# Augmented dodecahedron (Johnson solid)

Augmented dodecahedron (Johnson solid) Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymAud
Coxeter diagramoxfoo ooofx&#xt
Elements
Faces5 triangles, 1+5+5 pentagons
Edges5+5+5+5+5+10
Vertices1+5+5+5+5
Vertex figures1 pentagon, edge length 1
5 kites, edge lengths 1 and (1+5)/2
5+5+5 triangles, edge lengths (1+5)/2
Measures (edge length 1)
Volume$\frac{95+43\sqrt5}{24} ≈ 7.96462$ Dihedral angles3–5: $\arccos\left(-\sqrt{\frac{65-2\sqrt5}{75}}\right) ≈ 153.94242°$ 3–3: $\arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°$ 5–5: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$ Central density1
Related polytopes
ArmyAud
RegimentAud
DualMonotruncated icosahedron
ConjugateMonoexcavated great stellated dodecahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The augmented dodecahedron is one of the 92 Johnson solids (J58). It consists of 5 triangles and 1+5+5 pentagons. It can be constructed by attaching a pentagonal pyramid to one of the faces of the regular dodecahedron.

## Vertex coordinates

An augmented dodecahedron of edge length 1 has vertices given by all even permutations of:

• $\left(±\frac{3+\sqrt5}{4},\,±\frac12,\,0\right),$ As well as:

• $\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$ • $\left(0,\,\frac{15+\sqrt5}{20},\,\frac{5+4\sqrt5}{10}\right).$ 