Icosagon

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Regular icosagon
Regular polygon 20 annotated.svg
A regular icosagon
Type Regular polygon
Edges and vertices 20
Schläfli symbol {20}
Coxeter diagram CDel node 1.pngCDel 20.pngCDel node.png
Symmetry group 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.

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.

Regular Icosagon[edit]

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[edit]

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]

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

Construction[edit]

A regular icosagon is constructible using a compass and straightedge:

Regular Icosagon Inscribed in a Circle.gif
Construction of a regular icosagon

Petrie polygons[edit]

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

A9 19-simplex t0.svg
19-simplex (19D)
BC10 10-cube t9.svg
10-orthoplex
10-cube t8.svg
Rectified 10-orthoplex
10-cube t7.svg
Birectified 10-orthoplex
10-cube t6.svg
Trirectified 10-orthoplex
10-cube t5.svg
Quadrirectified 10-orthoplex
10-cube t4.svg
Quadrirectified 10-cube
10-cube t3.svg
Trirectified 10-cube
10-cube t2.svg
Birectified 10-cube
10-cube t1.svg
Rectified 10-cube
D11 11-demicube.svg
11-demicube
E8
[20]
4 21 t0 p20.svg
(421)
4 21 t1 p20.svg
t1(421)
4 21 t2 p20.svg
t2(421)
4 21 t3 p20.svg
t3(421)
4 21 t4 p20.svg
t4(421)
2 41 t0 p20.svg
t0(241)
2 41 t1 p20.svg
t1(241)
1 42 t0 p20.svg
t0(142)
H4
[20]
600-cell t1 p20.svg
Rectified 600-cell
600-cell t0 p20.svg
600-cell

References[edit]