CMI2016-A-5

Question

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

• A. $60$ edges and $20$ vertices
• B. $30$ edges and $20$ vertices
• C. $60$ edges and $50$ vertices
• D. $30$ edges and $50$ vertices

Answer 1

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

Answer 2