In the adjoining figure O is the centre of the circle with radius 4. AB,CD and EF are the diameters of the circle. Angle OAF= angle OCB=60.

What is the area of the shaded region ? (Correct to two decimal places )

The answer is 4.34.

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The shaded portion is composed of three congruent segment of a circle. The area of a segment of a circle is equal to the area of a sector minus area of the triangle. So we have

$A_{shaded}=\dfrac{60}{360}(\pi)(4^2)-\dfrac{1}{2}(4^2)(\sin 60)(3)\approx 4.34813$