# Small rhombicosidodecahedron

The small rhombicosidodecahedron, or srid, also commonly known as simply the rhombicosidodecahedron, is one of the 13 Archimedean solids. It consists of 20 triangles, 30 squares, and 12 pentagons, with 1 triangle, 2 squares, and 1 pentagon meeting at each vertex. It can be obtained by cantellation of the dodecahedron or icosahedron, or equivalently by expanding either polyhedron's faces outward.

Small rhombicosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSrid
Coxeter diagramx5o3x ()
Elements
Faces20 triangles, 30 squares, 12 pentagons
Edges60+60
Vertices60
Vertex figureIsosceles trapezoid, edge lengths 1, 2, (1+5)/2, 2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{11+4\sqrt5}}{2} ≈ 2.23295}$
Volume${\displaystyle \frac{60+29\sqrt5}{3} ≈ 41.61532}$
Dihedral angles4–3: ${\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484^\circ}$
5–4: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253^\circ}$
Central density1
Number of external pieces62
Level of complexity4
Related polytopes
ArmySrid
RegimentSrid
DualDeltoidal hexecontahedron
ConjugateQuasirhombicosidodecahedron
Abstract & topological properties
Flag count480
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

## Vertex coordinates

A small rhombicosidodecahedron of edge length 1 has vertex coordinates given by all permutations of

• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12\right),}$

along with all even permutations of

• ${\displaystyle \left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4}\right).}$

## Representations

A small rhombicosidodecahedron has the following Coxeter diagrams:

• x5o3x (full symmetry)
• oxxFofxx5xxfoFxxo&#xt (H2 axial, pentagon-first)
• xx(oF)fVxF(Vx)fo3of(Vx)FxVf(oF)xx&#xt (A2 symmetry, triangle-first)

## Semi-uniform variant

The small rhombicosidodecahedron has a semi-uniform variant of the form x5o3y that maintains its full symmetry. This variant has 12 pentagons of side length x, 20 triangles of size length y, and 30 rectangles as faces.

With edges of length a (of pentagons) and b (of triangles), its circumradius is given by ${\displaystyle \sqrt{\frac{9a^2+5b^2+8ab+(3a^2+b^2+4ab)\sqrt5}{8}}}$  and its volume is given by ${\displaystyle \frac{15a^3+45a^2b+15ab^2+5b^2}{4}+(21a^3+45a^2b+45ab^2+5b^3)\frac{\sqrt5}{12}}$ .

## Related polyhedra

The small rhombicosidodecahedron is the colonel of a three-member regiment that also includes the small dodecicosidodecahedron and the small rhombidodecahedron.

It is possible to cut off a pentagonal cupola cap from the rhombicosidodecahedron to diminish it, or to gyrate any such cap by 36° (so squares connect to other squares, and triangles connect to pentagons). The various combinations of diminishings and gyrations lead to a total of 12 Johnson solids:

o5o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Dodecahedron doe {5,3} x5o3o
Truncated dodecahedron tid t{5,3} x5x3o
Icosidodecahedron id r{5,3} o5x3o
Truncated icosahedron ti t{3,5} o5x3x
Icosahedron ike {3,5} o5o3x
Small rhombicosidodecahedron srid rr{5,3} x5o3x
Great rhombicosidodecahedron grid tr{5,3} x5x3x
Snub dodecahedron snid sr{5,3} s5s3s