What is the relation between the sides of regular $n$- and $m$-gons inscribed inside a unit radius circle?

Question

Suppose a regular $m$-gon is inscribed inside a unit circle. And suppose a regular $n$-gon is inscribed in another unit circle. What's the relation between sides of $m$ and $n$?

I know $$l_{2n}=\sqrt{2R^2 - R\sqrt{4R^2-l_{n}^2}}$$

where $l_n$ is the side of a regular $n$-sided polygon inscribed in a circle with $R$ radius.

But how can I use this equation to derive a relation between $m$ and $n$?

Answer 1

If you consider the triangle of 2 consecutive vertices and the centerpoint of a regular $n$-gon, then it is clear that the centri-angle will be $\varphi_n=2\pi/n$. Thus the side length of a regular $n$-gon is being given by $$s_n=2R\sin(\varphi_n/2)=2R\sin(\pi/n)$$ Therefrom you clearly get $$\frac{s_n}{s_m}=\frac{\sin(\pi/n)}{\sin(\pi/m)}$$ --- rk