2025 AMC 8 Problem 18

Question

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?

\sqrt{2}

2

2\sqrt{2}

4

4\sqrt{2}

Answer: B

Concepts:similarity power scaling of length, area, and volume

Difficulty rating: 1310

Small hint:

Match the shaded gap areas by thinking about scaling

Big hint:

Areas scale as the square of the length scale factor

Video solution:

Click to load, then click again to play

Written solution:

Only one quarter of the gap between the circle and square is shaded on the right. For that shaded area to equal the whole gap on the left, the entire gap on the right must have

4

times the area of the gap on the left.

The two circle-and-square figures are similar, and their areas scale as the square of their radii. Thus

R^2 = 4 \cdot 1^2,

so

R = 2,

which is choice B.