# 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:

### Base of Parallelepiped

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

The opposite face to that face is also referred to as one of the **bases**.

It is usual to choose one of the **bases** to be the one which is conceptually on the bottom.

In the above, $ABCD$ and $EFGH$ would conventionally be identified as being the **bases**.

### Lateral Face of Parallelepiped

The faces of the parallelepiped which are not the **bases** are called the **lateral faces** of the parallelepiped.

In the above, the **lateral faces** are $ABHG$, $BCFG$, $CDEF$ and $ADEH$.

### Height of Parallelepiped

The **height** of a **parallelepiped** is the length of the perpendicular between the planes of the bases.

In the above diagram, $h$ is the **height** of the parallelepiped one of whose bases is $AB$.

## Types of Parallelepiped

### Right Parallelepiped

A **right parallelepiped** is a **parallelepiped** whose lateral faces are square or rectangular.

### Rectangular Parallelepiped

A **rectangular parallelepiped** is better known as a **cuboid**:

A **cuboid** is a parallelepiped whose faces are all rectangular.

### Oblique Parallelepiped

An **oblique parallelepiped** is a **parallelepiped** such that the sides of the lateral faces are not perpendicular to the bases.

## Also known as

Some sources give this as **parallelopiped**.

## Also see

- Results about
**parallelepipeds**can be found**here**.

## 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 Elements*: Book $\text{XI}$: Proposition $24$)

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.*

(*The Elements*: Book $\text{XI}$: Proposition $25$)

## 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 **parallel opiped** is also considered by some authorities to be incorrect.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**parallelepiped** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**parallelepiped** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**parallelepiped** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**parallelepiped**