# Pentacontagon

polygon with 50 sides

In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon.[1][2] The sum of any pentacontagon's interior angles is 8640 degrees.

Regular pentacontagon
A regular pentacontagon
TypeRegular polygon
Edges and vertices50
Schläfli symbol{50}, t{25}
Coxeter diagram
Symmetry groupDihedral (D50), order 2×50
Internal angle (degrees)172.8°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

## 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​45°, meaning that one exterior angle would be 7​15°.

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

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

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

The circumradius of a regular pentacontagon is

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

Since 50 = 2 × 52, 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]

## 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.[5] 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.

## References

1. Gorini, Catherine A. (2003). The Facts on File Geometry Handbook. Infobase Publishing. p. 120. ISBN 978-1-4381-0957-2.
2. Peirce, Charles Sanders (1976). The New Elements of Mathematics: Algebra and geometry. Mouton Publishers. p. 298.
3. Constructible Polygon
4. "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.CS1 maint: archived copy as title (link)
5. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141