Heptacontagon

Regular heptacontagon

A regular heptacontagon
Type Regular polygon
Edges and vertices 70
Schläfli symbol {70}
t{35}
Coxeter diagram
Symmetry group Dihedral (D70), order 2×70
Internal angle (degrees) ≈174.9°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a heptacontagon (or hebdomecontagon from Ancient Greek ἑβδομήκοντα, seventy[1]) is a seventy-sided polygon.[2][3] The sum of any heptacontagon's interior angles is 12240 degrees.

A regular heptacontagon is represented by Schläfli symbol {70} and can also be constructed as a quasiregular truncated 35-gon, t{35}, which alternates two types of edges.

Regular heptacontagon properties

One interior angle in a regular heptacontagon is 1746⁄7°, meaning that one exterior angle would be 51⁄7°.

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

A = \frac{35}{2}t^2 \cot \frac{\pi}{70}

and its inradius is

r = \frac{1}{2}t \cot \frac{\pi}{70}

The circumradius of a regular heptacontagon is

R = \frac{1}{2}t \csc \frac{\pi}{70}

A regular heptacontagon is not constructible using a compass and straightedge,[4] but is constructible if the use of an angle trisector is allowed.[5]

Heptacontagram

A heptacontagram is a 70-sided star polygon. There are 11 regular forms given by Schläfli symbols {70/3}, {70/9}, {70/11}, {70/13}, {70/17}, {70/19}, {70/23}, {70/27}, {70/29}, {70/31}, and {70/33}, as well as 23 regular star figures with the same vertex configuration.

Regular star polygons {70/k}
Picture
{70/3}
{70/9}
{70/11}
{70/13}
{70/17}
{70/19}
Interior angle ≈164.571° ≈133.714° ≈123.429° ≈113.143° ≈92.5714° ≈82.2857°
Picture
{70/23}
{70/27}
{70/29}
{70/31}
{70/33}		  
Interior angle ≈61.7143° ≈41.1429° ≈30.8571° ≈20.5714° ≈10.2857°  

References

  1. ↑ Greek Numbers and Numerals (Ancient and Modern) by Harry Foundalis
  2. ↑ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 77, ISBN 9781438109572.
  3. ↑ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  4. ↑ Constructible Polygon
  5. ↑ http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf