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, the solid figure (i.e. the inside or interior of the collection of 6 polygons), it is a zonohedron.

The volume of a parallelepiped is the length of any edge times the lengths of the altitudes of the two faces that have that edge as base. Where the available facilities provide for it, this can be calculated most easily using the determinants, or equivalently via the scalar triple product or cross products.

Lexicography

In the two terms, the OED puts the main accent on, respectively,

• the fourth syllable, i.e. parallelEPiped, and
• the fifth syllable, i.e. paralleloPIPedon.

It also describes "parallelipiped" and "parallelopiped" explicitly as incorrect forms.

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