Pentadecagon

A pentadecagon or quindecagon or 15-gon is a shape with 15 sides and 15 corners.

Regular pentadecagon
A regular pentadecagon
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

Regular pentadecagon

All sides of a regular pentadecagon are the same length. Each corner is 156°. All corners added together equal 2340°.

Area

The amount of space a regular pentadecagon takes up is

${\begin{aligned}A={\frac {15}{4}}a^{2}\cot {\frac {\pi }{15}}&={\frac {15}{4}}{\sqrt {7+2{\sqrt {5}}+2{\sqrt {15+6{\sqrt {5}}}}}}a^{2}\\&={\frac {15a^{2}}{8}}\left({\sqrt {3}}+{\sqrt {15}}+{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}\right)\\&\simeq 17.6424\,a^{2}.\end{aligned}}$

a is the length of one of its sides.

Uses

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