Snub Disphenoid Calculator

Name: Snub Disphenoid - Geometry Calculator
Author: Jürgen Kummer

Calculations at a snub disphenoid, trigonal dodecahedron or dodecadeltahedron, the Johnson solid J₈₄. This is a deltahedron of 12 equilateral triangles.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

A = 3 \cdot \sqrt{3} \cdot a^{2}

r = 1 + 2 \sqrt{\frac{7}{3}} \cdot \cos [\frac{1}{3} \cdot \arccos (- \frac{3}{7} \sqrt{\frac{3}{7}}) - \frac{2 π}{3}]

r \approx 1.2891685464483

V = \frac{r \cdot a^{3} \cdot (6 \sqrt{3 - r^{2}} + r \sqrt{4 - 2 r^{2}})}{12}

The length has an one-dimensional 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.

The name snub disphenoid refers to the disphenoid. Trigonal dodecahedron and dodecadeltahedron are names derived from Ancient Greek. Trigonal dodecahedron signifies a twelve-faced polyhedron with triangular faces. Dodecadeltahedron is the more precise term, as a deltahedron is defined by having congruent equilateral triangles as faces, so incorporating the specific type of triangle into its name.

The formula for deriving the volume of this shape is exceptionally complicated. The volume formula for the trigondodecahedron can be derived using a three-dimensional coordinate system. First, the vertices are selected in a way that takes advantage of the solid's symmetry. The condition that all edges are of equal length leads to the the cubic equation r³ - 3r² - 4r + 8 = 0. The middle real solution to this equation serves as the auxiliary variable r in the formula. Subsequently, the volume is calculated based on the coordinates of the vertices.

The trigondodecahedron is one of eight convex deltahedra. The other seven comprise the Platonic solids tetrahedron, octahedron and icosahedron and four Johnson solids. Those are the two Johnson bipyramids, the triaugmented triangular prism J₅₁, and the gyroelongated square dipyramid. The trigondodecahedron is the only one of the eight convex deltahedra that is neither regular, nor a bipyramid, nor a solid formed by attaching pyramids to a prism. It is also the only one of these shapes lacking reflectional symmetry, thus occupying a unique position among these deltahedra.

Shapes resembling the trigondodecahedron can occur in chemistry in the form of nanoclusters.

Calculation of volume and other dimensions: snub disphenoid

Last updated on 07/04/2026. Author: Jürgen Kummer

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