smc

The circles have radii of $1$ and $2$. Draw the triangle shown in the figure above and write expressions in terms of $s$ (length of the side of the square) for the sides of the triangle. Because $AO$ is the radius of the larger circle, which is equal to $2$, we can write the Pythagorean Theorem. $$\begin{array}{rl}{\left(\frac{s}{2}\right)}^2+(s+1{)}^2& =2^2\\ \frac{1}{4}{s}^2+s^2+2s+1& =4\\ \frac{5}{4}{s}^2+2s-3& =0\\ 5s^2+8s-12& =0\end{array}$$ Use the quadratic formula. $$s=\frac{-8+\sqrt{8^2-4(5)(-12)}}{10}=\frac{-8+\sqrt{304}}{10}=\frac{-8+4\sqrt{19}}{10}=\boxed{\frac{2\sqrt{19}-4}{5}}$$