# Tetrahedron

A tetrahedron is a space figure with four triangular faces. The prefix "tetra" means four. In the figure below are 2 examples.

## Properties of a tetrahedron

A tetrahedron is a pyramid with one triangular base and three triangular sides, called lateral faces. The lateral faces share a common vertex called the apex. We usually think of the apex as the "top" of the tetrahedron. An edge is a line segment formed by the intersection of two adjacent faces. A tetrahedron has 4 faces, 6 edges, and 4 vertices. It is the polyhedron that can be formed with the fewest number of faces.

Any cross section that is parallel to the base of a tetrahedron forms a triangle that is similar to its base.

The two triangles, shaded in blue, formed by cross sections parallel to the base of the tetrahedron above, are similar (same shape but not the same size) to the tetrahedron's triangular base.

## Right and oblique tetrahedrons

A tetrahedron can be classified as either a right tetrahedron or an oblique tetrahedron. If an apex of the tetrahedron is directly above the center of the base, it is a right tetrahedron. If not, it is an oblique tetrahedron. The line segment from the apex to the center of the base of a right tetrahedron is perpendicular to the base, and is the height of the tetrahedron.

Right tetrahedron | Oblique tetrahedron |
---|---|

The apex is directly above the center of the base. | The apex is not directly above the center of the base. |

## Regular and irregular tetrahedrons

A tetrahedron can also be categorized as regular or irregular. If the four faces of a tetrahedron are equilateral triangles, the tetrahedron is a regular tetrahedron. Otherwise, it is irregular. All edges of a regular tetrahedron are equal in length and all faces of a tetrahedron are congruent to each other. A regular tetrahedron is also a right tetrahedron. An oblique tetrahedron is also an irregular tetrahedron.

Regular tetrahedron | Irregular tetrahedron |
---|---|

All faces are equilateral triangles. | Not all faces are equilateral triangles. |

Regular tetrahedrons are one of five Platonic solids. A Platonic solid is a type of a regular polyhedron.

## Height of a regular tetrahedron

Any face opposite a vertex can be considered a base of the tetrahedron. Any height or altitude drawn from the vertex to the opposite face of a regular tetrahedron is equal in length and any height of a regular tetrahedron goes through its center. The height, h, of a regular tetrahedron has a value of

where e is the length of the edge.

## Surface area of a regular tetrahedron

We can find the area of one of the faces and multiply it by four to find the total surface area of a regular tetrahedron. An equilateral triangle with side length e (also the length of the edges of a regular tetrahedron) has an area, A, of

The total surface area, S, of a regular tetrahedron in terms of its edges, e, is,

## Volume of a tetrahedron

The volume, V, of tetrahedron is

where B is the area of the base and h is the height of the tetrahedron.

If the tetrahedron is a regular tetrahedron, its volume is

where e is the edge length of the regular tetrahedron.

Example:

If the total surface area of a regular tetrahedron is , what is its volume?

We can find e by substituting the given value in for the total surface area to get

25 = e^{2}

e = 5

Substituting the length of the edge into the volume formula: