Rhombohedron

Solid geometry

For a unit (i.e.: with side length 1) rhombohedron, with rhombic acute angle ${\displaystyle \theta }$, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are

e₁ : ${\displaystyle (1,0,0),}$

e₂ : ${\displaystyle (\cos \theta ,\sin \theta ,0),}$

e₃ : ${\displaystyle (\cos \theta ,\frac{\cos \theta -{\cos }^2\theta }{\sin \theta },\frac{\sqrt{1-3{\cos }^2\theta +2{\cos }^3\theta }}{\sin \theta }).}$

The 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 ${\displaystyle V}$ of a rhombohedron, in terms of its side length ${\displaystyle a}$ and its rhombic acute angle ${\displaystyle \theta }$, is a simplification of the volume of a parallelepiped, and is given by

${\displaystyle V=a^3(1-\cos \theta )\sqrt{1+2\cos \theta }=a^3\sqrt{(1-\cos \theta {)}^2(1+2\cos \theta )}=a^3\sqrt{1-3{\cos }^2\theta +2{\cos }^3\theta }.}$

We can express the volume ${\displaystyle V}$ another way :

${\displaystyle V=2\sqrt{3}{a}^3{\sin }^2\left(\frac{\theta }{2}\right)\sqrt{1-\frac{4}{3}{\sin }^2\left(\frac{\theta }{2}\right)}.}$

As the area of the (rhombic) base is given by ${\displaystyle a^2\sin \theta }$, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height ${\displaystyle h}$ of a rhombohedron in terms of its side length ${\displaystyle a}$ and its rhombic acute angle ${\displaystyle \theta }$ is given by

${\displaystyle h=a\frac{(1-\cos \theta )\sqrt{1+2\cos \theta }}{\sin \theta }=a\frac{\sqrt{1-3{\cos }^2\theta +2{\cos }^3\theta }}{\sin \theta }.}$

Note:

${\displaystyle h=az}$₃ , where ${\displaystyle z}$₃ is the third coordinate of e₃ .

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.

Relation to orthocentric tetrahedra

Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, and all orthocentric tetrahedra can be formed in this way.

Rhombohedral lattice

The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron: