# 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.

## 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.