In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—parallelepiped and cube in three dimensions, parallelogram and square in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only parallelograms and parallelepipeds exist. Three equivalent definitions of parallelepiped are a polyhedron with six faces (hexahedron), each of which is a parallelogram, a hexahedron with three pairs of parallel faces, and a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. "Parallelepiped" is now usually pronounced or ; traditionally it was in accordance with its etymology in Greek παραλληλεπίπεδον parallelepipedon, a body "having parallel planes". Parallelepipeds are a subclass of the prismatoids. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four parallel edges; the edges within each set are of equal length. Parallelepipeds result from linear transformations of a cube (for the non-degenerate cases: the bijective linear transformations). Since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry (see also triclinic). Each face is, seen from the outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not. A space-filling tessellation is possible with congruent copies of any parallelepiped. Volume A parallelepiped can be considered as an oblique prism with a parallelogram as base. Hence the volume of a parallelepiped is the product of the base area and the height (see diagram). With (where is the angle between vectors and ), and (where is the angle between vector and the normal to the base), one gets: The mixed product of three vectors is called triple product. It can be described by a determinant. Hence for the volume is: Another way to prove () is to use the scalar component in the direction of of vector : The result follows. An alternative representation of the volume uses geometric properties (angles and edge lengths) only: where , , , and are the edge lengths. Corresponding tetrahedron