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Home » Thus, the circumradius of the triangle is \(\boxed8.5\). - Aurero

Thus, the circumradius of the triangle is \(\boxed8.5\). - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 The volume \( V_{\text{tetrahedron}} \) of a regular tetrahedron with side length \( b \) is:

📌 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{b \sqrt{6}}{12} \right)^3 = \frac{4}{3} \pi \frac{6\sqrt{6} b^3}{1728} = \frac{\pi \sqrt{6} b^3}{864}

📌 Now, the ratio of the volume of the sphere to the volume of the tetrahedron is:

📌 \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}

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