One can himself create this formula,by taking small polygons :

No. of sides in polygon                               No. of diagonals

3                                                                        0              = 3*0

4                                                                        2              = 4*1/2

5                                                                       5               = 5*1

6                                                                       9               = 6*3/2

7                                                                       14            = 7*2

Hence we see that D = n(n-3)/2

In the problem,

n(n-3)/2 = 275.

Solving we get, n = 25