A nonagon, or enneagon, is a polygon with nine sides and nine vertices, and it has 27 distinct diagonals. The formula for determining the number of diagonals of an n-sided polygon is n(n - 3)/2; thus, a nonagon has 9(9 - 3)/2 = 9(6)/2 = 54/2 = 27 diagonals.

The diagonal of a polygon is any line segment joining two nonadjacent vertices. A nonagon has nine vertices, and each vertex has six other vertices that are not adjacent; thus, each vertex forms six diagonals. For each vertex, there are three fewer diagonals than there are vertices. This is the first part of the procedure for finding the number of diagonals of a polygon: n(n - 3) = 9 x 6 = 54. Each diagonal has two end points, so in order to not count a duplicate diagonal, the final step is to divide 54 by 2, which results in 27 diagonals.