# 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.[1][2] 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 25-gon, 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)

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

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

The circumradius of a regular pentacontagon is

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

A regular pentacontagon is not constructible using a compass and straightedge,[3] and is not constructible even if the use of an angle trisector is allowed.[4]

## 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.

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