Find the area of the shaded region if the rad | vedclass

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(C) The centers of the three circles form an equilateral triangle $\Delta ABC$ with each side length $a = 1 + 1 = 2\, cm$.The area of an equilateral t

Vedclass · Accepted Answer

$\sqrt{3}-\frac{\pi}{2}$

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(C) The centers of the three circles form an equilateral triangle $\Delta ABC$ with each side length $a = 1 + 1 = 2\, cm$.The area of an equilateral triangle is given by $\frac{\sqrt{3}}{4} a^2$.Area of $\Delta ABC = \frac{\sqrt{3}}{4} 	imes (2)^2 = \sqrt{3}\, cm^2$.Each interior angle of an equilateral triangle is $60^{\circ}$.The area of a circular sector with radius $r$ and angle $	heta$ is $\frac{	heta}{360^{\circ}} 	imes \pi r^2$.Here,$r = 1\, cm$ and $	heta = 60^{\circ}$.Area of one sector = $\frac{60^{\circ}}{360^{\circ}} 	imes \pi (1)^2 = \frac{1}{6} \pi\, cm^2$.Since there are three such sectors inside the triangle,the total area of the three sectors = $3 	imes \frac{\pi}{6} = \frac{\pi}{2}\, cm^2$.The area of the shaded region is the area of the triangle minus the sum of the areas of the three sectors.Area of shaded region = $\sqrt{3} - \frac{\pi}{2}\, cm^2$.

Find the area of the shaded region if the radius of each of the circles is $1\, cm$.

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