Triacontadigon

Regular triacontadigon
A regular triacontadigon
Type	Regular polygon
Edges and vertices	32
Schläfli symbol	{32}, t{16}, tt{8}, ttt{4}
Coxeter diagram
Symmetry group	Dihedral (D32), order 2×32
Internal angle (degrees)	168.75°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a triacontadigon (or triacontakaidigon) or 32-gon is a thirty-two-sided polygon. In Greek, the prefix triaconta- means 30 and di- means 2. The sum of any triacontadigon's interior angles is 5400 degrees.

An older name is tricontadoagon. Another name is icosidodecagon, suggesting a (20 and 12)-gon, in parallel to the 32-faced icosidodecahedron, which has 20 triangles and 12 pentagons.

The regular triacontadigon can be constructed as a truncated hexadecagon, t{16}, a twice-truncated octagon, tt{8}, and a thrice-truncated square. A truncated triacontadigon, t{32}, is a hexacontatetragon, {64}.

One interior angle in a regular triacontadigon is 168 ¹⁄₄°, meaning that one exterior angle would be 11 ¹⁄₄°.

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

and its inradius is

The circumradius of a regular triacontadigon is

As 32 = 2⁵ (a power of two), the regular triacontadigon is a constructible polygon. It can be constructed by an edge-bisection of a regular hexadecagon.

The regular triacontadigon has Dih₃₂dihedral symmetry, order 64, represented by 32 lines of reflection. Dih₃₂ has 5 dihedral subgroups: Dih₁₆, Dih₈, Dih₄, Dih₂ and Dih₁ and 6 more cyclic symmetries: Z₃₂, Z₁₆, Z₈, Z₄, Z₂, and Z₁, with Z_n representing π/n radian rotational symmetry.