# Icosagon

Regular icosagon
A regular icosagon
Type Regular polygon
Edges and vertices 20
Schläfli symbol {20}
t{10}
Coxeter diagram
Symmetry group Dihedral (D20), order 2×20
Internal angle (degrees) 162°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an icosagon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

The regular icosagon is a constructible polygon, by an edge-bisection of a regular decagon, and can be seen as a truncated decagon, t{10}. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°.

## Regular Icosagon

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

$A={5}t^2(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}) \simeq 31.56875757 t^2.$

## Uses

The Big Wheel on the popular US game show The Price Is Right is an icosagon.

The Globe, the outdoor theater used by William Shakespeare's acting company, was discovered to have been built on an icosagonal foundation when a partial excavation was done in 1989.[1]

As a golygonal path, the swastika is considered to be an irregular icosagon.[2]

A regular square, pentagon, and icosagon can completely fill a plane vertex.

### Construction

A regular icosagon is constructible using a compass and straightedge:

Construction of a regular icosagon

## Related polygons

An icosagram is a 20-sided star polygon, represented by symbol {20/n}. There are 3 regular forms given by Schläfli symbols: {20/3}, {20/7}, and {20/9}. There are also 5 regular star figures (compounds) using the same vertex arrangement: 2{10}, 4{5}, 5{4}, 2{10/3}, 4{5/2}, and 10{2}.

n 1 2 3 4 5
Form Convex polygon Compound Star polygon Compound
Image
{20/1} = {20}

{20/2} = 2{10}

{20/3}

{20/4} = 4{5}

{20/5} = 5{4}
Interior angle 162° 144° 126° 108° 90°
n 6 7 8 9 10
Form Compound Star polygon Compound Star polygon Compound
Image
{20/6} = 2{10/3}

{20/7}

{20/8} = 4{5/2}

{20/9}

{20/10} = 10{2}
Interior angle 72° 54° 36° 18°

Deeper truncations of the regular decagon and decagram can produce isogonal (vertex-transitive) intermediate icosagram forms with equally spaced vertices and two edge lengths.[3]

A regular icosagram, {20/9}, can be seen as a quasitruncated decagon, t{10/9}={20/9}. Similarly a decagram, {10/3} has a quasitruncation t{10/7}={20/7}, and finally a simple truncation of a decagram gives t{10/3}={20/3}.

Quasiregular Quasiregular t{10}={20} t{10/9}={20/9} t{10/3}={20/3} t{10/7}={20/7}

## Petrie polygons

The regular icosagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in orthogonal projections in Coxeter planes:

A19 B10 D11 E8 H4 2H2

19-simplex

10-orthoplex

10-cube

11-demicube

(421)

600-cell

10-10 duopyramid

10-10 duoprism

It is also the Petrie polygon for the icosahedral 120-cell, small stellated 120-cell, great icosahedral 120-cell, and great grand 120-cell.

## References

1. ^ [1]
2. ^
3. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum