What the Hex-agon?

Using the Cosine rule, a side of length $20$ subtends an angle $a = \cos^{-1}\left(1 - \tfrac{200}{R^2}\right)$ at the centre, while a side of length $18$ subtends an angle $b = \cos^{-1}\left(1 - \tfrac{162}{R^2}\right)$ at the centre, where $R$ is the radius of the circle. Thus we require that $4a + 2b = 2\pi$ , and hence $a = \tfrac12\pi - \tfrac12b$ , so that $\cos a = \sin\tfrac12b$ . Thus we deduce that $\begin{aligned} 1 - \tfrac{200}{R^2} & = \; \tfrac{9}{R} \\ R^2 - 9R - 200 & = \; 0 \end{aligned}$ and hence the only possible value of $R > 0$ is $\tfrac12(9 + \sqrt{881})$ , making the answer $9 + 881 + 2 = \boxed{892}$ .