Heptacontagon

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Regular heptacontagon
Regular polygon 70.svg
A regular heptacontagon
Type Regular polygon
Edges and vertices 70
Schläfli symbol {70}
t{35}
Coxeter diagram CDel node 1.pngCDel 7.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 3x.pngCDel 5.pngCDel node 1.png
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] A regular heptacontagon is represented by Schläfli symbol {70} and can be constructed as a quasiregular truncated 35-gon, t{35}, which alternates two types of edges.

The sum of any heptacontagon's interior angles is 12240 degrees.

Regular heptacontagon properties[edit]

One interior angle in a regular heptacontagon is 17467°, meaning that one exterior angle would be 517°.

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[edit]

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}
Star polygon 70-3.svg
{70/3}
Star polygon 70-9.svg
{70/9}
Star polygon 70-11.svg
{70/11}
Star polygon 70-13.svg
{70/13}
Star polygon 70-17.svg
{70/17}
Star polygon 70-19.svg
{70/19}
Star polygon 70-23.svg
{70/23}
Star polygon 70-27.svg
{70/27}
Star polygon 70-29.svg
{70/29}
Star polygon 70-31.svg
{70/31}
Star polygon 70-33.svg
{70/33}

References[edit]