Hexacontagon

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

In geometry, a hexacontagon or hexecontagon or 60-gon is a sixty-sided polygon. The sum of any hexacontagon's interior angles is 10440 degrees.

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°.

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

and its inradius is

The circumradius of a regular hexacontagon is

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

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

The regular hexacontagon has Dih₆₀dihedral symmetry, order 120, represented by 60 lines of reflection. Dih₆₀ has 11 dihedral subgroups: (Dih₃₀, Dih₁₅), (Dih₂₀, Dih₁₀, Dih₅), (Dih₁₂, Dih₆, Dih₃), and (Dih₄, Dih₂, Dih₁). And 12 more cyclic symmetries: (Z₆₀, Z₃₀, Z₁₅), (Z₂₀, Z₁₀, Z₅), (Z₁₂, Z₆, Z₃), and (Z₄, Z₂, Z₁), with Z_n representing π/n radian rotational symmetry.