10. In a square lawn with side length $a$, there are sprinkler devices installed at each of the four corners. The sprinkler devices can rotate to spray water at a $90^{\circ}$ angle, and each sprinkler can rotate from one side of its corner to the other side, with an effective range of $a$. What is the proportion of the lawn that can be covered simultaneously by all four sprinklers to the entire lawn? $\qquad$ .
Solve:
As shown in the figure, $S_{\text {sector } A E F}=\frac{1}{2} \cdot \frac{\pi}{6} a^{2}, S_{\triangle A E F}=\frac{1}{2} \cdot \frac{1}{2} a^{2} \Rightarrow S_{\text {shape } E F}=\frac{\pi-3}{12} a^{2}$.
Also, $E F^{2}=2 a^{2}-2 a^{2} \cdot \frac{\sqrt{3}}{2}=(2-\sqrt{3}) a^{2}$, so the proportion of the lawn area simultaneously covered by the four sprinklers is $2-\sqrt{3}+\frac{\pi-3}{3}=\frac{\pi+3-3 \sqrt{3}}{3}$.