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Question:

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

Explanation:

Let O and P be the centres of the circles.

OR = OP = PR = 1cm

∴ ∆PRO is an equilateral triangle.

∴ m ∠ROP = 60°

∴ m ∠ROS = 120°

Now, area of the intersecting region = 2(area of sector O-RPS – area of ∆ORS)
                                                         = 2(area of sector O-RPS – area of ∆PRO) [area of ∆PRO = area of ∆ORS]

Area of sector O - RPS = $\frac{120}{360}\left(\pi \right)$ = $\frac{\pi }{3}$

Area of ∆PRO = $\frac{\sqrt{3}}{4}\left(1^2\right)$     

∴ Area of the intersecting region = $2\left(\frac{\pi }{3}\right)$ - $2\left(\frac{\sqrt{3}}{4}\right)$ = $\frac{2\pi }{3}-\frac{\sqrt{3}}{2}$

Hence, option (e).