Rollin' Stones

Geometry Level 5

Find the area of the hatched region, if the radii of the two circles in the figure above are $1$ and $\sqrt[5]{162+27\sqrt[5]{162+27\sqrt[5]{\cdots}}}$ .

If your answer is in the form of $a\sqrt{b}-\frac{c}{2b}\pi$ for some positive pairwise coprime integers $a, b, c$ . Find $a+b+c$ .