So the general idea is to find with compass and pencil the exact needed separation angle that can be used to bounce around the interior circumference of a circl

So the general idea is to find with compass and pencil the exact needed separation angle that can be used to bounce around the interior circumference of a circle with perfect overlap. You then join up the bounces to their neighbours to get the desired N(prime)-sided polygon. The easier Heptadecagon (17-sided) was illuminating here: https://en.wikipedia.org/wiki/Heptadecagon https://en.wikipedia.org/wiki/Heptadecagon