Question

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

Answer 1

Hint: Construct a square by joining the centre of each circle. The required area is the area between the pieces and can be found out by subtracting the area covered by the boxes on each circle from the area of the square constructed.

Complete step-by-step solution:
The following is the schematic diagram of the square which is joined with the centre of each circle.

Given:
The radius of one circle is $7\;{\rm{cm}}$.

The required area is the shaded region in the figure.

In the square $PQRS$, the length of each side is $14\;{\rm{cm}}$.
Hence the area of square $PQRS$ is given as:
\[
Ar\left( {PQRS} \right) = {\rm{side}} \times {\rm{side}}\\
 = {\rm{14}}\;{\rm{cm}} \times 14\;{\rm{cm}}\\
 = 1{\rm{96}}\;{\rm{c}}{{\rm{m}}^2}
\]
From the figure it is clear that $\dfrac{1}{4}$th of each circle is covered inside the box.
Hence the total area covered by the boxes in the 4 circles is given as,
$
A = \dfrac{1}{4} \times 4\\
A = 1
$

Therefore, the area covered by the boxes in the 4 circles is equal to the area of one circle.

 The area of circle is,
${{\rm{A}}_1} = \pi {r^2}$
Substitute \[7\;{\rm{cm}}\] for $r$ in the above expression.
$
{{\rm{A}}_1} = \dfrac{{22}}{7} \times {\left( {7\;{\rm{cm}}} \right)^2}\\
 = 154\;{\rm{c}}{{\rm{m}}^{\rm{2}}}
$
Now, the area of shaded region is given as,
\[
{\rm{Area of shaded region}} = 196\;{\rm{c}}{{\rm{m}}^{\rm{2}}} - 154\;{\rm{c}}{{\rm{m}}^{\rm{2}}}\\
 = 42\;{\rm{c}}{{\rm{m}}^2}
\]

Hence, the area of the portion enclosed between these pieces is $42\;{\rm{c}}{{\rm{m}}^{\rm{2}}}$.

Note:The area enclosed between the pieces is the area which is shaded in the figure. Make sure to make a diagram for the calculation. The $\dfrac{1}{4}$ portion of each circle is covered inside the box is the important point to find the area.