Pentacontagon

Template:Regular polygon db 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 172Template:Frac°, meaning that one exterior angle would be 7Template:Frac°.

The area of a regular pentacontagon is (with Template:Nowrap)

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

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.

Dissection

50-gon with 1200 rhombs

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. 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.

Examples

References

• Template:Mathworld
• Naming Polygons and Polyhedra
• Template:Cite book

Template:Math-stub