A dodecahedron is having 20 vertices and 30 edges. How many faces are there?

Here:
  Number of vertices (V) = 20
     Number of edges (E) = 30
Let the number of faces are F. Then using Euler's formula, we have
                  F + V = E + 2                                                      ...(1)
∴         Substituting the values of V and E in (1), we get
                   F + 20 = 30 + 2
              F + 20 = 32
                     F = 32 - 20
                     F = 12
   Thus, the required number of faces = 12.

Number of vertices (V) = 8
   Number of edges (E) = 12
Let the number of faces = V
Now, using the Euler's formula
                   F + V = E + 2
we have
                   F + 8 = 12 + 2
              F + 8 = 14
                  F =  14 - 8 
                 F = 6
  Thus, the required number of faces = 6.

If a polyhedron is having number of faces as F, number of edges as E and the number of vertices as V, then the relationship F + V = E + 2 is known as Euler’s formula. Following figure is a solid pentagonal prism.


It has:
 Number of faces (F) = 7
 Number of edges (E) = 15
 Number of vertices (V) = 10
Substituting the values of F, E and V in the relation,
                     F + V = E + 2
we have  
                     7 + 10 = 15 + 2
                     17  = 17
Which is true, the Euler’s formula is verified. 

∵ The ends (bases) of the given solid are congruent rectilinear figure each of six sides.
∴ It is a hexagonal prism.
In a hexagonal prism, we have:
The number of faces = 8
The number of edges = 18
The number of vertices = 12