Octacontagon

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Regular octacontagon
Regular polygon 80.svg
A regular octacontagon
Type Regular polygon
Edges and vertices 80
Schläfli symbol {80}
t{40}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel 0x.pngCDel node 1.png
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[edit]

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[edit]

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.

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

References[edit]