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📅 May 11, 2026 👤 scraface

May 11, 2026

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V_{\text{tetrahedron}} = \frac{b^3}{6\sqrt{2}}
The radius \( r \) of the inscribed sphere in a regular tetrahedron is...
r = \frac{b \sqrt{6}}{12}
The volume \( V_{\text{sphere}} \) of the inscribed sphere is:
V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left( \frac...
Now, the ratio of the volume of the sphere to the volume of the tetrah...
\text{Ratio} = \frac{V_{\text{sphere}}}{V_{\text{tetrahedron}}} = \fra...
\sqrt{6} \cdot \sqrt{2} = \sqrt{12} = 2\sqrt{3}
\text{Ratio} = \frac{\pi \cdot 6 \cdot 2\sqrt{3}}{864} = \frac{12\pi \...
Thus, the ratio is \(\boxed{\frac{\pi \sqrt{3}}{72}}\).

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