Euler Characteristic

K-12: Materials at high school level.

In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces F, edges E, and vertices V of a polyhedron:

He found that V - E + F = 2

Let's check this formula on some of the shapes below.

Tetrahedron

A tetrahedon is a simple shape that is made up of 4 triangles. Below you see a picture with labels on the vertices (V) and edges (E).

How many vertices ("corners") V do you see? _______

How may edges E do you see? ______

How many faces ("sides") F do you see? ______

Now find V - E + F = ___ - ___ + ___ =

Did the answer come out to 2? ______

Octahedron

A tetrahedon is a simple shape that is made up of 8 triangles. Below you see two pictures, the one on the left is given with labels on the vertices (V) and edges (E).

How many vertices ("corners") V do you see? _______

How may edges E do you see? ______

How many faces ("sides") F do you see? This may be easier to count in the figure on the right. ______

Now find V - E + F = ___ - ___ + ___ =

Did the answer come out to 2? ______

Cube

A cube is a simple shape that is made up of 6 squares. Below you see a picture with labels on the vertices (V) and edges (E).

How many vertices ("corners") V do you see? _______

How may edges E do you see? ______

How many faces ("sides") F do you see? ______

Now find V - E + F = ___ - ___ + ___ =

Did the answer come out to 2? ______