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.
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.
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 = e2
e = 5
Substituting the length of the edge into the volume formula: