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2015-07-28T20:05:33+05:30

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Let n be the number of vertices of the polygon.
then there are n  sides.

N = number of  all connections between all the vertices = 
{}^nC_2
N = n(n-1)/2

Number of diagonals =  N - n = n (n - 3)/2    =  27  given
           n²  - 3 n  - 54 = 0
        (n - 9) (n + 6) = 0
           n = 9

so there are  9 sides and 9 vertices
1 5 1