Quiz 11         MAT 118, Spring 2003

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1. A dodecahedron has 12 faces, each of which is a regular pentagon. How many edges does it have? How many vertices does it have? Write your answers in the spaces below.

   
   A dodecahedron has   30   edges and   20   vertices.

   Solution: There are 12 faces, each of which is a regular pentagon. Each face has 5 edges, but each edge is shared by exactly one other face. This means that there are $12\times 5 / 2 = 30$ edges in all.

   To get the number of vertices, we use Euler's formula, filling in what we know, and then solving for $V$.
   

   $$\begin{array}{lcr} V - E + F &=& 2 \\ V - 30 + 12 &=& 2 \\ V - 18 &=& 2 \\ V &=& 20 \end{array}$$
   
   

   
   

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