Question

In the given figure, ABCD is a square of side 14 cm. Find the area of the shaded portion.

Answer 1

Hint: First we need the area of square using the formula, (side$ \times $side). And then using the side of the square we can determine the diameter of the circle and from that we can get the radius of the circle. With the radius we can calculate the area of the circle. Now to find the area of the shaded region we need to remove the area of circles from the area of the square.

Complete step-by-step answer:
Given:
Side of a square, a = 14 cm
Using this side value of square we need to calculate the radius of the circle.
If we observe all the circles must be identical to fit in the square. Hence the diameter of the circle will be half the side of the square
Therefore, the diameter of circle is $\dfrac{{14}}{2} = 7cm$
$\therefore $Radius of each circle, $r = \dfrac{{diameter}}{2} = \dfrac{7}{2}cm$

Area of a shaded portion = Area of a square - 4$ \times $Area of a circle
Area of square =${{\text{a}}^2}$
$ \Rightarrow 14 \times 14{\text{ c}}{{\text{m}}^2}$
$ \Rightarrow 196{\text{ c}}{{\text{m}}^2}$
Area of circle =$\pi {r^2}$
$ \Rightarrow \dfrac{{22}}{7} \times \dfrac{7}{2} \times \dfrac{7}{2}$
As radius of the circle$ = \dfrac{7}{2}cm$
$ \Rightarrow \dfrac{{154}}{4} = \dfrac{{77}}{2}{\text{ c}}{{\text{m}}^2}$
Therefore, Area of a shaded portion = Area of a square - 4$ \times $Area of a circle
$ \Rightarrow {{\text{a}}^2} - 4\left( {\pi {r^2}} \right)$
$ \Rightarrow 196 - 4 \times \dfrac{{77}}{2}$
$ \Rightarrow 196 - 2 \times 77$
$ = 196 - 154 = 42$
Therefore, Area of the shaded portion when the side of the square is given as 14 cm and 4 identical circles are fitted in it is, $ = 42c{m^2}$

Note: Using the square properties we can be able to calculate the shaded regions when different shapes of geometry are placed. As the square is a regular quadrilateral, where all sides and angles are equal. And to calculate the perimeter of a square we can use the formula, 4$ \times $side of the square. As all sides of a square are equal.