Pentacontagon

Regular pentacontagon
A regular pentacontagon
Type	Regular polygon
Edges and vertices	50
Schläfli symbol	{50}, t{25}
Coxeter diagram
Symmetry group	Dihedral (D50), order 2×50
Internal angle (degrees)	172.8°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon. The sum of any pentacontagon's interior angles is 8640 degrees.

A regular pentacontagon is represented by Schläfli symbol {50} and can be constructed as a quasiregular truncated icosipentagon, t{25}, which alternates two types of edges.

One interior angle in a regular pentacontagon is 172 ⁴⁄₅°, meaning that one exterior angle would be 7 ¹⁄₅°.

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

and its inradius is

The circumradius of a regular pentacontagon is

Since 50 = 2 × 5², a regular pentacontagon is not constructible using a compass and straightedge, and is not constructible even if the use of an angle trisector is allowed.

The regular pentacontagon has Dih₅₀dihedral symmetry, order 100, represented by 50 lines of reflection. Dih₅₀ has 5 dihedral subgroups: Dih₂₅, (Dih₁₀, Dih₅), and (Dih₂, Dih₁). It also has 6 more cyclic symmetries as subgroups: (Z₅₀, Z₂₅), (Z₁₀, Z₅), and (Z₂, Z₁), with Z_n representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. 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.