# Icosagon

polygon with 20 sides

An icosagon is a shape with 20 sides and 20 corners. It has interior angles of 162 and exterior angles of 198.

Regular icosagon
A regular icosagon
TypeRegular polygon
Edges and vertices20
Schläfli symbol{20}, t{10}, tt{5}
Coxeter diagram
Symmetry groupDihedral (D20), order 2×20
Internal angle (degrees)162°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

## Regular icosagon

The regular icosagon has Schläfli symbol {20}, and can also be constructed as a truncated decagon, t{10}, or a twice-truncated pentagon, tt{5}.

### Area

The amount of space a regular icosagon takes up is

${\displaystyle {\text{Area}}={5}a^{2}(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}).}$

a is the length of one of its sides.

## Uses

The Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section.

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.

## Dissection

Coxeter states that every parallel-sided 2m-gon can be divided into m(m-1)/2 rhombs. For the icosagon, m=10, and it can be divided into 45: 5 squares and 4 sets of 10 rhombs. This decomposition is based on a Petrie polygon projection of a 10-cube, with 45 of 11520 faces. [3] The list A006245 enumerates the number of solutions as 18,410,581,880, including up to 20-fold rotations and chiral forms in reflection.

## References

1. Muriel Pritchett, University of Georgia "To Span the Globe", see also Editor's Note, retrieved on 10th January 2016
2. Eric W. Weisstein, Icosagon at MathWorld.
3. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141