagon" title="Nonagon">Nonagon/Enneagon (9)</a></li>

<li><a href="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="Hendecagon" title="Hendecagon">Hendecagon (11)</a></li>

<li><a href="Dodecagon" title="Dodecagon">Dodecagon (12)</a></li>

<li><a href="Tridecagon" title="Tridecagon">Tridecagon (13)</a></li>

<li><a href="Tetradecagon" title="Tetradecagon">Tetradecagon (14)</a></li>

<li><a href="Pentadecagon" title="Pentadecagon">Pentadecagon (15)</a></li>

<li><a href="Hexadecagon" title="Hexadecagon">Hexadecagon (16)</a></li>

<li><a href="Heptadecagon" title="Heptadecagon">Heptadecagon (17)</a></li>

<li><a href="Octadecagon" title="Octadecagon">Octadecagon (18)</a></li>

<li><a href="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="Icositrigon" title="Icositrigon">Icositrigon (23)</a></li>

<li><a href="Icositetragon" title="Icositetragon">Icositetragon (24)</a></li>

<li><a href="Triacontagon" title="Triacontagon">Triacontagon (30)</a></li>

<li><a href="Chiliagon" title="Chiliagon">Chiliagon (1000)</a></li>

<li><a href="Myriagon" title="Myriagon">Myriagon (10,000)</a></li>

<li><a href="Megagon" title="Megagon">Megagon (1,000,000)</a></li>

<li><a href="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="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="Pentagram" title="Pentagram">Pentagram</a></li>

<li><a href="Hexagram" title="Hexagram">Hexagram</a></li>

<li><a href="Heptagram" title="Heptagram">Heptagram</a></li>

<li><a href="Octagram" title="Octagram">Octagram</a></li>

<li><a href="Enneagram_(geometry)" title="Enneagram (geometry)">Enneagram</a></li>

<li><a href="Decagram_(geometry)" title="Decagram (geometry)">Decagram</a></li>

<li><a href="Hendecagram" title="Hendecagram">Hendecagram</a></li>

<li><a href="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 href="Convex_polygon" title="Convex polygon">Convex</a></li>

<li><a href="Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">Cyclic</a></li>

<li><a href="Equiangular_polygon" title="Equiangular polygon">Equiangular</a></li>

<li><a href="Equilateral_polygon" title="Equilateral polygon">Equilateral</a></li>

<li><a href="Infinite_skew_polygon" title="Infinite skew polygon">Infinite skew</a></li>

<li><a href="Isogonal_figure" title="Isogonal figure">Isogonal</a></li>

<li><a href="Isotoxal_figure" title="Isotoxal figure">Isotoxal</a></li>

<li><a href="Magic_polygon" title="Magic polygon">Magic</a></li>

<li><a href="Pseudotriangle" title="Pseudotriangle">Pseudotriangle</a></li>

<li><a href="Rectilinear_polygon" title="Rectilinear polygon">Rectilinear</a></li>

<li><a href="Regular_polygon" title="Regular polygon">Regular</a></li>

<li><a href="Reinhardt_polygon" title="Reinhardt polygon">Reinhardt</a></li>

<li><a href="Simple_polygon" title="Simple polygon">Simple</a></li>

<li><a href="Star-shaped_polygon" title="Star-shaped polygon">Star-shaped</a></li>

<li><a href="Tangential_polygon" title="Tangential polygon">Tangential</a></li>

<li><a href="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></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Concave_polygon"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="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="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-22" href="https://en.wikipedia.org/wiki/?title=Concave_polygon&oldid=1291597159">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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