# Definition:Parallelepiped

## Definition

A parallelepiped is a polyhedron formed by three pairs of parallel planes:

In the above example, the pairs of parallel planes are:

$ABCD$ and $HGFE$
$ADEH$ and $BCFG$
$ABGH$ and $DCFE$

### Opposite Face of Parallelepiped

The opposite face of the face $F$ of a parallelepiped $P$ is the face of $P$ which is parallel to $F$.

In the above example, the pairs of parallel planes are:

Face $ABCD$ is opposite $HGFE$
Face $ADEH$ is opposite $BCFG$
Face $ABGH$ is opposite $DCFE$

### Base of Parallelepiped

One of the faces of the parallelepiped is chosen arbitrarily, distinguished from the others and called the base of the parallelepiped.

It is usual to choose the base to be the one which is conceptually on the bottom.

In the above, $ABCD$ would conventionally be identified as being the base

### Height of Parallelepiped

The height of a parallelepiped is the length of the perpendicular from the plane of the base to the plane opposite.

In the above diagram, $h$ is the height of the parallelepiped whose base is $AB$.

## Also known as

Some sources give this as parallelopiped.

## Historical Note

The term parallelepiped is never actually defined by Euclid, although he specifies the concept:

In the words of Euclid:

If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.

The proposition that follows this one uses the term in a proof:

In the words of Euclid:

If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.

## Linguistic Note

The word parallelepiped is divided into syllables and stressed as par-al-lel-ep-i-ped.

The pronunciation par-al-lel-e-pi-ped is technically incorrect.

The occasionally-seen spelling parallelopiped is also considered by some authorities to be incorrect.