Question

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?

Accepted Answer

E. Distance between centers $PQ = 2$. $AB = 2 \sqrt{2^2 - 1^2} = 2\sqrt{3}$. Circle R has radius $r_R = AB/2 = \sqrt{3}$. Overlap area = Semi-circle area of R + Area of segment ANBOA. Area of semi-circle R = $\frac{\pi (\sqrt{3})^2}{2} = \frac{3\pi}{2}$. Area of segment = Area of sector ANBM - Area of $	riangle 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 = $\frac{3\pi}{2} + \frac{4\pi}{3} - \sqrt{3} = \frac{17\pi}{6} - \sqrt{3}$.

Area of Overlap Between Circles

Explanation

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