Tetrahedron (triangular pyramid)

Solid shapes--- cube --- tetrahedron --- octahedron --- icosahedron --- other shapes --- Euler's formula --- glossary --- for teachers


There are three different solids that you can make with triangles. The first is the triangular pyramid, made of four sides, all triangles. Usually pyramids have five sides, four triangles and a square, such as the Great Pyramid in Giza, Egypt. This is an interesting shape, but it isn't a regular shape, since it uses a square as well as triangles. To stop confusion between the two sorts of pyramids, mathematicians use the word tetrahedron to describe a triangular pyramid. 'Tetra' means four, and the tetrahedron has four sides. If you were going to be very pedantic, you could describe a cube as a hexahedron, but people tend not to! The mathematical word for a solid shape like these is a polyhedron (poly means many). By the way, the plural is polyhedra. A regular tetrahedron has its faces as equilateral triangles, where all sides are equal, and the angle between them is 60°

Model of tetrahedron

Model of a tetrahedron

This model (right) was made from a kit with magnetic connections. Since it just shows the edges, you can see through the model, which means that it is easier to count the vertices and edges. How many vertices (corners) and edges are there? See Euler's formula.

Game with nets of tetrahedra

See if you can work out which nets will make a tetrahedron. There are two correct nets, and they will change colour as you click on them. Click on New go for another go.

        2 nets left to find        

Net of a tetrahedron

Here is one net for a tetrahedron. Print it out, stick it on thin card, score along the lines and fold them, form the shape, then stick it together with small amounts of glue.

For more details, see the notes for the net of a cube.

Volume of a pyramid

The volume of a pyramid is a third the area of the base times the height. This is true of a tetrahedron as well as a Egyptian pyramid.

Pyramid packaging

You can get tetrahedral packaging, usually for liquids such as fruit juice. These are made in a clever way, which you can try for yourself. Make a cylinder of paper and glue the edge down. Pinch one end, and glue that. Now pinch the other end in the opposite direction, and glue that. It will naturally form a tetrahedron, although you might need to play around with the dimensions of the cylinder to get a regular tetrahedron.


Tetrahedra are not very common. They do have one useful property; they are very stable. A caltrop is an unpleasant medieval weapon. It has four sharp points, arranged at the vertices (corners) of a tetrahedron. Whichever way you throw it, one point will always point straight up. Anyone treading on it will get a spike through their foot. This is a modern version for puncturing car tyres instead.

Moving tetrahedron

Click on Move or Backwards to make tetrahedron move and Stop to stop it.

Other regular solids with triangular faces?

A tetrahedron has three triangles at each vertex (corner). Unlike a cube, it is possible to have more than three triangles at the vertex of a regular solid. These make an octahedron or an icosahedron.