Faces, edges and vertices: Euler's formula

Count the faces, subtract the edges and add the vertices of any ordinary polyhedron and the answer is always 2.

Count the pyramid before reading the rule

Turn on Vertices and edges in the model and count the square pyramid yourself. Faces first, then edges, then the corner points, which are called vertices. Write the three numbers down. You should have 5 faces (the square and four triangles), 8 edges (four around the base and four up to the apex) and 5 vertices. Now work out F − E + V. It comes to 5 − 8 + 5 = 2. Remember that 2. Switch to the cube: 6 − 12 + 8 = 2. Then the triangular prism: 5 − 9 + 6 = 2.

The one total that never moves

Those three solids are polyhedra: solids with flat polygon faces, straight edges and vertices. Cylinders, cones and spheres are not, because they have curved surfaces. The rule keeps working for a pentagonal prism (7 − 15 + 10), a hexagonal pyramid (7 − 12 + 7), a soccer-ball shape with 32 faces, and any other polyhedron without holes. The three counts change a lot from one solid to the next. This one combination never does. Mathematicians call a number that stays fixed while everything else changes an invariant.

Finding a missing count

Rearrange the formula and two counts give you the third: E = F + V − 2. A polyhedron with 8 faces and 12 vertices must have 8 + 12 − 2 = 18 edges, and a hexagonal prism is one. A polyhedron with 20 faces and 12 vertices has 30 edges; that is the icosahedron. The formula catches counting slips, too. If you count a solid by hand and get 1 or 3, you have missed something or counted it twice.

Prisms and pyramids follow simple patterns. A prism whose base has n sides has n + 2 faces, 3n edges and 2n vertices, and (n + 2) − 3n + 2n = 2 whatever n is. A pyramid with an n-sided base has n + 1 faces, 2n edges and n + 1 vertices: (n + 1) − 2n + (n + 1) = 2. Writing the counts with n is a small proof in itself. It covers every prism and every pyramid in one line.