Number of diagonals of a polygon of n sides is given by the formula

n*(n-3)/2  where n >= 3

(I'm not giving the proof for this formula here - it has already been described
as part of other solutions in this site)

Here since it is given that the polygon has 275 diagonals

 or 

 which is a standard quadratic equation.

Solving this by factorization







Solving, n = 25 or n = -22

Since n cannot be negative, the polygon has  sides.

:)