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
- [math]\displaystyle{ \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} }[/math]
a is the length of one of its sides.
Uses
A regular triangle, decagon, and pentadecagon can completely fill a plane vertex.
Pentadecagon Media
Drawing of a regular pentadecagon inscribed in a circle showing animated strightedge and compass
Pentadecagon
Pentadecagon, animation
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.
en:Star polygon
Regular star figure 3(5,1)
en:Star polygon
Regular star figure 5(3,1)
Regular star figure 3(5,2)
en:Star polygon
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