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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Thus, the final answer is \(\boxed{\frac{1}{3}}\).

📌 A regular tetrahedron has a side length of $s$. Determine the ratio of the volume of the tetrahedron to the volume of the sphere circumscribed around it.

📌 The volume of a regular tetrahedron with side length $s$ is given by:

📌 V_{\text{tetrahedron}} = \frac{\sqrt{2}}{12} s^3

📌 The radius of the circumscribed sphere (circumradius) of a regular tetrahedron is:

📌 R = \frac{s \sqrt{6}}{4}

📌 The volume of the sphere is:

📌 V_{\text{sphere}} = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi \left(\frac{s \sqrt{6}}{4}\right)^3 = \frac{4}{3} \pi \frac{6\sqrt{6}}{64} s^3 = \frac{\pi \sqrt{6}}{8} s^3

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