Enneacontagon

Regular enneacontagon

A regular enneacontagon
Type Regular polygon
Edges and vertices 90
Schläfli symbol {90}
t{45}
Coxeter diagram
Symmetry group Dihedral (D90), order 2×90
Internal angle (degrees) 176°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an enneacontagon or enenecontagon (from Ancient Greek ἑννενήκοντα, ninety[1]) is a ninety-sided polygon.[2][3] The sum of any enneacontagon's interior angles is 15840 degrees.

A regular enneacontagon is represented by Schläfli symbol {90} and can be constructed as a quasiregular truncated 45-gon, t{45}, which alternates two types of edges.

Regular enneacontagon properties

One interior angle in a regular enneacontagon is 176°, meaning that one exterior angle would be 4°.

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

A = \frac{45}{2}t^2 \cot \frac{\pi}{90}

and its inradius is

r = \frac{1}{2}t \cot \frac{\pi}{90}

The circumradius of a regular enneacontagon is

R = \frac{1}{2}t \csc \frac{\pi}{90}

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

Enneacontagram

An enneacontagram is a 90-sided star polygon. There are 11 regular forms given by Schläfli symbols {90/7}, {90/11}, {90/13}, {90/17}, {90/19}, {90/23}, {90/29}, {90/31}, {90/37}, {90/41}, and {90/43}, as well as 33 regular star figures with the same vertex configuration.

Regular star polygons {90/k}
Pictures
{90/7}
{90/11}
{90/13}
{90/17}
{90/19}
{90/23}
Interior angle 152° 136° 128° 112° 104° 88°
Pictures
{90/29}
{90/31}
{90/37}
{90/41}
{90/43}		  
Pictures 64° 56° 32° 16° 8°  

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. 57, 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