Pentacontagon

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Regular pentacontagon
Regular polygon 50.svg
A regular pentacontagon
Type Regular polygon
Edges and vertices 50
Schläfli symbol {50}
t{25}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 2x.pngCDel 5.pngCDel node 1.png
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[edit]

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[edit]

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 Star polygon 50-3.svg
{50/3}
Star polygon 50-7.svg
{50/7}
Star polygon 50-9.svg
{50/9}
Star polygon 50-11.svg
{50/11}
Star polygon 50-13.svg
{50/13}
Interior angle 158.4° 129.6° 115.2° 100.8° 86.4°
Picture Star polygon 50-17.svg
{50/17}
Star polygon 50-19.svg
{50/19}
Star polygon 50-21.svg
{50/21}
Star polygon 50-23.svg
{50/23}
 
Interior angle 57.6° 43.2° 28.8° 14.4°  

References[edit]