Pentadecagon facts for kids

Quick facts for kids 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

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

Regular pentadecagon

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

The amount of space a regular pentadecagon takes up is

\begin{align} 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{align}

a is the length of one of its sides.

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

The symmetries of a regular pentadecagon as shown with colors on edges and vertices. Lines of reflections are blue. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions.