Relation between radius of circles

Question

A circle of radius R is given. Inside it there are N circles of radius r touching circumference of bigger circle.

What is the relation between radius R and r?

Answer 1

The angle between the centers of the small circles is $\frac{2\pi}{n}$(radians), so the angle from one of their centers to their point of tangency is half of that, $\frac{\pi}{n}$. Then

$$\sin\frac\pi n = \frac r x$$ But we don't know what $x$ is yet. We can get it from $$x + r = R$$ and combine the two equations: $$x = R - r = \frac{r}{\sin\frac\pi n}$$ $$R = r \left(1 + \frac{1}{\sin\frac\pi n} \right)$$ $$R \sin\frac\pi n = r \left(\sin\frac\pi n + 1 \right)$$