jmc

An $n$-gon has $n(n-3)/2$ diagonals. To see this, subtract the $n$ consecutive pairs of vertices from the $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ pairs of vertices: $$\begin{array}{rl}\left(\genfrac{}{}{0ex}{}{n}{2}\right)-n& =\frac{n(n-1)}{2}-n\\ & =\frac{n^2-n}{2}-\frac{2n}{2}\\ & =\frac{n^2-n-2n}{2}\\ & =\frac{n^2-3n}{2}\\ & =\frac{n(n-3)}{2}.\end{array}$$ An octagon has $8(8-3)/2=\boxed{20}$ diagonals.