Icosihenagon

In geometry, an icosihenagon or 21-gon is a twenty-one-sided polygon. The sum of any icosihenagon's interior angles is 3420 degrees. An icosihenagon has 189 diagonals.

Regular icosihenagon
A regular icosihenagon
Type	Regular polygon
Edges and vertices	21
Schläfli symbol	{21}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D21), order 2×21
Internal angle (degrees)	≈162.857°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

Regular form

An angle of a regular icosihenagon is ${\frac {1140}{7}}=162.857142857...$ degrees.

Since $\varphi$(21) = 12 is 3-smooth but not power of 2, thus the regular icosihenagon is constructible using neusis, or an angle trisector, but not constructible using a compass and straightedge.

The area of a regular icosihenagon with edge length a is

$S={\frac {21}{4}}a^{2}\cot {\frac {\pi }{21}}\simeq 34.83147a^{2}$

$\cos(2\pi /21)$ can be written using only square roots and cube roots.

$\cos {\frac {2\pi }{21}}=\cos \left({\frac {2\pi }{3}}-{\frac {4\pi }{7}}\right)$

$\cos {\frac {2\pi }{21}}={\frac {1+{\sqrt {21}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(42{\sqrt {3}}-18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(18{\sqrt {7}}-42{\sqrt {3}}\right)i}}}{12}}$, see Trigonometric constants expressed in real radicals.

Related figures

Below is a table of five regular icosihenagrams, or star 21-gons, labeled with their respective Schläfli symbol {21/q}, 2 ≤ q ≤ 10 where gcd(q,21) = 1.

{21/2}

{21/4}

{21/5}

{21/8}

{21/10}

See also

• Heptagon
• Tetradecagon (14-sided)
• Icosioctagon (28-sided)
• Tetracontadigon (42-sided)