# Octacontagon

Regular octacontagon
A regular octacontagon
Type Regular polygon
Edges and vertices 80
Schläfli symbol {80}
t{40}
Coxeter diagram
Symmetry group Dihedral (D80), order 2×80
Internal angle (degrees) 175.5°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an octacontagon (or ogdoëcontagon from Ancient Greek ὁγδοήκοντα, eighty[1]) is an eighty-sided polygon.[2][3] The sum of any octacontagon's interior angles is 14040 degrees.

A regular octacontagon is represented by Schläfli symbol {80} and can also be constructed as a quasiregular truncated tetracontagon, t{40}, which alternates two types of edges.

## Regular octacontagon properties

One interior angle in a regular octacontagon is 175.5°, meaning that one exterior angle would be 4.5°.

The area of a regular octacontagon is (with t = edge length)

$A = 20t^2 \cot \frac{\pi}{80}$

and its inradius is

$r = \frac{1}{2}t \cot \frac{\pi}{80}$

The circumradius of a regular octacontagon is

$R = \frac{1}{2}t \csc \frac{\pi}{80}$

A regular octacontagon is constructible using a compass and straightedge.[4] As a truncated tetracontagon, it can be constructed by an edge-bisection of a regular tetracontagon. This means that the trigonometric functions of π/80 can be expressed in radicals:

$\sin\frac{\pi}{80}=\sin 2.25^\circ=\frac{1}{8}(1+\sqrt{5})(-\sqrt{2-\sqrt{2+\sqrt{2}}}-\sqrt{(2+\sqrt{2})(2-\sqrt{2+\sqrt{2}})})$
$+\frac{1}{4}\sqrt{\frac{1}{2}(5-\sqrt{5})}(\sqrt{(2+\sqrt{2})(2+\sqrt{2+\sqrt{2}})}-\sqrt{2+\sqrt{2+\sqrt{2}}})$
$\cos\frac{\pi}{80}=\cos 2.25^\circ=\sqrt{\frac{1}{2}+\frac{1}{4}\sqrt{\frac{1}{2}(4+\sqrt{2(4+\sqrt{2(5+\sqrt{5})})})}}$

## Octacontagram

An octacontagram is an 80-sided star polygon. There are 15 regular forms given by Schläfli symbols {80/3}, {80/7}, {80/9}, {80/11}, {80/13}, {80/17}, {80/19}, {80/21}, {80/23}, {80/27}, {80/29}, {80/31}, {80/33}, {80/37}, and {80/39}, as well as 24 regular star figures with the same vertex configuration.

 Picture Interior angle Picture Interior angle {80/3} {80/7} {80/9} {80/11} {80/13} {80/17} {80/19} {80/21} 166.5° 148.5° 139.5° 130.5° 121.5° 103.5° 94.5° 85.5° {80/23} {80/27} {80/29} {80/31} {80/33} {80/37} {80/39} 76.5° 58.5° 49.5° 40.5° 31.5° 13.5° 4.5°

## References

1. ^ Greek Numbers and Numerals (Ancient and Modern) by Harry Foundalis
2. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 110, ISBN 9781438109572.
3. ^
4. ^ Constructible Polygon