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<div>{{unreferenced|date=August 2012}}<br />
{{Even polygon db|Even polygon stat table|p16}}<br />
In mathematics, a '''hexadecagon''' (sometimes called a hexakaidecagon) is a [[polygon]] with 16 [[Edge (geometry)|sides]] and 16 [[Vertex (geometry)|vertices]].<ref name=Weisstein2002>{{cite book|last=Weisstein|first=Eric W.|title=CRC Concise Encyclopedia of Mathematics, Second Edition|year=2002|publisher=CRC Press|isbn=9781420035223|page=1365}}</ref><br />
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==Regular hexadecagon==<br />
A regular hexadecagon is [[constructible polygon|constructible]] with a [[Compass and straightedge constructions|compass and straightedge]].<br />
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Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any hexadecagon is 2520 degrees.<br />
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===Construction===<br />
A regular hexadecagon is [[constructible polygon|constructible]] using [[compass and straightedge]]:<br />
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[[File:Regular_Hexadecagon_Inscribed_in_a_Circle.gif]]<br>Construction of a regular hexadecagon<br />
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==Area==<br />
The [[area]] of a regular hexadecagon is: (with ''t'' = edge length)<br />
:<math>A = 4t^2 \cot \frac{\pi}{16} = 4t^2 (\sqrt{2}+1)(\sqrt{4-2\sqrt{2}}+1)</math><br />
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Because the hexadecagon has a number of sides that is a [[power of two]], its area can be computed in terms of the [[circumradius]] ''r'' by truncating [[Viète's formula]]:<br />
:<math>A=r^2\cdot\frac{2}{1}\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}=4r^2\sqrt{2-\sqrt{2}}.</math><br />
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===Petrie polygons===<br />
The regular hexadecagon is the [[Petrie polygon]] for many higher dimensional polytopes, shown in these skew [[orthogonal projection]]s, including:<br />
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{| class=wikitable width=80<br />
|- align=center valign=top<br />
!valign=center|A<sub>15</sub><br />
|[[File:15-simplex_t0.svg|100px]]<br>[[15-simplex]]<br />
|- align=center valign=top<br />
!valign=center|B<sub>8</sub><br />
|[[File:8-cube_t7.svg|100px]]<br>[[8-orthoplex]]<br />
|[[File:8-cube_t6.svg|100px]]<br>[[Rectified 8-orthoplex]]<br />
|[[File:8-cube_t5.svg|100px]]<br>[[Birectified 8-orthoplex]]<br />
|[[File:8-cube_t4.svg|100px]]<br>[[Trirectified 8-orthoplex]]<br />
|[[File:8-cube_t3.svg|100px]]<br>[[Trirectified 8-cube]]<br />
|[[File:8-cube_t2.svg|100px]]<br>[[Birectified 8-cube]]<br />
|[[File:8-cube_t1.svg|100px]]<br>[[Rectified 8-cube]]<br />
|[[File:8-cube_t0.svg|100px]]<br>[[8-cube]]<br />
|- align=center valign=top<br />
!valign=center|D<sub>9</sub><br />
|[[File:9-cube_t8_B8.svg|100px]]<br>[[9-orthoplex|t<sub>7</sub>(1<sub>61</sub>)]]<br />
|[[File:9-cube_t7_B8.svg|100px]]<br>[[rectified 9-orthoplex|t<sub>6</sub>(1<sub>61</sub>)]]<br />
|[[File:9-cube_t6_B8.svg|100px]]<br>[[birectified 9-orthoplex|t<sub>5</sub>(1<sub>61</sub>)]]<br />
|[[File:9-cube_t5_B8.svg|100px]]<br>[[trirectified 9-orthoplex|t<sub>4</sub>(1<sub>61</sub>)]]<br />
|[[File:9-cube_t4_B8.svg|100px]]<br>[[quadrirectified 9-cube|t<sub>3</sub>(1<sub>61</sub>)]]<br />
|[[File:9-cube_t3_B8.svg|100px]]<br>[[trirectified 9-cube|t<sub>2</sub>(1<sub>61</sub>)]]<br />
|[[File:9-cube_t2_B8.svg|100px]]<br>[[birectified 9-cube|t<sub>1</sub>(1<sub>61</sub>)]]<br />
|[[File:9-demicube.svg|100px]]<br>[[9-demicube]]<br>(1<sub>61</sub>)<br />
|}<br />
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==References==<br />
{{reflist}}<br />
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==External links==<br />
*{{MathWorld|title=Hexadecagon|urlname=Hexadecagon}} <br />
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{{Polygons}}<br />
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[[Category:Polygons]]</div>161.28.81.69