\text{Ratio} = \frac{V_{\text{sphere}}}{V_{\text{tetrahedron}}} = \frac{\frac{\pi \sqrt{6} b^3}{864}}{\frac{b^3}{6\sqrt{2}}} = \frac{\pi \sqrt{6} b^3}{864} \cdot \frac{6\sqrt{2}}{b^3} = \frac{\pi \sqrt{6} \cdot 6\sqrt{2}}{864}