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Area of Overlap Between Circles

XAT 2021 · Quantitative Ability & Data Interpretation · Hard · Geometry

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Two circles P and Q, each of radius 2 cm, pass through each other's centres. They intersect at points A and B. A circle R is drawn with diameter AB. What is the area of overlap (in square cm) between the circles R and P?

  1. A.

    8π3−23\frac{8\pi}{3} - 2\sqrt{3}

  2. B.

    8π3\frac{8\pi}{3}

  3. C.

    13π3−3\frac{13\pi}{3} - \sqrt{3}

  4. D.

    17π6−23\frac{17\pi}{6} - 2\sqrt{3}

  5. E.

    17π6−3\frac{17\pi}{6} - \sqrt{3}

Answer

E

Explanation

Distance between centers PQ=2PQ = 2. AB=222−12=23AB = 2 \sqrt{2^2 - 1^2} = 2\sqrt{3}. Circle R has radius rR=AB/2=3r_R = AB/2 = \sqrt{3}. Overlap area = Semi-circle area of R + Area of segment ANBOA. Area of semi-circle R = π(3)22=3π2\frac{\pi (\sqrt{3})^2}{2} = \frac{3\pi}{2}. Area of segment = Area of sector ANBM - Area of △AMB=120∘360∘π(22)−12(23)(1)=4π3−3\triangle AMB = \frac{120^\circ}{360^\circ} \pi (2^2) - \frac{1}{2}(2\sqrt{3})(1) = \frac{4\pi}{3} - \sqrt{3}. Total overlap area = 3π2+4π3−3=17π6−3\frac{3\pi}{2} + \frac{4\pi}{3} - \sqrt{3} = \frac{17\pi}{6} - \sqrt{3}.

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