The trapezo-rhombic dodecahedron, also called the rhombo-trapezoidal dodecahedron, is the Voronoi cell of a sphere
center in ideal hexagonal close packing.
Its six identical rhombi and six identical isosceles
trapezoids form a convex, space-filling dodecahedron with 
symmetry.
It is implemented in Wolfram Language as PolyhedronData["TrapezoRhombicDodecahedron"].
Its net is illustrated above.
A skeleton based on projecting from along the 
symmetry axis
and rotating the inner and central points is illustrated
above.
The Voronoi cells of the sphere centers in face-centered cubic packing are rhombic dodecahedra, while those in ideal hexagonal close packing are trapezo-rhombic dodecahedra (Steinhaus 1999, p. 206).
The trapezo-rhombic dodecahedron can be obtained from the rhombic dodecahedron by cutting through its center perpendicular
to a threefold symmetry axis
and rotating one half 
about that axis. The short
and long edges of its trapezoids
have lengths 2/3 and 4/3 times the edge length
of the original rhombic faces.
The top and bottom edge lengths of the constituent trapezoids are 
and 
,
with side lengths (corresponding to the rhombi edge
lengths) of 
. The acute angles
of the trapezoids and rhombi
therefore have angle measure of
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
(OEIS A137914).
The surface area and volume are given by
 |  |  |
(4)
|
 |  |  |
(5)
|
and the central moment of inertia tensor of the solid with uniform density and mass 
by
 |
(6)
|
where 
is the 
identity matrix. For unit mass and shortest edge
length 
, this is 
.
