Subject: General Aptitude · Chapter: Area · Exam: · Marks: · Difficulty:
A circle is circumscribed around a square as shown in the figure. The area of one of the four shaded portions is equal to $\frac{4}{7}$. The radius of the circle is (R.R.B., 2006) $\sqrt{2}$
(b)$\frac{1}{\sqrt{2}}$
(c)2
(d)3
Answer
Answer (as printed): A
Explanation
Let the radius of the circle be $r$ and the side of the square be $a$. Then, diagonal of the square $$=2 r \Rightarrow \quad \sqrt{2} a=2 r \Rightarrow a=\frac{2}{\sqrt{2}} r=\sqrt{2} r .$$ Area of one shaded portion $=\frac{1}{4}\left[\pi r^{2}-(\sqrt{2} r)^{2}\right]=\frac{1}{4}(\pi-2) r^{2}$. $$\begin{aligned} \therefore \frac{1}{4}(\pi-2) r^{2}=\frac{4}{7} & \Rightarrow\left(\frac{22}{7}-2\right) r^{2}=\frac{16}{7} \Rightarrow r^{2}=\frac{16}{7} \times \frac{7}{8} \times 2 \\ & \Rightarrow r=\sqrt{2} \end{aligned}$$
Explanation as extracted from the printed page; notation may be imperfect.