Regular 120-gon | |
---|---|

A regular 120-gon | |

Type | Regular polygon |

Edges and vertices | 120 |

Schläfli symbol | {120}, t{60}, tt{30}, ttt{15} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{120}), order 2×120 |

Internal angle (degrees) | 177° |

Dual polygon | Self |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

In geometry, a **120-gon** is a polygon with 120 sides. The sum of any 120-gon's interior angles is 21240 degrees.

Alternative names include **dodecacontagon** and **hecatonicosagon**.^{[1]}

## Contents

## Regular 120-gon properties

A *regular 120-gon* is represented by Schläfli symbol {120} and also can be constructed as a truncated hexacontagon, t{60}, or a twice-truncated triacontagon, tt{30}, or a thrice-truncated pentadecagon, ttt{15}.

One interior angle in a regular 120-gon is 177°, meaning that one exterior angle would be 3°.

The area of a regular 120-gon is (with *t* = edge length)

and its inradius is

The circumradius of a regular 120-gon is

This means that the trigonometric functions of π/120 can be expressed in radicals.

### Constructible

Since 120 = 2^{3} × 3 × 5, a regular 120-gon is constructible using a compass and straightedge.^{[2]} As a truncated hexacontagon, it can be constructed by an edge-bisection of a regular hexacontagon.

## Symmetry

The *regular 120-gon* has Dih_{120} dihedral symmetry, order 240, represented by 120 lines of reflection. Dih_{120} has 15 dihedral subgroups: (Dih_{60}, Dih_{30}, Dih_{15}), (Dih_{40}, Dih_{20}, Dih_{10}, Dih_{5}), (Dih_{24}, Dih_{12}, Dih_{6}, Dih_{3}), and (Dih_{8}, Dih_{4}, Dih_{2}, Dih_{1}). And 16 more cyclic symmetries: (Z_{120}, Z_{60}, Z_{30}, Z_{15}), (Z_{40}, Z_{20}, Z_{10}, Z_{5}), (Z_{24}, Z_{12}, Z_{6}, Z_{3}), and (Z_{8}, Z_{4}, Z_{2},Z_{1}), with Z_{n} representing π/*n* radian rotational symmetry.

These 32 symmetries are related to 44 distinct symmetries on the 120-gon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[3]} He gives **d** (diagonal) with mirror lines through vertices, **p** with mirror lines through edges (perpendicular), **i** with mirror lines through both vertices and edges, and **g** for rotational symmetry. **a1** labels no symmetry.

These lower symmetries allows degrees of freedom in defining irregular 120-gons. Only the **g120** symmetry has no degrees of freedom but can seen as directed edges.

## Dissection

Coxeter states that every zonogon (a 2*m*-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 *regular 120-gon*, *m*=60, and it can be divided into 1770: 30 squares and 29 sets of 60 rhombs. This decomposition is based on a Petrie polygon projection of a *60-cube*.

## 120-gram

A 120-gram is a 120-sided star polygon. There are 15 regular forms given by Schläfli symbols {120/7}, {120/11}, {120/13}, {120/17}, {120/19}, {120/23}, {120/29}, {120/31}, {120/37}, {120/41}, {120/43}, {120/47}, {120/49}, {120/53}, and {120/59}, as well as 44 compound star figures with the same vertex configuration.

Picture | {120} |
{120/7} |
{120/11} |
{120/13} |
{120/17} |
{120/19} |
{120/23} |
{120/29} |
---|---|---|---|---|---|---|---|---|

Interior angle | 177° | 159° | 147° | 141° | 129° | 123° | 111° | 93° |

Picture | {120/31} |
{120/37} |
{120/41} |
{120/43} |
{120/47} |
{120/49} |
{120/53} |
{120/59} |

Interior angle | 87° | 69° | 57° | 51° | 39° | 33° | 21° | 3° |

## References

**^**Norman Johnson, Geometries and Transformations (2018), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5*full polychoric groups***^**Constructible Polygon**^**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)**^**Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141