Question

In fig., $ABCD$ is a square of side $14cm$. With centers $A,B,C$ and $D$, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region

Answer 1

The square $ABCD$ touches $4$ circles and forms $4$ quadrants which can be seen in the figure.
Since, $AE=ED=AF=FB=BG=GC=CH=HD=\frac{1}{2}\times$ Side of Square$=$ Common radius
$\therefore$ Radius $=7cm$
Since, Area of Square ABCD $=14\times 14=196c{m}^2$
And, area of Quadrants(Sectors) $=\frac{\theta }{360}\times \pi \times r^2$

$=4\times \frac{90}{360}\times \pi \times 7\times 7$
$=154c{m}^2$
$\therefore$ Area of Shaded region $=$ Area of ABCD $-$ Area of 4 Quadrants
$=196-154=42c{m}^2$