Hexacontagon

A hexacontagon or 60-gon is a shape with 60 sides and 60 corners.^[1][2]

Regular hexacontagon
A regular hexacontagon
Type	Regular polygon
Edges and vertices	60
Schläfli symbol	{60}, t{30}, tt{15}
Coxeter diagram
Symmetry group	Dihedral (D60), order 2×60
Internal angle (degrees)	174°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

Regular hexacontagon

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

One interior angle in a regular hexacontagon is 174°, meaning that one exterior angle would be 6°.

Area

The area of a regular hexacontagon is (with t = edge length)

${\displaystyle A=15t^{2}\cot {\frac {\pi }{60}}}$

and its inradius is

${\displaystyle r={\frac {1}{2}}t\cot {\frac {\pi }{60}}}$

The circumradius of a regular hexacontagon is

${\displaystyle R={\frac {1}{2}}t\csc {\frac {\pi }{60}}}$

This means that the trigonometric functions of π/60 can be expressed in radicals.

Constructible

Since 60 = 2² × 3 × 5, a regular hexacontagon is constructible using a compass and straightedge.^[3] As a truncated triacontagon, it can be constructed by an edge-bisection of a regular triacontagon.

Dissection

60-gon with 1740 rhombs

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 regular hexacontagon, m=30, and it can be divided into 435: 15 squares and 14 sets of 30 rhombs. This decomposition is based on a Petrie polygon projection of a 30-cube.

Examples