Pentadecagon

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Regular pentadecagon
Regular polygon 15 annotated.svg
A regular pentadecagon
Type Regular polygon
Edges and vertices 15
Schläfli symbol {15}
Coxeter diagram CDel node 1.pngCDel 15.pngCDel node.png
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) is any 15-sided, 15-angled, polygon.

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

Regular pentadecagon[edit]

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


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

Uses[edit]

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

Construction[edit]

A regular pentadecagon is constructible using compass and straightedge:

Regular Pentadecagon Inscribed in a Circle.gif
Construction of a regular pentadecagon

Construction of a regular pentadecagon is Proposition XVI of Book IV of Euclid's Elements.[1]

Pentadecagrams[edit]

There are 3 regular star polygons: {15/2}, {15/4}, {15/7}, constructed from the same 15 vertices of a regular pentadecagon, but connected by skipping every second, fourth, or seventh vertex respectively.

There are also three regular star figures: {15/3}, {15/5}, {15/6}, the first being a compound of 3 pentagons, the second a compound of 5 equilateral triangles, and the third is a compound of 3 pentagrams.

Picture Regular star polygon 15-2.svg
{15/2}
CDel node 1.pngCDel 15.pngCDel rat.pngCDel 2x.pngCDel node.png
Regular star figure 3(5,1).svg
{15/3} or 3{5}
Regular star polygon 15-4.svg
{15/4}
CDel node 1.pngCDel 15.pngCDel rat.pngCDel 4.pngCDel node.png
Regular star figure 5(3,1).svg
{15/5} or 5{3}
Regular star figure 3(5,2).svg
{15/6} or 3{5/2}
Regular star polygon 15-7.svg
{15/7}
CDel node 1.pngCDel 15.pngCDel rat.pngCDel 7.pngCDel node.png
Interior angle 132° 108° 84° 60° 36° 12°

Petrie polygons[edit]

The regular pentadecagon is the Petrie polygon for some higher-dimensional polytopes, projected in a skew orthogonal projection:

14-simplex t0.svg
14-simplex (14D)

It is also the Petrie polygon for the great 120-cell and grand stellated 120-cell.

References[edit]

  1. ^ William Dunham, Journey Through Genius: The Great Theorems of Mathematics, Penguin, 1991, ISBN 014014739X, p. 65.

External links[edit]