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{{Multiple issues|{{unreferenced|date=September 2018}}{{notability|date=September 2018}}}}
{{Regular polygon db|Regular polygon stat table|p21}}
In [[geometry]], an '''icosihenagon''' or '''21-gon''' is a twenty-one-sided [[polygon]]. The sum of any icosihenagon's interior angles is 3420 [[degree (angle)|degrees]]. An icosihenagon has 189 diagonals.
==Regular form==
An angle of a regular icosihenagon is <math>\frac{1140}{7}=162.857142857...</math> degrees.
Since [[Euler's totient function|<math>\varphi</math>]](21) = 12 is 3-[[smooth number|smooth]] but not [[power of 2]], thus the regular icosihenagon is [[constructible polygon|constructible]] using [[neusis]], or an [[Angle trisection|angle trisector]], but not constructible using a [[compass and straightedge]].
The [[area]] of a regular icosihenagon with edge length ''a'' is
:<math>S = \frac{21}{4}a^2 \cot \frac{\pi}{21} \simeq 34.83147 a^2</math>
<math>\cos (2\pi/21)</math> can be written using only [[square root]]s and [[cube root]]s.
:<math>\cos\frac{2\pi}{21} = \cos \left(\frac{2\pi}{3}-\frac{4\pi}{7}\right) </math>
:<math>\cos\frac{2\pi}{21} = \frac{1+\sqrt{21}+\sqrt[3]{154-30\sqrt{21}+\left(42\sqrt{3}-18\sqrt{7}\right)i}+\sqrt[3]{154-30\sqrt{21}+\left(18\sqrt{7}-42\sqrt{3}\right)i}}{12}</math>, see [[Trigonometric constants expressed in real radicals#List of trigonometric constants of 2π/n|Trigonometric constants expressed in real radicals]].
==Related figures==
Below is a table of five regular icosihenagrams, or [[Star polygons|star]] 21-gons, labeled with their respective [[Schläfli symbol]] {21/q}, 2 ≤ q ≤ 10 where gcd(q,21) = 1.
{| class="wikitable"
|- align=center
|[[File:Regular star polygon 21-2.svg|85px]]<br>{21/2}
|[[File:Regular star polygon 21-4.svg|85px]]<br>{21/4}
|[[File:Regular star polygon 21-5.svg|85px]]<br>{21/5}
|[[File:Regular star polygon 23-8.svg|85px]]<br>{21/8}
|[[File:Regular star polygon 23-10.svg|85px]]<br>{21/10}
|}
==See also==
* [[Heptagon]]
* [[Tetradecagon]] (14-sided)
* [[Icosioctagon]] (28-sided)
* [[Tetracontadigon]] (42-sided)
{{Polygons}}
{{Geometry-stub}}
[[Category:Polygons]]
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