In geometry, a rhombohedron (also called a rhombic hexahedron[1][2] or, inaccurately, a rhomboid[a]) is a special case of a parallelepiped in which all six faces are congruent rhombi.
A rhombohedron has two opposite apices at which all face angles are equal; a prolate rhombohedron has this common angle acute, and an oblate rhombohedron has an obtuse angle at these vertices.
A cube is a special case of a rhombohedron with all sides square.
There are two general forms of the rhombohedron: oblate (flattened) and prolate (stretched).
Certain proportions of the rhombs give rise to some well-known special cases.
These typically occur in both prolate and oblate forms.
For a unit (i.e.: with side length 1) rhombohedron,[4] with rhombic acute angle
of a rhombohedron, in terms of its side length
of a rhombohedron in terms of its side length
is given by Note: The body diagonal between the acute-angled vertices is the longest.
By rotational symmetry about that diagonal, the other three body diagonals, between the three pairs of opposite obtuse-angled vertices, are all the same length.
[6] The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron[citation needed]: