Pentadecagon

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

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.

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

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

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

As 15 = 3 × 5, a regular pentadecagon is constructible using compass and straightedge: The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's Elements.

Comparison the construction according Euclid in this image: Pentadecagon

In the construction for given circumcircle: ${\displaystyle {\overline {FG}}={\overline {CF}}{\text{,}}\;{\overline {AH}}={\overline {GM}}{\text{,}}\;|E_{1}E_{6}|}$ is a side of equilateral triangle and ${\displaystyle |E_{2}E_{5}|}$ is a side of a regular pentagon. The point ${\displaystyle H}$ divides the radius ${\displaystyle {\overline {AM}}}$ in golden ratio: ${\displaystyle {\frac {\overline {AH}}{\overline {HM}}}={\frac {\overline {AM}}{\overline {AH}}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618{\text{.}}}$