There are nC2 ways to choose two vertices to join.  Every diagonal
is formed by joining two vertices. The only ways to join two vertices 
which will not determine a diagonal is to join two adjacent vertices,
which will form a side, not a diagonal.  So we only need to subtract 
the n sides from the nC2 ways to join two vertices and we get 



diagonals.  Since there are 12n diagonals we have this equation:





Clear of fractions by multiplying through by 2







Divide through by n, which is OK since n can't be 0.





The polygon has 27 sides.

Edwin