Rhombohedron

In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) 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 $\theta$.
There are two general forms of the rhombohedron: oblate (flattened) and prolate (stretched).

In the oblate case $\theta > 90^\circ$ and in the prolate case $\theta < 90^\circ$. For $\theta = 90^\circ$ 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.

Solid geometry

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'' : $\biggl(\cos\theta, , \biggr).$

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

:$V = a^3(1-\cos\theta)\sqrt = a^3\sqrt = a^3\sqrt~.$

We can express the volume $V$ another way :

:$V = 2\sqrt ~ a^3 \sin^2\left(\frac\right) \sqrt~.$

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

:$h = a~ = a~~.$

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

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

See also

* Lists of shapes

Notes

References

External links

*
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Volume Calculator https://rechneronline.de/pi/rhombohedron.php
{{Polyhedron navigator

Prismatoid polyhedra
Space-filling polyhedra
Zonohedra