Euler was able to relate the number of faces (F), vertices (V), and edges (E) of a polyhedron by the following equation,F + V = E + 2.
Students can discover or verify this formula by examining three dimensional polyhedra. There are many ways to create inexpensive models. Particularly useful materials for this activity are gumdrops and toothpicks. (Mini-marshmallows can be substituted for the gumdrops.) Students work alone or in groups to build the polyhedra. The gumdrops clearly represent the vertices and the toothpicks clearly represent the edges.
If you prefer, you can have students assemble polyhedra nets. The gumdrops (or marshmallows), however, seem to more clearly emphasize vertices and edges, which are harder for some students to count correctly.
Ask students to record the number of faces, vertices, and edges as they build their polyhedra. You can specify which polyhedra are to be built or you can let students decide for themselves. After the data is collected, have students examine the results for any relationships between their findings. You may need to give a hint if no students are seeing a relationship.
Hint: Be sure to purchase enough gumdrops to allow for nibbling!