# Definition:Tetrahedron

## Definition

A **tetrahedron** is a polyhedron which has $4$ (triangular) faces.

Thus a **tetrahedron** is a $3$-simplex.

### Base of Tetrahedron

One of the faces of a tetrahedron can be chosen arbitrarily, distinguished from the others and identified as the **base** of the tetrahedron.

In the above diagram, $ABC$ is the **base** of the tetrahedron $ABCD$.

### Apex of Tetrahedron

Once the base of a tetrahedron has been identified, the vertex which does not lie on the base is called the **apex** of the tetrahedron.

In the above diagram, given that the base of the tetrahedron $ABCD$ is the triangle $ABC$, the **apex** is $D$.

## Regular Tetrahedron

A **regular tetrahedron** is a tetrahedron whose $4$ faces are all congruent equilateral triangles.

## Also known as

A **tetrahedron** is also known as a **triangular pyramid**.

In Euclid's *The Elements*, a **tetrahedron** is referred to as a **pyramid (with / which has) a triangular base**.

The first reference to it is in Proposition $3$ of Book $\text{XII} $: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms:

In the words of Euclid:

*Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.*

(*The Elements*: Book $\text{XII}$: Proposition $3$)

## Also see

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

## Linguistic Note

The word **tetrahedron** derives from the Classical Greek **τετράεδρόν**:

**tetrás**(**τετράς**), meaning**four****hedron**(a form of**ἕδρα**), meaning**base**or**seat**.

The technically correct plural of **tetrahedron** is **tetrahedra**, but the word **tetrahedrons** can often be found.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $4$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**tetrahedron** - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $4$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**pyramid** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**tetrahedron**(*plural***tetrahedra**) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**pyramid** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**tetrahedron**(*plural***tetrahedra**) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**tetrahedron (tetrahedra)**