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 is a fifty-sided polygon or 50-gon. 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.

Regular pentacontagon properties

One interior angle in a regular pentacontagon is 172.8°, meaning that one exterior angle would be 7.2°.

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

$A = \frac{25}{2}t^2 \cot \frac{\pi}{50}$

and its inradius is

$r = \frac{1}{2}t \cot \frac{\pi}{50}$

The circumradius of a regular pentacontagon is

$R = \frac{1}{2}t \csc \frac{\pi}{50}$

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.

Symmetry

The symmetries of a regular pentacontagon. Light blue lines show subgroups of index 2. The 3 boxed subgraphs are positionally related by index 5 subgroups.

The regular pentacontagon has Dih₅₀ dihedral 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.

These lower symmetries allows degrees of freedom in defining irregular pentacontagons. Only the g50 subgroup has no degrees of freedom but can seen as directed edges.

Pentacontagram

A pentacontagram is a 50-sided star polygon. There are 9 regular forms given by Schläfli symbols {50/3}, {50/7}, {50/9}, {50/11}, {50/13}, {50/17}, {50/19}, {50/21}, and {50/23}, as well as 16 compound star figures with the same vertex configuration.

Regular star polygons {50/k}

Picture	{50/3}	{50/7}	{50/9}	{50/11}	{50/13}
Interior angle	158.4°	129.6°	115.2°	100.8°	86.4°
Picture	{50/17}	{50/19}	{50/21}	{50/23}
Interior angle	57.6°	43.2°	28.8°	14.4°

References

• Weisstein, Eric W., "Pentacontagon", MathWorld.
• Naming Polygons and Polyhedra
• Pentecontagon