# How to find the side length of an n sided regular polygon that is inscribed by a circle of radius r? [closed]

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?

• Welcome to MSE. Which image? Aug 7, 2023 at 10:18
• A hexagon is very easy, so possibly not the best example (you probably know the answer if you think about it) .... For the general case, how much trigonometry do you know? It is quite a basic exercise and worth working out for yourself if you can, as it will give you confidence with the ideas involved. Aug 7, 2023 at 10:50
• Isn't that one of the first applications of the cosine-rule? :-) Aug 7, 2023 at 14:41
• @JoséCarlosSantos sorry the image didn't upload. It's there now. Aug 8, 2023 at 9:30
• @MarkBennet ok I guess for the hexagon the edge length is the same as the radius. In general, my trigonometry is rusty. Aug 8, 2023 at 9:31

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
$$2R\sin\left(\frac{2\pi}{2n}\right) = 2R\sin\left(\frac{\pi}{n}\right) = l$$