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Regular icosagon
Regular polygon 20 annotated.svg
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

In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

Regular icosagon[edit]

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}.

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.


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.


As 20 = 22 × 5, regular icosagon is constructible using a compass and straightedge, or by an edge-bisection of a regular decagon, or a twice-bisected regular pentagon:

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

The golden ratio in icosagon[edit]

  • In the construction with given side length the circular arc around C with radius CD, shares the segment E20F in ratio of the golden ratio.
Icosagon with given side length, animation (The construction is very similar to that of decagon with given side length)


Symmetries of a regular icosagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.

The regular icosagon has Dih20 symmetry, order 40. There are 5 subgroup dihedral symmetries: (Dih10, Dih5), and (Dih4, Dih2, and Dih1), and 6 cyclic group symmetries: (Z20, Z10, Z5), and (Z4, Z2, Z1).

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, an 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.


20-gon with 180 rhombs
20-gon rhombic dissection-size2.svg
Isotoxal 20-gon rhombic dissection-size2.svg

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. 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. The list OEISA006245 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
20-gon-dissection.svg 20-gon rhombic dissection2.svg 20-gon rhombic dissectionx.svg 20-gon-dissection-random.svg

Related polygons[edit]

An icosagram is a 20-sided 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 Regular polygon 20.svg
{20/1} = {20}
Regular star figure 2(10,1).svg
{20/2} = 2{10}
Regular star polygon 20-3.svg
Regular star figure 4(5,1).svg
{20/4} = 4{5}
Regular star figure 5(4,1).svg
{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 Regular star figure 2(10,3).svg
{20/6} = 2{10/3}
Regular star polygon 20-7.svg
Regular star figure 4(5,2).svg
{20/8} = 4{5/2}
Regular star polygon 20-9.svg
Regular star figure 10(2,1).svg
{20/10} = 10{2}
Interior angle 72° 54° 36° 18°

Deeper truncations of the regular decagon and decagram can produce isogonal (vertex-transitive) intermediate icosagram forms with equally spaced vertices and two edge lengths.[5]

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}.

Icosagrams as truncations of a regular decagons and decagrams, {10}, {10/3}
Quasiregular Quasiregular
Regular polygon truncation 10 1.svg
Regular polygon truncation 10 2.svg Regular polygon truncation 10 3.svg Regular polygon truncation 10 4.svg Regular polygon truncation 10 5.svg Regular polygon truncation 10 6.svg
Regular star truncation 10-3 1.svg
Regular star truncation 10-3 2.svg Regular star truncation 10-3 3.svg Regular star truncation 10-3 4.svg Regular star truncation 10-3 5.svg Regular star truncation 10-3 6.svg

Petrie polygons[edit]

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

A19 B10 D11 E8 H4 ½2H2 2H2
19-simplex t0.svg
10-cube t9.svg
10-cube t0.svg
4 21 t0 p20.svg
600-cell t0 p20.svg
Grand antiprism 20-gonal orthogonal projection.png
Grand antiprism
10-10 duopyramid ortho-3.png
10-10 duopyramid
10-10 duoprism ortho-3.png
10-10 duoprism

It is also the Petrie polygon for the icosahedral 120-cell, small stellated 120-cell, great icosahedral 120-cell, and great grand 120-cell.


  1. ^ Muriel Pritchett, University of Georgia "To Span the Globe", see also Editor's Note, retrieved on 10 January 2016
  2. ^ Weisstein, Eric W. "Icosagon". MathWorld.
  3. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
  4. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  5. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

External links[edit]