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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Correct derivation: The height $ h $ of an equilateral triangle is $ rac{\sqrt{3}}{2}s $. The centroid (which coincides with the circumcenter) is at $ rac{2}{3} $ of the height from a vertex, so:

📌 R = rac{2}{3} \cdot rac{\sqrt{3}}{2}s = rac{s}{\sqrt{3}} \cdot rac{\sqrt{3}}{\sqrt{3}} = rac{s\sqrt{3}}{3}

📌 R = rac{s\sqrt{3}}{3}

📌 The area $ A $ of the circular dome is:

📌 A = \pi R^2 = \pi \left( rac{s\sqrt{3}}{3}

📌 ight)^2 = \pi \cdot rac{3s^2}{9} = rac{\pi s^2}{3}

📌 Thus, the area of the dome is $ oxed{\dfrac{\pi s^2}{3}} $.

📌 Question: A cartographer uses satellite data to map a circular forest reserve. A straight river crosses the reserve forming a chord of length 14 km, and the shortest distance from the forestâs center to the river is 5 km. What is the area of the forest in square kilometers?

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