question_answer

Three circles of equal radii touch each other as shown in the figure. The radius of each circle is 1 cm. What is the area of the shaded region?

A)  \[\left( \frac{2\sqrt{3}-\pi }{2} \right)c{{m}^{2}}\]         

B)  \[\left( \frac{3\sqrt{2}-\pi }{3} \right)c{{m}^{2}}\]

C)  \[\frac{2\sqrt{3}}{\pi }c{{m}^{2}}\]                

D)  \[\frac{\sqrt{6}}{2\pi }c{{m}^{2}}\]

Correct Answer: A

Solution :

(a): Area of triangle \[=\frac{\sqrt{3}}{4}\times {{(2)}^{2}}=\sqrt{3}\,\,c{{m}^{2}}\] Area of part of three circles (indicated as circles p, q, r in the adjoining figure) enclosed by the triangle.             \[=3\times \pi \times {{(1)}^{2}}\times \frac{60}{360}=\frac{\pi }{2}c{{m}^{2}}\] Area of shaded region\[=\sqrt{3}-\frac{\pi }{2}=\frac{\left( 2\sqrt{3}-\pi  \right)}{2}c{{m}^{2}}\] Additional Insight If radius of circle is 'a', then, area of \[\Delta =\sqrt{3}{{a}^{2}}\]  and area of part of three circles (indicated as circles p, q, r in the adjoining figure) enclosed by \[\Delta =3\times {{a}^{2}}\times \frac{60}{360}=\frac{\pi {{a}^{2}}}{2}\] \[\therefore \]  Area of shaded region, \[=\frac{{{a}^{2}}}{2}\left( 2\sqrt{3}-\pi  \right)\]