Polyhedron Investigation: Euler’s Formula (Faces + Vertices — Two = Edges)

The children investigated the properties of polygons and polyhedron my using manipulative that they could touch and make observations. We focused on the definition of these terms and the prefixes from the Latin and Greek stems as they are named by the number of sides or faces, respectively. We also looked at the distinction between convex and concave polygon and polyhedron. 

They learned how to draw analogies to the definitions of vertices (V), faces (F) and edges (E) of polyhedron and could locate examples of each in the room. After developing strategies for counting the number of V, F and E for each of the Platonic solids (tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron), I asked them to look for patterns the the relationship of the solutions. When they added the F and V, they would always conclude that subtracting two from the sum would result in the number of edges. Thus, F + V — 2 = E. This is Euler’s Formula and I asked them to verify Euler for other polyhedron. The answer always was yes. However, Euler’s Formula only is conclusive for convex polyhedron; it will work for many concave polyhedron but not all and I showed them examples of concave polyhedron that do not  work. If you take a convex polyhedron and connect two opposite vertices until they touch, you have effectively subtracted one vertex from the mix throwing off Euler’s Formula.

They were given 19 polyhedron to experiment with. First, they were instructed to record the type of polygon of which it is constructed. Second, they were challenged to find the number of faces and vertices. They would then be able to add these two numbers and subtract two to get the number of edges.

During the week they are to find household polyhedron such as a table, room, book, eraser, box, soccer ball, television, kitchen appliances and then record the number of F, V and E. Did Euler work for each one? See attached pdf for all of the above.

I gave the 4-6th graders an assessment that would provide a given number of F and V and they would have to generate the missing number. There are nine questions with visual answers that follow.  See attached pdf.