In a regular polygon, a diagonal is a line segment joining a vertex to any non-neighbouring vertex. For example, a regular hexagon has 9 diagonals. If a regular polygon with $n$ sides has 90 diagonals, what is the value of $n$ ?
Consider one of the $n$ vertices of a regular polygon with $n$ sides.

This vertex will be part of $(n-3)$ diagonals - one joining to every vertex other than itself and its two neighbours.

This means that there are $\frac{1}{2} n(n-3)$ diagonals in total, since each of $n$ vertices is part of $n-3$ diagonals, and each diagonal is counted twice in this way.

For there to be 90 diagonals, then $\frac{1}{2} n(n-3)=90$ or $n^{2}-3 n-180=0$.

Thus, $(n-15)(n+12)=0$.

Since $n>0$, then $n=15$.