Parallelepiped

In 3-D, a parallelepiped is a Prism whose faces are all Parallelograms. The volume of a 3-D parallelepiped is given by the Scalar Triple Product

$$V_{\rm parallelepiped} = \vert{\bf A}\cdot ({\bf B}\times {\bf C})\vert$$

$$= \vert{\bf C}\cdot ({\bf A}\times {\bf B})\vert = \vert{\bf B}\cdot ({\bf C}\times {\bf A})\vert.$$

In $n$-D, a parallelepiped is the Polytope spanned by $n$ Vectors ${\bf v}_1$, ..., ${\bf v}_n$ in a Vector Space over the reals,

$$\mathop{\rm span}({\bf v}_1, \ldots, {\bf v}_n)=t_1 {\bf v}_1 + \ldots + t_n {\bf v}_n,$$

where $t_i \in [0,1]$ for $i=1$, ..., $n$. In the usual interpretation, the Vector Space is taken as Euclidean Space, and the Content of this parallelepiped is given by

$$\mathop{\rm abs}(\mathop{\rm det} ({\bf v}_1, \ldots, {\bf v}_n)),$$

where the sign of the determinant is taken to be the ``orientation'' of the ``oriented volume'' of the parallelepiped.

See also Prismatoid, Rectangular Parallelepiped, Zonohedron

References

Phillips, A. W. and Fisher, I. Elements of Geometry. New York: Amer. Book Co., 1896.