120-gon


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.

Regular 120-gon properties

A regular 120-gon is represented by Schläfli symbol and also can be constructed as a truncated hexacontagon, t, or a twice-truncated triacontagon, tt, or a thrice-truncated pentadecagon, ttt.
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
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 = 23 × 3 × 5, a regular 120-gon is constructible using a compass and straightedge. As a truncated hexacontagon, it can be constructed by an edge-bisection of a regular hexacontagon.

Symmetry

The regular 120-gon has Dih120 dihedral symmetry, order 240, represented by 120 lines of reflection. Dih120 has 15 dihedral subgroups:,,, and. And 16 more cyclic symmetries:,,, and, with Zn 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. He gives d with mirror lines through vertices, p with mirror lines through edges, 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

states that every zonogon can be dissected into m/2 parallelograms. 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,,,,,,,,,,,,,, and, as well as 44 compound star figures with the same vertex configuration.
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Interior angle177°159°147°141°129°123°111°93°
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Interior angle87°69°57°51°39°33°21°3°