Rhombitruncated Icosidodecahedron20 hexagonal faces
12 10-sided faces 30 square faces |
Rhombicosidodecahedron20 triangular faces
12 pentagonal faces 30 square faces |

If, when truncating the vertices of the icosidodecahedron, we move in exactly half way, the result is the rhombicosidodecahedron.

In the figure above, we can see that the 12 purple pentagonal faces and the 20 green triangular faces are what is left of the faces of the cubocahedron, after truncating the corners of that solid. The 30 tan square faces arise from the truncation procedure.