Type Regular polygon
Edges and vertices 15
Schläfli symbol {15}
Coxeter diagram
Symmetry group Dihedral (D15), order 2×15
Internal angle (degrees) 156°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

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

A regular pentadecagon is represented by Schläfli symbol {15}.

## Contents

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}

## Uses

A regular triangle, decagon, and pentadecagon can completely fill a plane vertex.

### Construction

A regular pentadecagon is constructible using compass and straightedge:

Construction of a regular pentadecagon is Proposition XVI of Book IV of Euclid's Elements.[1]

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, fourth, 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.

 Picture Interior angle {15/2} {15/3} or 3{5} {15/4} {15/5} or 5{3} {15/6} or 3{5/2} {15/7} 132° 108° 84° 60° 36° 12°

### Petrie polygons

The regular pentadecagon is the Petrie polygon for some higher-dimensional polytopes, projected in a skew orthogonal projection:

 14-simplex (14D)

It is also the Petrie polygon for the great 120-cell and grand stellated 120-cell.

## References

1. ^ William Dunham, Journey Through Genius: The Great Theorems of Mathematics, Penguin, 1991, ISBN 014014739X, p. 65.