The rhombic triacontahedron is a zonohedron which is the dual polyhedron of the icosidodecahedron (Holden 1971, p. 55). It is composed of 30 golden rhombi joined at 32 vertices. It is a zonohedron and one of the five golden isozonohedra. It is illustrated above together with a wireframe version and a net that can be used for its construction.
It is Wenninger dual 
.
The intersecting edges of the dodecahedron-icosahedron compound form the diagonals of 30 rhombi which comprise the triacontahedron. The cube 5-compound has the 30 facial planes of the rhombic triacontahedron and its interior is a rhombic triacontahedron (Wenninger 1983, p. 36; Ball and Coxeter 1987).
More specifically, a tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the rhombic triacontahedron (E. Weisstein, Dec. 25-27, 2009).
The rhombic triacontahedron is implemented in the Wolfram Language as PolyhedronData["RhombicTriacontahedron"].
The short diagonals of the faces of the rhombic triacontahedron give the edges of a dodecahedron, while the long diagonals give the edges of the icosahedron (Steinhaus 1999, pp. 209-210).
Taken together, the dodecahedron and icosahedron give a dodecahedron-icosahedron compound.
The rhombic triacontahedron is the convex hull of the dodecahedron-icosahedron compound and the hull of the first icosidodecahedron stellation.
The rhombs of the rhombic triacontahedron generated from an icosidodecahedron of unit edge lengths have dimensions
 |  |  |
(1)
|
 |  |  |
(2)
|
giving a ratio
 |
(3)
|
where 
is the golden ratio, making them golden
rhombi. The edge length are therefore
 |
(4)
|
The rhombs are tangential quadrilaterals with inradius
 |
(5)
|
The dihedral angle between adjacent faces is 
,
while the dihedral angle between faces sharing
only a single point in common is 
.
12 rhombic triacontahedra can be fit together about a central rhombic hexecontahedron, as illustrated above (Kabai 2002, p. 173).
The rhombic triacontahedron has inradius
 |
(6)
|
A rhombic triacontahedron with edge length 
has surface area and volume
given by
 |  |  |
(7)
|
 |  |  |
(8)
|
and inertia tensor given by
![I=[1/(75)(35+12sqrt(5))Ma^2 0 0; 0 1/(75)(35+12sqrt(5))Ma^2 0; 0 0 1/(75)(35+12sqrt(5))Ma^2].](/images/equations/RhombicTriacontahedron/NumberedEquation5.svg) |
(9)
|
