Icosagon
Regular icosagon  

A regular icosagon  
Type  Regular polygon 
Edges and vertices  20 
Schläfli symbol  {20}, t{10}, tt{5} 
Coxeter diagram 

Symmetry group  Dihedral (D_{20}), order 2×20 
Internal angle (degrees)  162° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, an icosagon or 20gon is a twentysided polygon. The sum of any icosagon's interior angles is 3240 degrees.
Regular icosagon
The regular icosagon has Schläfli symbol {20}, and can also be constructed as a truncated decagon, t{10}, or a twicetruncated pentagon, tt{5}.
One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°.
The area of a regular icosagon with edge length t is
In terms of the radius R of its circumcircle, the area is
since the area of the circle is the regular icosagon fills approximately 98.36% of its circumcircle.
Uses
The Big Wheel on the popular US game show The Price Is Right has an icosagonal crosssection.
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
As 20 = 2^{2} × 5, regular icosagon is constructible using a compass and straightedge, or by an edgebisection of a regular decagon, or a twicebisected regular pentagon:
Construction of a regular icosagon 
Construction of a regular decagon 
The golden ratio in icosagon
 In the construction with given side length the circular arc around C with radius CD, shares the segment E_{20}F in ratio of the golden ratio.
Symmetry
The regular icosagon has Dih_{20} symmetry, order 40. There are 5 subgroup dihedral symmetries: (Dih_{10}, Dih_{5}), and (Dih_{4}, Dih_{2}, and Dih_{1}), and 6 cyclic group symmetries: (Z_{20}, Z_{10}, Z_{5}), and (Z_{4}, Z_{2}, Z_{1}).
These 10 symmetries can be seen in 16 distinct symmetries on the icosagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.^{[3]} Full symmetry of the regular form is r40 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g20 subgroup has no degrees of freedom but can seen as directed edges.
The highest symmetry irregular icosagons are d20, a isogonal icosagon constructed by ten mirrors which can alternate long and short edges, and p20, an isotoxal icosagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosagon.
Related polygons
An icosagram is a 20sided star polygon, represented by symbol {20/n}. There are three regular forms given by Schläfli symbols: {20/3}, {20/7}, and {20/9}. There are also five 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°  0° 
Deeper truncations of the regular decagon and decagram can produce isogonal (vertextransitive) intermediate icosagram forms with equally spaced vertices and two edge lengths.^{[4]}
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 higherdimensional polytopes, shown in orthogonal projections in Coxeter planes:
A_{19}  B_{10}  D_{11}  E_{8}  H_{4}  2H_{2}  

19simplex 
10orthoplex 
10cube 
11demicube 
(4_{21}) 
600cell 
1010 duopyramid 
1010 duoprism 
It is also the Petrie polygon for the icosahedral 120cell, small stellated 120cell, great icosahedral 120cell, and great grand 120cell.
References
 ↑ Muriel Pritchett, University of Georgia "To Span the Globe", see also Editor's Note, retrieved on 10th January 2016
 ↑ Weisstein, Eric W. "Icosagon". MathWorld.
 ↑ John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275278)
 ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum