Hexacontagon

Regular hexacontagon

A regular hexacontagon
Type Regular polygon
Edges and vertices 60
Schläfli symbol {60}
t{30}
Coxeter diagram
Symmetry group Dihedral (D60), order 2×60
Internal angle (degrees) 174°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a hexacontagon or hexecontagon is a sixty-sided polygon.[1][2] The sum of any hexacontagon's interior angles is 10440 degrees.

A regular hexacontagon is represented by Schläfli symbol {60} and also can be constructed as a quasiregular truncated triacontagon, t{30}, which alternates two types of edges.

Regular hexacontagon properties

One interior angle in a regular hexacontagon is 174°, meaning that one exterior angle would be 6°.

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

A = 15t^2 \cot \frac{\pi}{60}

and its inradius is

r = \frac{1}{2}t \cot \frac{\pi}{60}

The circumradius of a regular hexacontagon is

R = \frac{1}{2}t \csc \frac{\pi}{60}

A regular hexacontagon is constructible using a compass and straightedge.[3] As a truncated triacontagon, it can be constructed by an edge-bisection of a regular triacontagon. This means that the trigonometric functions of π/60 can be expressed in radicals:

\sin\frac{\pi}{60}=\sin 3^\circ=\tfrac{1}{16} \left[2(1-\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3+1)\right]\,
\cos\frac{\pi}{60}=\cos 3^\circ=\tfrac{1}{16} \left[2(1+\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3-1)\right]\,
\tan\frac{\pi}{60}=\tan 3^\circ=\tfrac{1}{4} \left[(2-\sqrt3)(3+\sqrt5)-2\right]\left[2-\sqrt{2(5-\sqrt5)}\right]\,
\cot\frac{\pi}{60}=\cot 3^\circ=\tfrac{1}{4} \left[(2+\sqrt3)(3+\sqrt5)-2\right]\left[2+\sqrt{2(5-\sqrt5)}\right]\,

Hexacontagram

A hexacontagram is a 60-sided star polygon. There are 7 regular forms given by Schläfli symbols {60/7}, {60/11}, {60/13}, {60/17}, {60/19}, {60/23}, and {60/29}, as well as 22 compound star figures with the same vertex configuration.

Regular star polygons {60/k}
Picture
{60/7}
{60/11}
{60/13}
{60/17}
{60/19}
{60/23}
{60/29}
Interior angle 138° 114° 102° 78° 66° 42° 6°

References

  1. ↑ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 78, ISBN 9781438109572.
  2. ↑ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  3. ↑ Constructible Polygon