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Euler's formula

25.5 Euler’s formula

A special relationship exists between the number of edges (\(e\)), the number of faces (\(f\)) and the number of vertices (\(v\)) for any polyhedron.

polyhedron
a 3D object that has polygons for faces, straight edges and sharp vertices

This relationship was first stated by mathematician Leonhard Euler in 1758 and is called Euler’s polyhedron formula.

The relationship between the faces, edges and vertices of the Platonic solids

Complete this investigation in pairs or small groups.

  1. For each Platonic solid shown in the first column, write down the number of edges (\(e\)), the number of faces (\(f\)) and the number of vertices (\(v\)).

      Number of faces (\(f\)) Number of edges (\(e\)) Number of vertices (\(v\))
    Hexahedron      
    Octahedron      
    Dodecahedron      
  2. Check your values with another group in the class. Make sure that you agree with each other’s answers.
  3. Look at the numbers in each row. Can you deduce a relationship between the number of faces, edges and vertices?
  4. Express this relationship mathematically using the symbols \(e\), \(f\) and \(v\).
  5. Now use the Platonic solids in the table below to test your formula. Do the values satisfy the equation?

      Number of faces (\(f\)) Number of edges (\(e\)) Number of vertices (\(v\))
    Tetrahedron      
    Icosahedron      
  6. For each Platonic solid, what do you notice about the number of faces at each vertex?
  7. For each vertex, what do you notice about the number of edges that meet at the vertex?
  8. Look at each Platonic solid. What do you notice about the lengths of the edges?

From the investigation, you should have discovered that the difference between the number of vertices (\(v\)) and the number of edges (\(e\)) added to the number of faces (\(f\)) is equal to \(2\).

We can express this mathematically as:

\[v - e + f = 2\]

or

\[v + f - e = 2\]

You should also have noticed the following properties for each of the five Platonic solids:

  • All faces are regular and congruent.
  • The same number of faces meet at each vertex.
  • All the edges are the same length.
  • Euler’s formula is satisfied.
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