Discovering the Relationships Between the Faces, Edges, and Vertices

Discovering The Relationships Between the Faces, Edges, and Vertices

Of The Platonic Solids

 f the number of faces of the polyhedron,  v the number of vertices of the polyhedron, and  e the number of edges of the polyhedron.

Construct the nets of each of the Platonic Solids. Next, construct the Platonic Solids from their respective nets. Find the number of faces, edges, and vertices of each of the Platonic Solids (be looking for creative ways of finding each):

Platonic Solid: f v e

Tetrahedron 4 4 6

Hexahedron (Cube) 6 8 12

Octahedron 8 6 12

Dodecahedron 12 20 30

Icosahedron 20 12 30

Do you see a unique relationship between the number of faces, edges, and vertices? _____ If yes, state your conjecture about the relationship: ______Next, find an algebraic formula that describes this relationship mathematically: F + V = E + 2

If you found the correct formula, it is called Euler’s Theorem or Euler’s Formula (depends on the text you read). Can you express Euler’s Theorem/Formula in several different ways? If yes, show below: F + V = E + 2 , F + V – E = 2 , F + V – E – 2 = 0 , F = -V + E + 2 (Note: “F, V, & E” are VARIABLES; “2” is a CONSTANT) Next, construct other nets that will produce convex solids and test your “formula” to see if it works on different convex solids other than the Platonic Solids (record your findings in the chart below):

Solid: # of Faces: # of Vertices: # of Edges: Does formula hold?

What are your findings? Do you believe that Euler’s Formula works for any convex solid or just the Platonic Solids? Why or why not? Explain:

Euler’s Theorem/Formula holds for ALL convex polyhedron solids!