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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 a 15-sided polygon or 15-gon.

Regular pentadecagon[edit]

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

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


3.10.15 vertex.png
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 construction of a regular pentadecagon is Proposition XVI of Book IV of Euclid's Elements. [1]

Comparison the construction according Euclid with this article: Pentadecagon

In the construction for a given circumscribed circle:  \overline{FG} = \overline{CF}\text{,} \; \overline{AH} = \overline{GM}\text{,} \; |E_1E_6| is a side of equilateral triangle and |E_2E_5| is a side of a regular pentagon. The point H divides the radius \overline{AM} in golden ratio: \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), both circle arc are in the following two pictures offset by 90 °, (from C to A).

Regular Pentadecagon Inscribed in a Circle.gif01-Fünfzehneck01-FünfzehneckAnimation

A compass and straightedge construction for a given side length. The construction is nearly equal to that of the pentagon at a given side,[2] then also the presentation is succeed by extension one side and it generates a segment, here \overline{FE_2}\text{,} which is divided according to the golden ratio. \frac{\overline{E_1 E_2}}{\overline{E_1 F}} = \frac{\overline{E_2 F}}{\overline{E_1 E_2}} = \frac{1+ \sqrt{5}}{2} = \Phi \approx 1.618 \text{.}

Construction as 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.

The regular pentadecagon has Dih15 dihedral symmetry, order 30, represented by 15 lines of reflection. Dih15 has 3 dihedral subgroups: Dih5, Dih3, and Dih1. And four more cyclic symmetries: Z15, Z5, Z3, and Z1, with Zn representing π/n radian rotational symmetry.

On the pentadecagon, there are 8 distinct symmetries. John Conway labels these symmetries with a letter and order of the symmetry follows the letter.[3] He gives r30 for the full reflective symmetry, Dih15. He gives d (diagonal) with reflection lines through vertices, p with reflection lines through edges (perpendicular), and for the odd-sided pentadecagon i with mirror lines through both vertices and edges, and g for cyclic symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular pentadecagons. Only the g15 subgroup has no degrees of freedom but can seen as directed edges.


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
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
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
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.


  1. ^ William Dunham. "Journey through Genius The Great Theorems of Mathematics" (PDF). in English (in German). Penguin, 1991. p. 65. Retrieved 2015-11-12.  from good authority: UNIVERSITY OF KENTUCKY College of Arts & Sciences Mathematics
  2. ^ A simplified version of the representation of: Der goldene Schnitt und das regelmäßige Fünfeck; 6th image; Susanne Mueller-Philipp; Uni Münster retrieved on 19. Dezember 2014.
  3. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)

External links[edit]