Calculations at a rhombic triacontahedron, the dual body of the icosidodecahedron. A rhombic triacontahedron is a polyhedron with rhombic faces, those have angles of 63.435° and 116.565°. It has twenty vertices with three edges and twelve vertices with five edges.
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The rhombic triacontahedron is a Catalan solid. Edge length and radiuses have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit -1.
Triacontahedron means a thirty-faced polyhedron. While there are other polyhedra with 30 faces, when the term triacontahedron is used without further qualification, it typically refers to the rhombic triacontahedron. The rhombuses of the rhombic triacontahedron are sometimes referred to as golden rhombuses, as the ratio of their diagonals corresponds to the Golden Ratio. Another Catalan solid featuring rhombic faces is the rhombic dodecahedron. In this case, however, the rhombuses exhibit different proportions.
A rhombic triacontahedron can be generated from a cube by subdividing each of the cube's faces into four equal squares and then distorting these squares in three dimensions into rhombuses in such a way that a new rhombus emerges in the center of each face, situated between the original four. The resulting five rhombuses on each face have to be all of equal size. Since the cube has six faces, and each face is transformed into five rhombuses, the resulting form consists of 30 equal-sized rhombuses.
By attaching right pyramids of appropriate height to its faces, the rhombic triacontahedron can be expanded to form a hexakisicosahedron or a deltoidal hexacontahedron.
The rhombic triacontahedron is occasionally used in art and design. There are lampshades and other objects based on this shape, and it is used as a thirty-sided die.