# Definition:Tetrahedron/Regular

## Definition

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

It has:

The **regular tetrahedron** is an example of a deltahedron.

## Also known as

It is commonplace for authors to refer to a **regular tetrahedron** as just a **tetrahedron**, glossing over the fact of its regularity.

## Also see

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

## Historical Note

The concept of a **regular tetrahedron** is not mentioned explicitly by Euclid in *The Elements*.

The first reference to it is in Proposition $13$ of Book $\text{XIII} $: Construction of Regular Tetrahedron within Given Sphere:

In the words of Euclid:

*To construct a pyramid, to comprehend it in a given sphere, and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid.*

(*The Elements*: Book $\text{XIII}$: Proposition $13$)

According to the Pythagorean tradition, the **regular tetrahedron** was the symbol for the element fire.

## 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): $5$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $8$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $8$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**tetrahedron** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: Euclid