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}

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}

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.

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

  1. ↑ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 120, ISBN 9781438109572.
  2. ↑ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  3. ↑ Constructible Polygon
  4. ↑ http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf