Rhombohedron
In geometry, a rhombohedron is a special case of a parallelepiped in which all six faces are congruent rhombi. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. 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.
Special cases
The common angle at the two apices is here given as.There are two general forms of the rhombohedron: oblate and prolate.
In the oblate case and in the prolate case. For the figure is a cube.
Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms.
| Form | Cube | √2 Rhombohedron | Golden Rhombohedron |
| Angle constraints | |||
| Ratio of diagonals | 1 | √2 | Golden ratio |
| Occurrence | Regular solid | Dissection of the rhombic dodecahedron | Dissection of the rhombic triacontahedron |
Solid geometry
For a unit rhombohedron, with rhombic acute angle, with one vertex at the origin, and with one edge lying along the x-axis, the three generating vectors areThe other coordinates can be obtained from vector addition of the 3 direction vectors: e1 + e2, e1 + e3, e2 + e3, and e1 + e2 + e3.
The volume of a rhombohedron, in terms of its side length and its rhombic acute angle, is a simplification of the volume of a parallelepiped, and is given by
We can express the volume another way :
As the area of the base is given by, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height of a rhombohedron in terms of its side length and its rhombic acute angle 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.