3. We are looking at a square, the circle passing through the vertices of the square, and the circle that touches the four sides of the square (see the left figure). The area of the ring between the two circles is divided into four dark pieces (inside the square) and four light pieces (outside the square). The area of the square is 60.
![](https://cdn.mathpix.com/cropped/2024_04_17_51d50f3426d3742e694cg-3.jpg?height=412&width=1194&top_left_y=1146&top_left_x=435)

What is the combined area of two dark pieces and one light piece, as shown in the right figure?
B3. 15

The diagonal of the square is $\sqrt{2}$ times as long as the side of the square.

The diameters of the two circles are in the ratio $1: \sqrt{2}$, and thus their areas are in the ratio $1: 2$ because the area scales with the square of the diameter.

Since the area of the large circle is twice that of the small circle, the area of the four light gray pieces and the four dark gray pieces together is equal to the area of the small circle.

The small circle and the four dark gray pieces together form the square and thus have a combined area of 60. From the earlier observation, we conclude that the area of the four light gray pieces and eight dark gray pieces together is 60.

The area of one light gray piece and two dark gray pieces, which is what is asked for, is therefore equal to a quarter of this: $\frac{60}{4}=15$.