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A parallelepiped is a prism that has parallelograms for its faces. Similarly, a parallelepiped is a hexahedron with six parallelogram faces. Specific parallelepipeds include the cube, the cuboid, and any rectangular prism.

Specific Cases

A parallelepiped with all rectangular faces is a cuboid, and a parallelepiped with six rhombus faces is known as a rhombohedron. In an $n-$dimensional space, a parallelepiped is sometimes referred to as an $n-$dimensional parallelepiped, or as an $n-$parallelepiped. A cube is a parallelepiped with all square faces.


The volume of a parallelepiped is the product of the area of one of its faces times the perpendicular distance to the corresponding top face. Alternately, if the three edges of a parallelepiped that meet at one vertex are defined as vector $a, b,$ and $c$ with the specific vertex as the origin, then the volume of the parallelepiped is the same as the scalar triple product of the vectors, or $a \cdot (b \times c)$. Suppose that $\bold{a} = a_1\bold{i}+a_2\bold{j}+a_3\bold{k}$, $\bold{b} = b_1\bold{i}+b_2\bold{j}+b_3\bold{k}$, $\bold{c} = c_1\bold{i}+c_2\bold{j}+c_3\bold{k}$. We then have the area of the parallelepiped is \[|\text{det}(\begin{bmatrix} c_1 & c_2 & c_3 \\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \\ \end{bmatrix})|.\]

See also