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2015 AMC 12B Problem 14

Problem 14 of 25IntermediateGeometry

A circle of radius 22 is centered at A.A. An equilateral triangle with side 44 has a vertex at A.A. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?

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Solution

Let zz be the area shared by the circle and triangle. The requested difference is (circle−z)−(triangle−z)=circle−triangle. \begin{aligned} &(\text{circle} - z) - (\text{triangle} - z) \\ &= \text{circle} - \text{triangle}. \end{aligned} The circle has area π⋅22=4π,\pi \cdot 2^2 = 4\pi, and the equilateral triangle has area 34⋅42=43.\dfrac{\sqrt3}{4}\cdot 4^2 = 4\sqrt3. The difference is 4π−43=4(π−3).4\pi - 4\sqrt3 = 4(\pi - \sqrt3). Thus, the correct answer is D.

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Concepts: area decomposition · circle area · equilateral triangle

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.