Correct solutions with proofs came from James Page, Hethersett High School, Norfolk, Ruoyi Sun, the NLCS Puzzle Club and from girls in years 8, 9 and 10 at the Mount School York. Well done all of you. The diagram here is by Daniella Yule, Sophie Brook and Hollie Jefferson.

Thank you Nisha Doshi, Bella Heesom, Daniella Yule, Sophie Brook and Hollie Jefferson for this solution.

If we use $k$ to stand for the number of sides, there are also k vertices. Each vertex has $(k - 3)$ diagonals coming from it. This is $(k - 3) \times k$ diagonals. However, this method counts each of the diagonals twice, so the correct result is $\frac{1}{2}k(k - 3)$.