jmc

If $r$ is the radius of the circle, we know that $x=\pi r^2$ and $y=2\pi r$. Thus, we have $$\begin{array}{rl}x+y& =80\pi \\ \pi r^2+2\pi r& =80\pi \\ r(r+2)& =80.\end{array}$$

We want to find two integers whose product is 80, such that one integer is two more than the other. We note that 80 can be factored as $8\cdot 10$. Therefore, $r=\boxed{8}$.