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

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 $$n$$?

• The side of the $7$-gon is not expressible using square roots and so I don't expect a simple relationship between $l_7$ and $l_4$ say.
– lhf
Commented Nov 27, 2019 at 11:51

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