Intersecting Circles Overlap Area

CAT 2008 Slot 1 · QA · Medium · Geometry

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Two circles, both of radii 1 cm1\text{ cm}, intersect such that the circumference of each one passes through the centre of the other. What is the area (in sq. cm.) of the intersecting region?

  1. A.

    π3−34\frac{\pi}{3} - \frac{\sqrt{3}}{4}

  2. B.

    2π3+32\frac{2\pi}{3} + \frac{\sqrt{3}}{2}

  3. C.

    4π3−32\frac{4\pi}{3} - \frac{\sqrt{3}}{2}

  4. D.

    4π3+32\frac{4\pi}{3} + \frac{\sqrt{3}}{2}

  5. E.

    2π3−32\frac{2\pi}{3} - \frac{\sqrt{3}}{2}

Answer

E

Explanation

The region consists of two circular segments formed by an angle of 120∘120^\circ (2π/32 \pi / 3).

Area of segment=Area of sector−Area of triangle\text{Area of segment} = \text{Area of sector} - \text{Area of triangle} Area of sector=120360×π(1)2=π3\text{Area of sector} = \frac{120}{360} \times \pi (1)^2 = \frac{\pi}{3} Area of triangle=12(1)(1)sin⁡120∘=34\text{Area of triangle} = \frac{1}{2} (1)(1) \sin 120^\circ = \frac{\sqrt{3}}{4}

Intersection Area =2×(π3−34)=2π3−32= 2 \times \left(\frac{\pi}{3} - \frac{\sqrt{3}}{4}\right) = \frac{2\pi}{3} - \frac{\sqrt{3}}{2}.

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