# Dodecagon

Dodecagon
Rank2
TypeRegular
Notation
Bowers style acronymDog
Coxeter diagramx12o ()
Schläfli symbol{12}
Elements
Edges12
Vertices12
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\approx 1.93185}$
Inradius${\displaystyle {\frac {2+{\sqrt {3}}}{2}}\approx 1.86603}$
Area${\displaystyle 3(2+{\sqrt {3}})\approx 11.19615}$
Angle150°
Central density1
Number of external pieces12
Level of complexity1
Related polytopes
ArmyDog
DualDodecagon
ConjugateDodecagram
Abstract & topological properties
Flag count24
Euler characteristic0
OrientableYes
Properties
SymmetryI2(12), order 24
Flag orbits1
ConvexYes
NatureTame

The dodecagon is a polygon with 12 sides. A regular dodecagon has equal sides and equal angles.

The only non-compound stellation of the dodecagon is the dodecagram. This makes it the largest polygon with a single non-compound stellation. The only other polygons with only one are the pentagon, the octagon, and the decagon.

The dodecagon is not found in the non-prismatic uniform polyhedra nor the Johnson solids. It does appear in regular-faced tilings such as the uniform truncated hexagonal tiling (since it's the uniform truncation of the hexagon) and is the largest polygon to do so.

The dodecagon appears as the face of several paracompact regular skew polyhedra, and is the largest planar polygon to appear in a spherical, euclidean, compact, or paracompact polyhedron in 3D space.

## Naming

The name decagon is derived from the Ancient Greek δώδεκα (12) and γωνία (angle), referring to the number of vertices.

Other names include:

• dog, Bowers style acronym, short for "dodecagon".

The combining prefix in BSAs is tw-, as in twip or twaddip.

## Vertex coordinates

Coordinates for a regular dodecagon of unit edge length, centered at the origin, are all permutations of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}}\right)}$.

## Representations

A dodecagon has the following Coxeter diagrams:

• x12o () (full symmetry)
• x6x () (G2 symmetry, generally a dihexagon)
• xy3yx&#zx (A2 symmetry, y = ${\displaystyle 1+{\sqrt {3}}}$)

## Variations

Two main variants of the dodecagon have hexagon symmetry: the dihexagon, with two alternating side lengths and equal angles, and the dual hexambus, with two alternating angles and equal edges. Other less regular variations with chiral hexagonal, square, triangular, rectangular, inversion, mirror, or no symmetry also exist.