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CMI2016-A-5

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A dodecahedron is a regular solid with $12$ faces, each face being a regular pentagon. How many edges are there? And how many vertices?

  1. $60$ edges and $20$ vertices
  2. $30$ edges and $20$ vertices
  3. $60$ edges and $50$ vertices
  4. $30$ edges and $50$ vertices

2 Answers

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B.) Given f=12, degree of each face = df = 5.

By Euler's formula, v-e+f = 2 

=> v - e + 12 =2 

=> 10 = e-v

Also, $\sum degree(face) = 2*e$

=> 5*12 = 2*e or e=30

Solving the equations, e=30 and v=20

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Answer: B

Each face has five edges, and there are 12 faces. But each edge is shared between two faces. Thus the number of edges is (12 × 5)/2 = 30. The number of vertices is given by Euler’s formula to be |E| − |F| + 2 = 30 − 12 + 2 = 20.

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