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you miss every shot you don't take quote - IX Labs

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 A_{\text{circle}} = \pi r^2 = \pi \left(\frac{s \sqrt{3}}{6}\right)^2 = \pi \frac{3s^2}{36} = \frac{\pi s^2}{12}

📌 Now, the ratio of the area of the circle to the area of the triangle is:

📌 \text{Ratio} = \frac{A_{\text{circle}}}{A_{\text{triangle}}} = \frac{\frac{\pi s^2}{12}}{\frac{\sqrt{3}}{4} s^2} = \frac{\pi s^2}{12} \cdot \frac{4}{\sqrt{3} s^2} = \frac{\pi}{3\sqrt{3}} = \frac{\pi \sqrt{3}}{9}

📌 Thus, the ratio of the area of the circle to the area of the triangle is \(\boxed{\frac{\pi \sqrt{3}}{9}}\).

📌 A regular tetrahedron has a side length of \( b \) units. What is the ratio of the volume of the inscribed sphere to the volume of the tetrahedron?

📌 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:

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