2025 AMC 8 Problems/Problem 18

Problem

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?

$\textbf{(A)}\ \sqrt2\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 2\sqrt2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 4\sqrt2$

Solution

The area of the shaded region in the circle on the left is the area of the circle minus the area of the square, or $\big(\pi-2)$ . The shaded area in the circle on the right is $\dfrac{1}{4}$ of the area of the circle minus the area of the square, or $\dfrac{\pi R^2-2R^2}{4}$ , which can be factored as $\dfrac{R^2(\pi-2)}{4}$ . Since the shaded areas are equal to each other, we have $\pi-2=\dfrac{R^2(\pi-2)}{4}$ , which simplifies to $R^2=4$ . Taking the square root, we have $R=\boxed{\text{(B)\ 2}}$

~mrtnvlknv

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2025 AMC 8 Problems/Problem 18

Problem

Solution