3. From a circular disk with radius 3, we cut out three circles with radius 1 as shown in the left figure. This causes the circular disk to fall apart into two pieces. We rotate the bottom piece a quarter turn and place it on the top piece as shown in the right figure. The area where the two pieces overlap is shaded darker.

What is the total area of this right figure (i.e., the light and dark gray parts together)?
A) $4 \pi$
B) $\frac{9}{2} \pi$
C) $\frac{19}{4} \pi$
D) $5 \pi$
E) $\frac{21}{4} \pi$
![](https://cdn.mathpix.com/cropped/2024_04_17_f11aeda05102501e0c70g-1.jpg?height=320&width=848&top_left_y=2310&top_left_x=615)
A3. D) $5 \pi$ In the figure on the right, we see that the large circle consists of five small circles and four dark gray areas. The large circle has an area of $9 \pi$ and each of the small circles has an area of $\pi$, so the four dark gray areas together have an area of $9 \pi - 5 \pi = 4 \pi$. Since the four dark gray areas have equal areas, each dark gray area thus has an area of $\pi$. We conclude that the total area of the picture in the problem is equal to $5 \pi$ (two small circles and three dark gray areas together).

![](https://cdn.mathpix.com/cropped/2024_04_17_732d64300714d2dca006g-1.jpg?height=320&width=316&top_left_y=1636&top_left_x=1521)