What is a Tetrahedron?

A tetrahedron is a type of polyhedron which has four faces, making it the smallest possible type of polyhedron. This geometric figure is the basis for a wide variety of geometry problems, and examples of tetrahedra can be seen in architecture, the arts, and even daily life. In fact, chances are very good that there's a tetrahedron in your vicinity.

To understand the tetrahedron, it is unfortunately necessary to discuss a few key terms in geometry. A polygon is a flat or “planar” shape created with a series of connecting line segments: a triangle, for example, is a polygon. A polyhedron is a three-dimensional object composed of multiple polygons which meet to form straight edges. A well-known example of a polyhedron is a cube, a six-sided polyhedron. If the edges are curved, as in the case of a cylinder, the shape is no longer a polyhedron.

In the case of a tetrahedron, the polygons are all triangles by default, because in order to create a three-dimensional object with four polygons, each polygon must have three sides to connect with the other three polygons. The triangles can come in a variety of styles: when equilateral triangles are used, a tetrahedron is known as a “regular tetrahedron.” Tetrahedra are also sometimes called triangular pyramids, because they include a flat base and an apex.

There are lots of ways to play with this shape in mathematics. Triangles themselves are very interesting shapes from a mathematical standpoint, so an assortment of triangles is all the more interesting. Tetrahedra can also be joined together to create numerous other polyhedra, especially in the case of regular tetrahedra.

The tetrahedron is an example of a convex polyhedron. This means that if you randomly select any two points on the tetrahedron and connect them with a line, the line will pass through the tetrahedron, and not stray outside of it. By contrast, in a non-convex polyhedron, the line would at some point travel outside the polyhedron. Generally, the more faces a polyhedron has, the harder it is to make it convex, and at a certain point, it must become non-convex to accommodate all the faces.

Some architects like to use this shape to add visual interest to their designs. Some cultures have also historically attached religious significance to this shape, or to collections of tetrahedra. The star tetrahedron, for example, is a polygon created by merging two tetrahedra which face in opposite directions, creating an eight-pointed star.