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📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 The area of a regular hexagon with side $ s $ is:

📌 The area of the moat (annulus between the circle and the hexagon) is:

📌 Now, the ratio of the moat area to the temple area is:

📌 Question: In a right triangle, the hypotenuse is $ h $ units and the inradius is $ r $. If the triangle represents a plot of land in a rural community where a circular garden is inscribed perfectly within the triangular boundary, find the ratio of the area of the garden to the area of the plot.

📌 Solution: Let the right triangle have legs $ a $, $ b $, and hypotenuse $ h $. The area $ A $ of the triangle is $

📌 angle $. Compute the cosine of the angle between them to assess alignment efficiency.

📌 Question: An archaeologist discovers a dome-shaped ancient structure in Southeast Asia, modeled as a hemisphere with radius $ 3r $. A smaller spherical artifact with radius $ r $ is found buried at the center. What is the ratio of the volume of the artifact to the volume of the dome?

📌 The dome is a hemisphere of radius $ 3r $. Its volume is half of a full sphere:

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