Circles in the Circle

Consider a general case of $n$ small circles of radius $r$ touching the circumference of a large circle internally. The the distance between centers of two adjacent small circles is $2r$ and the angle extended by the two centers and the center $O$ of the large circle is (\frac {2\pi}n}. Then the radius of the large circle is given by:

$OQ = PQ + OP = r + \frac r{\sin \frac \pi n} = \boxed{r\left(1+\cosec \frac \pi n\right)}$

The centers of $n$ small circles form a regular $n$ -gon whose each internal angle is $(2n-4)\dfrac{π}{2n}$ . The distance of any vertex of this $n$ -gon from it's centre is $r\sec (\dfrac{π}{2}-\dfrac{π}{n})=r\cosec \dfrac{π}{n}$ . Hence, the radius of the big circle is $r+r\cosec \dfrac{π}{n}=\boxed {r(1+\cosec \dfrac{π}{n})}$