Icosagon

Regular icosagon
A regular icosagon
Type	general type of this shape
Edges and vertices	20
Schläfli symbol	{20} t{10}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D20)
Area	$A = 5t^2 \cot \frac{\pi}{20}$ $\simeq 31.5688\,t^2$ (with t = edge length)
Internal angle (degrees)	162°
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.

One interior angle in a regular icosagon is 162° meaning that one exterior angle would be 18°.

The regular icosagon is a constructible polygon, by an edge-bisection of a regular decagon, and can be seen as a truncated decagon.

Uses

One of the most famous uses of an icosagon is the Showcase Showdown wheel on the popular US game show The Price Is Right.

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.

Construction

A regular icosagon is constructible using a compass and straightedge:

Construction of a regular icosagon

Petrie polygons

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

A9	19-simplex (19D)
BC10	10-orthoplex	Rectified 10-orthoplex	Trirectified 10-orthoplex	Quadrirectified 10-orthoplex	Quintirectified 10-cube	Quadrirectified 10-cube	Trirectified 10-cube	Birectified 10-cube	Rectified 10-cube
D11	11-demicube
E8 [20]	(421)	t1(421)	t2(421)	t3(421)	t4(421)	t0(241)	t1(241)	t0(142)
H4 [20]	Rectified 600-cell	600-cell

References

• Naming Polygons and Polyhedra