2025 AMC 8 Problem 18

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?

\textbf{(A)} \sqrt{2} \qquad \textbf{(B)} 2 \qquad \textbf{(C)} 2\sqrt{2} \qquad \textbf{(D)} 4 \qquad \textbf{(E)} 4\sqrt{2}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.