Random circle lying fully inside a square

Take a square with side $20$. Pick a point uniformly at random inside it to be the centre of a circle, and independently choose the radius $R$ uniformly from $(0,10)$. What is the probability that the circle does not touch or cross the boundary of the square?

$\tfrac13$. For radius $r$, the centre must lie at least $r$ from every side, i.e. in a concentric square of side $20-2r$, with probability $\left(\tfrac{20-2r}{20}\right)^2=(1-\tfrac r{10})^2$. Averaging over $r$ uniform on $(0,10)$ gives $\int_0^1(1-u)^2\,du=\tfrac13$.