In the diagram, two circles are centred at $O$. The smaller circle has a radius of 1 and the larger circle has a radius of 3. Points $P$ and $Q$ are placed on the larger circle so that the areas of the two shaded regions are equal. If $\angle P O Q=x^{\circ}$, what is the value of $x$ ?

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-011.jpg?height=369&width=374&top_left_y=1084&top_left_x=1298)
Since the shaded regions are equal in area, then when the unshaded sector in the small circle is shaded, the area of the now fully shaded sector of the larger circle must be equal to the area of the smaller circle.
![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-018.jpg?height=380&width=790&top_left_y=1136&top_left_x=728)

The smaller circle has radius 1 and so it has area $\pi \times 1^{2}=\pi$.

The larger circle has radius 3 and so it has area $\pi \times 3^{2}=9 \pi$.

This means that the area of the shaded sector in the larger circle is $\pi$, which means that it must be $\frac{1}{9}$ of the larger circle.

This means that $\angle P O Q$ must be $\frac{1}{9}$ of a complete circular angle, and so $\angle P O Q=\frac{1}{9} \times 360^{\circ}=40^{\circ}$. Thus, $x=40$.

ANSWER: 40