For a unit (i.e.: with side length 1) rhombohedron, with rhombic acute angle \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 : \biggl(1, 0, 0\biggr), :
e2 : \biggl(\cos\theta, \sin\theta, 0\biggr), :
e3 : \biggl(\cos\theta, {\cos\theta-\cos^2\theta\over \sin\theta}, {\sqrt{1-3\cos^2\theta+2\cos^3\theta} \over \sin\theta} \biggr). 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 V of a rhombohedron, in terms of its side length a and its rhombic acute angle \theta~, is a simplification of the volume of a
parallelepiped, and is given by :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 V another way : :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 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 h of a rhombohedron in terms of its side length a and its rhombic acute angle \theta is given by :h = a~{(1-\cos\theta)\sqrt{1+2\cos\theta} \over \sin\theta} = a~{\sqrt{1-3\cos^2\theta+2\cos^3\theta} \over \sin\theta}~. Note: :h = a~z
3 , where 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 ==