Tridecagon

Regular tridecagon
A regular tridecagon
Type	Regular polygon
Edges and vertices	13
Schläfli symbol	{13}
Coxeter diagram
Symmetry group	Dihedral (D13), order 2×13
Internal angle (degrees)	≈152.308°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

A tridecagon or triskaidecagon or trisdecagon or 13-gon is a shape with 13 sides and 13 corners.

Regular tridecagon

All sides of a regular tridecagon are the same length. Each corner is about 152.3°. All corners added together equal 1980°.

Symmetry

Symmetries of a regular tridecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices and edge. Gyration orders are given in the center.

The regular tridecagon has Dih<sub id="mwYQ">13</sub> symmetry, order 26. Since 13 is a prime number there is one subgroup with dihedral symmetry: Dih₁, and 2 cyclic group symmetries: Z₁₃, and Z₁.

These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r26 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g13 subgroup has no degrees of freedom but can be seen as directed edges.

Area

The amount of space a regular tridecagon takes up is

$A={\frac {13}{4}}a^{2}\cot {\frac {\pi }{13}}\simeq 13.1858\,a^{2}.$

a is the length of one of its sides.

Numismatic use

The regular tridecagon is used as the shape of the Czech 20 korun coin.

Related pages

• Triskaidekaphobia

Other websites

• Eric W. Weisstein, Tridecagon at MathWorld.