The circle shown below on the left has a radius of $1$ unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius $R,$ in units, of the circle on the right?

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Written solution:

The diagram on the right is similar to the diagram on the left, but the corresponding areas in the diagram on the right are $4$ times the areas on the left. So, each length on the right is $\sqrt{4} = 2$ times the corresponding length on the left. This gives $R = 2,$ which is choice B.