How many vertices does a polygon have with 54 diagonals?

1. Each vertex of a convex n-gon can be connected by a diagonal with n – 3 vertices – we exclude the same one and two neighboring vertices.

2. Since each diagonal involves two vertices, the number of diagonals of an n-gon is equal to:

N (n) = n (n – 3) / 2.

3. By the condition of the problem, we have:

n (n – 3) / 2 = 54;
n (n – 3) = 108;
n ^ 2 – 3n – 108 = 0;
D = 3 ^ 2 + 4 * 108 = 9 + 432 = 441;
n = (3 ± √441) / 2 = (3 ± 21) / 2;
n1 = (3 – 21) / 2 = -18/2 = -9 – does not fit within the meaning of the problem;
n2 = (3 + 21) / 2 = 24/2 = 12.
Answer: 12 peaks.