Rhombic Dodecahedron Calculator

Name: Rhombic Dodecahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a rhombic dodecahedron, the dual body of the cuboctahedron. A rhombic dodecahedron is a polyhedron with rhombic faces, those have angles of 70.529° and 109.471°. It has eight vertices with three edges and six vertices with four edges.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

A = 8 \cdot a^{2} \cdot \sqrt{2}

V = \frac{16}{9} \cdot a^{3} \cdot \sqrt{3}

r_{m} = \frac{2}{3} \cdot a \cdot \sqrt{2}

r_{i} = \frac{a}{3} \cdot \sqrt{6}

A / V = \frac{9 \cdot \sqrt{2}}{2 \cdot a \cdot \sqrt{3}}

The rhombic dodecahedron is a Catalan solid. Edge length and radiuses have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

Space can be tiled seamlessly using rhombic dodecahedra of equal size. This is the only Catalan solid with this property. In nature, the rhombic dodecahedron is a typical crystal form of gemstones from the garnet group. A three-dimensional projection of the four-dimensional tesseract, when viewed obliquely, has the projection surface of a rhombic dodecahedron.
The rhombic dodecahedron finds application in materials science and crystallography because its geometric structure allows for the efficient packing of atoms or molecules in certain crystal systems. Furthermore, the rhombic dodecahedron plays a role in architecture and design, where its distinctive shape is used for decorative elements or structural solutions. In mathematics, it is studied as an example of a zonohedral polyhedron, which arises from the superposition of planes of symmetry and exhibits interesting properties regarding space filling and symmetry.

If a rhombic dodecahedron is bisected in the middle by the long diagonals of four rhombuses and the resulting rhombuses are extended to form regular hexagons, then an elongated rhombic dodecahedron is created. The rhombic dodecahedron has the same vertex and edge structure as the tricylinder.

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