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 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​⁴⁄₅°, meaning that one exterior angle would be 7​¹⁄₅°.

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

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

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

• Eric W. Weisstein, Pentacontagon at MathWorld.
• Naming Polygons and Polyhedra
• Finkel, Benjamin Franklin (1888). A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems: Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. Kibler.