The gyrate rhombicosidodecahedron can be constructed similarly as rhombicosidodecahedron: it is constructed from parabidiminished rhombicosidodecahedron by attaching two regular pentagonal cupolas onto its decagonal faces. As a result, these pentagonal cupolas cover its dodecagonal faces, so the resulting polyhedron has 20 equilateral triangles, 30 squares, and 10 regular pentagons as its faces. The difference between those two polyhedrons is that one of two pentagonal cupolas from the gyrate rhombicosidodecahedron is rotated through 36°.[2] A convex polyhedron in which all faces are regular polygons is called the Johnson solid, and the gyrate rhombicosidodecahedron is among them, enumerated as the 72th Johnson solid .[3]
The difference between rhombicosidodecahedron (left) and gyrate rhombicosidodecahedron (right) by their construction
Properties
Because the two aforementioned polyhedrons have similar construction, they have the same surface area and volume. A gyrate rhombicosidodecahedron with edge length has a surface area by adding all of the area of its faces:[2]
Its volume can be calculated by slicing it into two regular pentagonal cupolas and one parabigyrate rhombicosidodecahedron, and adding their volumes:[2]