# Parallelepiped

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In geometry, a parallelepiped or parallelopipedon is a 3-dimensional figure that can be thought of as an (impractical) box that, while not necessarily having right angles, still has its upper surface level whenever it rests its lower surface on something level. Rigorously specified, it is a polyhedron with six faces, each a parallelogram. (The word is also sometimes used for the higher-dimensional analogues.)

## Properties

It follows from the parallelogram faces that opposite faces are parallel. Since each face has point symmetry, a parallelepiped is a zonohedron.

The volume of a parallelepiped is the product of the area of the base with the height. Here, the base is one of the six parallelograms that make up the parallelepiped, and the height is the length of the altitude from one of the vertices that does not lie on the chosen parallelogram. Alternatively, if the vectors a = (a1, a2, a3), b = (b1, b2, b3) and c = (c1, c2, c3) represent three edges that meet at one vertex, then the volume of the parallelepiped equals the absolute value of the scalar triple product a · (b × c), or, equivalently, the determinant

[itex] \left| \det \begin{bmatrix}
       a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{bmatrix} \right|. [itex]


## Lexicography

The 1989 edition of the Oxford English Dictionary describes "parallelipiped" and "parallelopiped" explicitly as incorrect forms, but these are listed without comment in the 2004 edition. Pronunciation has the emphasis consistently on the fifth syllable.

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