Question

Four circular cardboard pieces of radii $7cm$ are placed on a paper in such a way that each pieces touches the other two. Find the area of the portion enclosed between these pieces.

Answer 1

The $4$ circles are placed in such a way that each piece touches the other two pieces.

So, by joining the centers of the circles by a line segment, we get a square $ABDC$ with $AB=BD=DC=CA=2\left(7\right)=14cm$.

Now, Area of square = $(side{)}^2=(14{)}^2=196$ $c{m}^2$

Since, $ABDC$ is a square, each angle has a measure of ${90}^o$.

$\angle A=\angle B=\angle C=\angle D={90}^o=\frac{\pi }{2}radians=\theta$

Also, radius of each sector = 7 cm

Area of sector with central angle A = $\frac{1}{2}{r}^2\theta =\frac{77}{2}c{m}^2$

Area of shaded portion = Area of square - Area of 4 sectors

$=196-(4\times \frac{77}{2})=42$ $c{m}^2$

Hence, required area is $42$ $c{m}^2$.