I'm wondering if there is a general formula to calculate the side length of a regular polygon of n sides that is inscribed by a circle of a given radius.
For example, in this image, if the circle has a radius of 100, what is the side length of the hexagon that is inscribed?
Consider a regular polygon with $n$ sides inscribed in a circle of radius $R$. Then, each side will subtend an angle of $\frac{2\pi}{n}$ at the centre of the circle. Joining the ends of the side to the centre of the circle, we get an isoceles triangle with 2 sides $R$, and one side $l$ where l is the length of a side of the polygon. By trigonometry, we have
$$ 2R\sin\left(\frac{2\pi}{2n}\right) = 2R\sin\left(\frac{\pi}{n}\right) = l$$