Trapezo-Rhombic Dodecahedron


The trapezo-rhombic dodecahedron, also called the rhombo-trapezoidal dodecahedron, is a general dodecahedron consisting of six identical rhombi and six identical isosceles trapezoids. It is convex, space-filling, and has D_(3h) symmetry.

It is implemented in Wolfram Language as PolyhedronData["TrapezoRhombicDodecahedron"].


Its net is illustrated above.


A skeleton based on projecting from along the C_3 symmetry axis and rotating the inner and central points is illustrated above.

If the spheres of face-centered cubic packing are expanded until they fill up the gaps, they form a solid rhombic dodecahedron, and if the spheres of hexagonal close packing are expanded, they form the trapezo-rhombic dodecahedron (Steinhaus 1999, p. 206).

The trapezo-rhombic dodecahedron can be obtained from the rhombic dodecahedron by slicing in half and rotating the two halves 60 degrees with respect to each other. The lengths of the short and long edges of the rotated dodecahedron have lengths 2/3 and 4/3 times the length of the rhombic faces.

The top and bottom edge lengths of the constituent trapezoids are a and 2a, with side lengths (corresponding to the rhombi edge lengths) of 3a/2. The acute angles of the trapezoids and rhombi therefore have angle measure of

=70.52877... degrees

(OEIS A137914).

The surface area and volume are given by


and the moment of inertia tensor by


where I_3 is the 3×3 identity matrix.

See also

Dodecahedron, Hexagonal Close Packing, Space-Filling Polyhedron, Spherical Code

Explore with Wolfram|Alpha


Sloane, N. J. A. Sequence AA137914 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 202-203, 1999.

Cite this as:

Weisstein, Eric W. "Trapezo-Rhombic Dodecahedron." From MathWorld--A Wolfram Web Resource.

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