\text{Ratio} = \frac{A_{\text{circle}}}{A_{\text{triangle}}} = \frac{\frac{\pi s^2}{12}}{\frac{\sqrt{3}}{4} s^2} = \frac{\pi s^2}{12} \cdot \frac{4}{\sqrt{3} s^2} = \frac{\pi}{3\sqrt{3}} = \frac{\pi \sqrt{3}}{9}