Enneacontagon

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Regular enneacontagon
Regular polygon 90.svg
A regular enneacontagon
Type Regular polygon
Edges and vertices 90
Schläfli symbol {90}
t{45}
Coxeter diagram CDel node 1.pngCDel 9.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel 5.pngCDel node 1.png
Symmetry group Dihedral (D90), order 2×90
Internal angle (degrees) 176°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an enneacontagon or enenecontagon (from Ancient Greek ἑννενήκοντα, ninety[1]) is a ninety-sided polygon.[2][3] A regular enneacontagon is represented by Schläfli symbol {90} and can be constructed as a quasiregular truncated 45-gon, t{45}, which alternates two types of edges.

The sum of any enneacontagon's interior angles is 15840 degrees.

Regular enneacontagon properties[edit]

One interior angle in a regular enneacontagon is 176°, meaning that one exterior angle would be 4°.

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

A = \frac{45}{2}t^2 \cot \frac{\pi}{90}

and its inradius is

r = \frac{1}{2}t \cot \frac{\pi}{90}

The circumradius of a regular enneacontagon is

R = \frac{1}{2}t \csc \frac{\pi}{90}

A regular enneacontagon is not constructible using a compass and straightedge,[4] but is constructible if the use of an angle trisector is allowed.[5]

Enneacontagram[edit]

An enneacontagram is a 90-sided star polygon. There are 11 regular forms given by Schläfli symbols {90/7}, {90/11}, {90/13}, {90/17}, {90/19}, {90/23}, {90/29}, {90/31}, {90/37}, {90/41}, and {90/43}, as well as 33 regular star figures with the same vertex configuration.

Regular star polygons {90/k}
Star polygon 90-7.svg
{90/7}
Star polygon 90-11.svg
{90/11}
Star polygon 90-13.svg
{90/13}
Star polygon 90-17.svg
{90/17}
Star polygon 90-19.svg
{90/19}
Star polygon 90-23.svg
{90/23}
Star polygon 90-29.svg
{90/29}
Star polygon 90-31.svg
{90/31}
Star polygon 90-37.svg
{90/37}
Star polygon 90-41.svg
{90/41}
Star polygon 90-43.svg
{90/43}

References[edit]