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

As 15 = 3 × 5, a product of distinct Fermat primes, 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.

Compare the construction according to 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: ${\displaystyle {\frac {\overline {AH}}{\overline {HM}}}={\frac {\overline {AM}}{\overline {AH}}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618{\text{.}}}$

Compared with the first animation (with green lines) are in the following two images the two circular arcs (for angles 36° and 24°) rotated 90° counterclockwise shown. They do not use the segment ${\displaystyle {\overline {CG}}}$, but rather they use segment ${\displaystyle {\overline {MG}}}$ as radius ${\displaystyle {\overline {AH}}}$ for the second circular arc (angle 36°).

A compass and straightedge construction for a given side length. The construction is nearly equal to that of the pentagon at a given side, then also the presentation is succeed by extension one side and it generates a segment, here ${\displaystyle {\overline {FE_{2}}}{\text{,}}}$ which is divided according to the golden ratio: