Euler's Formula for Polyhedra

Leonhard Euler (1707-1783) was a Swiss mathematician. He married twice and had 13 children, all but five of them died young. He proved many theorems and formulas in his life. This theorem helps us count the faces, vertices, and edges of 3-D polyhedra.

ACTIVITY: consider the Platonic solids below. Count the number of faces, vertices, and edges for each figure and record them on the chart.

Solid
Faces
Vertices
Edges
     
     
     
     
     

Can you find the relationship between the three numbers for each figure? This is what Euler's formula is all about. It is often written V + F - E= 2.

Why does this work? What is going on with each polyhedra that makes the number two come up over and over? Think about one face of a polyhedra, and how many points and lines it takes to make just one face. Now think about how many other faces are connected to the points and lines of the first face.

Paper nets for polyhedra
Examples
Lesson plan
Another lesson plan
Uniform polyhedra
Buckyballs