In the diagram, each of the two circles has centre $O$. Also, $O P: P Q=1: 2$. If the radius of the larger circle is 9 , what is the area of the shaded region?
(A) $12 \pi$
(B) $36 \pi$
(C) $54 \pi$
(D) $72 \pi$
(E) $81 \pi$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-070.jpg?height=406&width=441&top_left_y=255&top_left_x=1292)
Since the radius of the larger circle is 9 , then $O Q=9$ and the area of the larger circle is $\pi 9^{2}=81 \pi$.

Since $O P: P Q=1: 2$ and $O Q=9$, then $O P=\frac{1}{3} O Q=3$.

Thus, the radius of the smaller circle is 3 and so the area of the smaller circle is $\pi 3^{2}=9 \pi$.

The area of the shaded region equals the area of the large circle minus the area of the small circle, or $81 \pi-9 \pi=72 \pi$.

ANswer: (D)