# Dodecagon

Dodecagon Rank2
TypeRegular
SpaceSpherical
Bowers style acronymDog
Info
Coxeter diagramx12o
Schläfli symbol{12}
SymmetryI2(12), order 24
ArmyDog
Elements
Edges12
Vertices12
Measures (edge length 1)
Circumradius$\frac{\sqrt2+\sqrt6}{2} ≈ 1.93185$ Inradius$\frac{2+\sqrt3}{2} ≈ 1.86603$ Area$3(2+\sqrt3) ≈ 11.19615$ Angle150°
Central density1
Euler characteristic0
Number of pieces12
Level of complexity1
Related polytopes
DualDodecagon
ConjugateDodecagram
Properties
ConvexYes
OrientableYes
NatureTame

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

The combining prefix is twa-, as in twaddip.

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.

Regular dodecagons dgenerally do not occur in higher spherical polytopes aside from prisms, though some Euclidean tilings with hexagonal tiling symmetry do use dodecagonal faces.

## 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"

## Vertex coordinates

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

• $\left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),$ • $\left(±\frac12,\,±\frac{2+\sqrt3}{2}\right).$ ## Representations

A dodecagon has the following Coxeter diagrams:

• x12o (full symmetry)
• x6x (G2 symmetry, generally a dihexagon)
• xy3yx&#zx (A2 symmetry, y = 1+√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 square, triangular, rectangular, mirror, or no symmetry also exist.