question_answer

O is the centre of a circle of diameter 4 cm and OABC is a square, if the shaded area is \[\frac{1}{3}\] area of the square, then the side of the square is _____.

A)  \[\pi \sqrt{3}\]                        

B)  \[\sqrt{3\pi }\]                        

C)  \[3\sqrt{\pi }\]                        

D)  \[3\pi \]           

Correct Answer: B

Solution :

Let the length of side of square =x c Then area of square \[={{x}^{2}}\,c{{m}^{2}}\]             Area of sector of circle \[=\frac{\theta }{{{360}^{o}}}\times \pi {{r}^{2}}\] [\[\because \]  angle of a square \[=\theta ={{90}^{o}}\]] \[\therefore \]  Shaded area \[=\frac{\pi \times 4}{4}=\pi \] According to question, Area of square \[=3\text{ }\times \]shaded area, \[\Rightarrow \]            \[3\pi ={{x}^{2}}\]   \[\therefore \]  \[x=\sqrt{3\pi }\,cm\]