A Delicious Golden Region

We can split our work on finding green and yellow area (picture above).

Green area is a half of the smaller circle's area, its radius is $\frac{9}{2}$ , therefore green area is $\frac{81 \pi}{8}$ .

Yellow area is a quarter of the bigger circle's area minus area of $\triangle ODA$ . The first one is $\frac{81 \pi}{8}$ and the latter one is $\frac{81}{4}$ .

Now, when we collect the green and the yellow area we get: $\frac{81 \pi}{8} + \frac{81 \pi}{8} + \frac{81}{4} = \frac{81}{4}(\pi - 1)$ , which is solution. The rest will be done by a calculator.

Also, very nice problem. All compliments.