V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left( \frac{b \sqrt{6}}{12} \right)^3 = \frac{4}{3} \pi \frac{6\sqrt{6} b^3}{1728} = \frac{\pi \sqrt{6} b^3}{864}