Platonic Solids aka Regular Polyhedra

Practice problems:

1. A dodecahedron has 12 faces. Each face is a pentagon (5 sides). Show how to use this information to find the number of edges a dodecahedron has

2. Explain the formula or process you used to find the number of edges a dodecahedron has. Include an explanation of why you divide by 2 as part of the formula

3. A dodecahedron has 12 faces. Each face is a pentagon (5 vertices), and at each vertex there are 3 faces that meet there. Show how to use this information to find the number of vertices a dodecahedron has

4. Tell what the dual of the Octahedron is, and explain how you would create the dual if you had an octahedron.

Answers and more problems

Explanations about polyhedra

A regular polyhedron has:

• All of its faces are the same regular polygon
• If you look at any vertex, the same number of faces (and/or edges) meet at each vertex
• It's a polygon: it meets up to make a vaguely ball-shaped thing

This means that one way of describing a regular polyhedron is to tell what shape each of its faces is, and how many of each face meet at a vertex.

There are only 5 regular polyhedra: Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron

Can you identify the regular polyhedron with the given properties?

If you know the number of faces on a regular polyhedron, and the number of sides on each face, you can figure out how many edges the polyhedron has by calculating:

(#faces)x(# edges per face)/2

(we divide by 2 because each edge is always where two faces meet, so it is counted exactly twice by those two faces)

If you know the number of faces on a regular polyhedron, and the number of vertices on each face, and the number of faces that meet at each vertex, you can figure out how many vertices the polyhedron has by calculating:

(#faces)x(# vertices per face)/(# faces per vertex)

(we divide by # faces per vertex because (#faces)x(# vertices per face) counts each vertex (# faces per vertex) times, because each vertex is counted by each of those faces)

Each convex polyhedron (and, in particular, each regular polyhedron) has a dual. You can geometrically find the dual by putting a dot in the center of each face of the polyhedron, and then connecting the vertices. See an applet here of what this looks like