2. Let the distance from point $P$ to plane $\alpha$ be $\sqrt{3}$, and point $Q$ lies on plane $\alpha$ such that the angle between line $P Q$ and plane $\alpha$ is no less than $30^{\circ}$ and no more than $60^{\circ}$. Then the area of the region formed by such points $Q$ is $\qquad$.
2. $8 \pi$.

Let point $P$ have its projection on plane $\alpha$ as $O$. From the given condition, we have
$$
\frac{O P}{O Q}=\tan \angle O Q P \in\left[\frac{\sqrt{3}}{3}, \sqrt{3}\right] \text {, }
$$

which means $O Q \in[1,3]$.
Therefore, the area of the required region is $\pi \cdot 3^{2}-\pi \cdot 1^{2}=8 \pi$.