# Euler's formula

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

At several places on this website, we have looked at the number of faces, edges and vertices (corners) for different shapes. Here is a table of them:

PolyhedronFacesVertices
(corners)
Vertices + FacesEdges
Tetrahedron4486
Cube681412
Octahedron861412
Icosahedron20123230
Dodecahedron12203230

Notice anything odd about these figures?

First, look at the column of Vertices + Faces and compare it to Edges. The Vertices + Faces is always two more than Edges. You can write this down as a formula:

V - E + F = 2

What's more, it's true for other polyhedra as well. Why not try it out on some other figures?

Another strange fact is that the edges for a cube are the same as the edges for an octahedron, and the faces of a cube are the same as the vertices of an octahedron, and the vertices of a cube are the same as the faces of an octahedron. What's more, there are 11 different nets for both a cube and octahedron. This makes us wonder, are they connected? The answer is, Yes. The octahedron is the dual of the cube. This means that they have the same symmetry. You can fit a smaller octahedron inside a cube so that all the vertices of the octahedron touch the centre of each face of the cube.

Now look at the icosahedron and the dodecahedron. They are duals as well. They have the same number of nets are well (43380!)

Finally, add up the edges of the cube, tetrahedron and octahedron. They come to the same number as the edges of an icosahedron (and, of course, as the dodecahedron as well). I don't think this has any mathematical significance, but it's quite fun!