Pentadecagon

Regular pentadecagon
A regular pentadecagon
Type	Regular polygon
Edges and vertices	15
Schläfli symbol	{15}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D15)
Internal angle (degrees)	156°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a pentadecagon (or pentakaidecagon) is any 15-sided, 15-angled, polygon.

Regular pentadecagon

A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by

$\begin{align} A & = \frac{15}{4}a^2 \cot \frac{\pi}{15} \\ & = \frac{15a^2}{8} \left( \sqrt{3}+\sqrt{15}+ \sqrt{2}\sqrt{5+\sqrt{5}} \right) \\ & \simeq 17.6424\,a^2. \end{align}$

Construction

A regular pentadecagon is constructible using compass and straightedge:

Construction of a regular pentadecagon

Pentadecagrams

There are 3 regular star polygons: {15/2}, {15/4}, {15/7}, constructed from the same 15 vertices of a regular pentadecagon, but connected by skipping every second, forth, or seventh vertex respectively.

There are also three regular star figures: {15/3}, {15/5}, {15/6}, the first being a compound of 3 pentagons, the second a compound of 5 equilateral triangles, and the third is a compound of 3 pentagrams.

Petrie polygons

The regular pentadecagon is the Petrie polygon for one higher dimensional polytope, projected in a skew orthogonal projection:

14-simplex (14D)

External links

• Weisstein, Eric W., "Pentadecagon" from MathWorld.