Question

$ABCD$ is a square with side $a$. With centers, $A,B,C$ and $D$ four circles are drawn such that each circle touches externally the two of the remaining three circles. Let $\delta$ be the area of the region in the interior of the square that is not covered by the circles. Then the value of $\delta$ is:

• A. $a^2(1-\pi )$
• B. $a^2\left(\frac{4-\pi }{4}\right)$
• C. $a^2(\pi -1)$
• D. $\frac{\pi a^2}{4}$

Answer 1

Answer: (B)

$a^2\left(\frac{4-\pi }{4}\right)$

Area of the shaded region $=$ Area of square $-$ Area of 1 Circle (4 quarter = 1 circle)

Area of Square ABCD $=a^2$

radius of circle $=\frac{a}{2}$

Area of circle $=\pi r^2=\pi \frac{a^2}{4}$

$\therefore$ Area of shaded region $=a^2-\pi \frac{a^2}{4}$

$=a^2(\frac{4-\pi }{4})$

Hence, option $B$ is correct.