Question

O is the center 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. $\pi \sqrt{3}cm$
• B. $\sqrt{3\pi }cm$
• C. $3\sqrt{\pi }cm$
• D. $3\pi cm$

Answer 1

The correct option is B $\sqrt{3\pi }cm$

Let the length of side of square be $xcm$

$\Rightarrow$ Area of square $=(side{)}^2=x^2$

Diameter of circle is $4cm$ then,

Radius $\left(r\right)=2cm$

$\Rightarrow$ Area of sector of circle $=\frac{\theta }{{360}^o}\times \pi r^2$

$=\frac{{90}^o}{{360}^o}\times \pi (2{)}^2$ [ Since, angle of a square $=\theta ={90}^o$ ]

$=\frac{1}{4}\times 4\pi$

$=\pi c{m}^2$

$\therefore$ Area of shaded part $=\pi c{m}^2$

According to the question,

$\Rightarrow$ Shaded area $=\frac{1}{3}\times$ Area of a square.

$\Rightarrow$ $\pi =\frac{1}{3}{x}^2$

$\Rightarrow$ $3\pi =x^2$

$\therefore$ $x=\sqrt{3\pi }cm$

$\therefore$ The length of side of square is $x=\sqrt{3\pi }cm$