Hectogon

Regular hectogon
A regular hectogon
Type Regular polygon
Edges and vertices 100
Schläfli symbol {100}
t{50}
Coxeter diagram
Symmetry group Dihedral (D100), order 2×100
Internal angle (degrees) 176.4°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a hectogon or hecatontagon[1][2] is a hundred-sided polygon.[3][4] A regular hectogon is represented by Schläfli symbol {100} and can be constructed as a quasiregular truncated pentacontagon, t{50}, which alternates two types of edges.

The sum of any hectogon's interior angles is 17640 degrees.

Regular hectogon properties

One interior angle in a regular hectogon is 176.4°, meaning that one exterior angle would be 3.6°.

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

$A = 25t^2 \cot \frac{\pi}{100}$

$r = \frac{1}{2}t \cot \frac{\pi}{100}$

The circumradius of a regular hectogon is

$R = \frac{1}{2}t \csc \frac{\pi}{100}$

A regular hectogon is not constructible using a compass and straightedge,[5] and is not constructible even if the use of an angle trisector is allowed.[6]

Hectogram

A hectogram is an 100-sided star polygon. There are 19 regular forms[7] given by Schläfli symbols {100/3}, {100/7}, {100/9}, {100/11}, {100/13}, {100/17}, {100/19}, {100/21}, {100/23}, {100/27}, {100/29}, {100/31}, {100/33}, {100/37}, {100/39}, {100/41}, {100/43}, {100/47}, and {100/49}, as well as 30 regular star figures with the same vertex configuration.

 {100/3} {100/7} {100/11} {100/13} {100/17} {100/19} {100/21} {100/23} {100/27} {100/29} {100/31} {100/37} {100/39} {100/41} {100/43} {100/47} {100/49}

References

1. ^ [1]
2. ^ [2]
3. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 110, ISBN 9781438109572.
4. ^
5. ^ Constructible Polygon
6. ^ http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf
7. ^ 19 = 50 cases - 1 (convex) - 10 (multiples of 5) - 25 (multiples of 2)+ 5 (multiples of 2 and 5)