Pentacontagon

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.

 Regular pentacontagon 
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.

 Area 
One interior angle in a regular pentacontagon is 172°, meaning that one exterior angle would be 7°.

The area of a regular pentacontagon is (with )

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,Constructible Polygon and is not constructible even if the use of an angle trisector is allowed.

 Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular pentacontagon, m=25, it can be divided into 300: 12 sets of 25 rhombs. This decomposition is based on a Petrie polygon projection of a 25-cube.

+ Examples160px160px160px160px

References


 Naming Polygons and Polyhedra 


50