parallelepiped

Parallelepiped is a six-faced polyhedron whose faces are parallelograms. It can be formed by translating a parallelogram along two directions, or equivalently by choosing three edge vectors a, b, c that meet at a vertex. The faces come in three pairs: parallelograms with sides a and b, b and c, and c and a. Opposite faces are parallel and congruent; corresponding edges are parallel in pairs.

Volume is V = |a · (b × c)|, the absolute value of the scalar triple product of a, b, c (the determinant with these vectors as columns). Equivalently, V = |(a × b) · c|. It also equals the base area |a × b| times the height (the component of c perpendicular to the base). Surface area is S = 2(|a × b| + |b × c| + |c × a|).

If the edges are mutually perpendicular, the figure is a rectangular parallelepiped (rectangular prism). If the three edge lengths are equal as well, it is a cube. If all edges have equal length but the angles are not right, it is a rhombohedron (rhombic parallelepiped).

The space diagonals connect opposite vertices; for example from the origin to a + b + c. There are four such diagonals in a parallelepiped, and their lengths may differ in general unless additional symmetry is present.

In coordinates and linear algebra, a parallelepiped is the image of a unit cube under the linear map with matrix having a, b, c as columns. It is a basic model in geometry, crystallography, and computer graphics, useful for computing volumes, areas, and moments under shear.