Platonic Solids and Euler's Formula

You will need the Platonic Solids Nets for this activity. Not provided at this site yet. (Under construction!)

See also the fabulous TOOL for investigating Platonic Solids at the NCTM site:

  1. A simple closed surface has exactly one interior. The interior is hollow. It separates space into exactly three sets of points: the interior points, the points on the surface, and the exterior points.
  2. Write examples of simple closed surfaces. __________________________________
  3. A solid is the union of a simple closed surface and its interior points. Name three everyday solids. _______________________________________________________
  4. A polyhedron is a simple closed surface made up of polygonal regions. (Poly means "many" and hedron means "flat surfaces") Each polyhedron has the following four features: base(s), lateral faces, edges, and vertices. These features are used to classify and name the polyhedron.
  5. A polyhedron is called a regular polyhedron if the faces of the polyhedron are congruent regular polygonal regions, and if each vertex is the intersection of the same number of edges. There are exactly five that can be formed! These polyhedra are often called the Platonic Solids in honor of the Greek philosopher Plato.
  6. Use your Platonic Solids to fill in the table. Once you are done, look for a relationship between the number of edges of each prism. Fill in the last column above. Find a relationship between the Edges, Vertices, and Faces. This relationship is called Euler's Formula (pronounced Oiler).

    Write your relationship here: ________________________________

      Faces Vertices Edges
    Tetrahedron 4 triangular    
    Hexahedron (Cube) 6 square    
    Octahedron 8 triangular    
    Dodecahedron 12 pentagonal    
    Icosahedron 20 triangular    

  7. Think of another solid with polygonal faces. Count the faces, vertices, and edges. Does Euler's Formula still hold?



Copyright 1998 by Margo Lynn Mankus