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Regular icosagon
A regular icosagon
TypeRegular polygon
Edges and vertices20
Schläfli symbol{20}, t{10}, tt{5}
Coxeter diagramCDel node 1.pngCDel 20.pngCDel node.png
CDel node 1.pngCDel 10.pngCDel node 1.png
Symmetry groupDihedral (D20), order 2×20
Internal angle (degrees)162°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

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

Regular icosagon[change | change source]

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[change | change source]

The amount of space a regular icosagon takes up is

a is the length of one of its sides.

Uses[change | change source]

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]

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

Dissection[change | change source]

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.

Dissection into 45 rhombs
10-cube.svg 20-gon-dissection.svg

References[change | change source]

  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