# Augmented dodecahedron (Johnson solid)

Augmented dodecahedron (Johnson solid)
Rank3
TypeCRF
Notation
Bowers style acronymAud
Coxeter diagramoxfoo ooofx&#xt
Stewart notationJ58
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${\displaystyle {\frac {95+43{\sqrt {5}}}{24}}\approx 7.96462}$
Dihedral angles3–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {65-2{\sqrt {5}}}{75}}}\right)\approx 153.94242^{\circ }}$
3–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
5–5: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density1
Number of external pieces16
Level of complexity14
Related polytopes
ArmyAud
RegimentAud
DualMonotruncated icosahedron
ConjugateMonoexcavated great stellated dodecahedron
Abstract & topological properties
Flag count140
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:

• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right)}$,

As well as:

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