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Home » Dukes of Legend? The Shocking Truth Behind the Oldest Video Game Icon! - inBeat

Dukes of Legend? The Shocking Truth Behind the Oldest Video Game Icon! - inBeat

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 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}

📌 \sqrt{6} \cdot \sqrt{2} = \sqrt{12} = 2\sqrt{3}

📌 \text{Ratio} = \frac{\pi \cdot 6 \cdot 2\sqrt{3}}{864} = \frac{12\pi \sqrt{3}}{864} = \frac{\pi \sqrt{3}}{72}

📌 Thus, the ratio is \(\boxed{\frac{\pi \sqrt{3}}{72}}\).

📌 A rectangle with dimensions \( 5 \) cm by \( 12 \) cm is inscribed in a circle. What is the number of centimeters in the circumference of the circle? Express your answer in terms of \( \pi \).

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