|Set of trapezohedra|
The name trapezohedron can be misleading as the faces are not trapezoids, but the alternative deltohedron is sometimes confused with the unrelated term deltahedron. It is quite possible, but not necessarily true, that the name trapezohedron could be derived from trapezium (a quadrilateral with no parallel sides) as opposed to trapezoid, since the faces are kites, a type of trapezium.
The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-gonal antiprism has two actual n-gon faces.
An n-gonal trapezohedron can be decomposed into two equal n-gonal pyramids and an n-gonal antiprism.
In the case of the dual of a regular triangular antiprism the kites are rhombi, hence these trapezohedra are also zonohedra. They are called rhombohedron. They are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces.
A special case of a rhombohedron is one of the which the rhombi which form the faces have angles of 60° and 120°. It can be decomposed into two equal regular tetrahedra and a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra and regular octahedra.
The rotation group is Dn of order 2n, except in the case of a cube, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups.