In the diagram, the smaller circles touch the larger circle and touch each other at the centre of the larger circle. The radius of the larger circle is 6 . What is the area of the shaded region?
(A) $27 \pi$
(B) $6 \pi$
(C) $9 \pi$
(D) $18 \pi$
(E) $36 \pi$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-081.jpg?height=342&width=339&top_left_y=1344&top_left_x=1343)
Label the centre of the larger circle $O$ and the points of contact between the larger circle and the smaller circles $A$ and $B$. Draw the radius $O A$ of the larger circle.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-155.jpg?height=342&width=415&top_left_y=1027&top_left_x=909)

Since the smaller circle and the larger circle touch at $A$, then the diameter through $A$ of the smaller circle lies along the diameter through $A$ of the larger circle. (This is because each diameter is perpendicular to the common tangent at the point of contact.)

Since $A O$ is a radius of the larger circle, then it is a diameter of the smaller circle.

Since the radius of the larger circle is 6 , then the diameter of the smaller circle is 6 , so the radius of the smaller circle on the left is 3 .

Similarly, we can draw a radius through $O$ and $B$ and deduce that the radius of the smaller circle on the right is also 3.

The area of the shaded region equals the area of the larger circle minus the combined area of the two smaller circles.

Thus, the area of the shaded region is $6^{2} \pi-3^{2} \pi-3^{2} \pi=36 \pi-9 \pi-9 \pi=18 \pi$.

ANSWER: (D)