In the given figure on a square handkerchief, nine circular designs each of radius $7cm$ are made. Find the area of the remaining of the handkerchief.

Answer
363.6k+ views
Hint: Find the relation between the radius of the circle and the side of the square and then calculate the required areas.
Let the radius of the circle is $r$ and the side of the square handkerchief is $S$. The radius is given in the question. So, we have:
$ \Rightarrow r = 7cm$
As we can see from the figure, each side of the square is completely covered by three circles.
Thus the side of the square will be the sum of the lengths of the diameter of these three circles. But the circles are of equal diameters, then we have:
$
\Rightarrow s = 2r + 2r + 2r, \\
\Rightarrow s = 2(7) + 2(7) + 2(7) \\
\Rightarrow s = 14 + 14 + 14, \\
\Rightarrow s = 42 \\
$
Thus the side of the square is $42 cm$ . And we know the formula for the area of square which is ${s^2}$. So we have:
$
\Rightarrow {A_{square}} = {s^2}, \\
\Rightarrow {A_{square}} = {\left( {42} \right)^2}, \\
\Rightarrow {A_{square}} = 1764c{m^2} \\
$
Area of circle is $\pi {r^2}$. And there are $9$ circles in the square. So, the total area of all the circles is:
$
\Rightarrow {A_{circles}} = 9\pi {r^2}, \\
\Rightarrow {A_{circles}} = 9 \times \dfrac{{22}}{7} \times {\left( 7 \right)^2}, \\
\Rightarrow {A_{circles}} = 1386c{m^2}. \\
$
Therefore, the area of the remaining part of the handkerchief is:
$
\Rightarrow {A_{remaining}} = {A_{square}} - {A_{circles}}, \\
\Rightarrow {A_{remaining}} = 1764 - 1386, \\
\Rightarrow {A_{remaining}} = 378c{m^2}. \\
$
Thus, the area of the remaining portion of the handkerchief is $378c{m^2}$.
Note: In such cases, when one of the standard geometrical figures is inscribed in another standard figure, finding the relation between the sides of both the figures is the key point to solve the question.
Let the radius of the circle is $r$ and the side of the square handkerchief is $S$. The radius is given in the question. So, we have:
$ \Rightarrow r = 7cm$
As we can see from the figure, each side of the square is completely covered by three circles.
Thus the side of the square will be the sum of the lengths of the diameter of these three circles. But the circles are of equal diameters, then we have:
$
\Rightarrow s = 2r + 2r + 2r, \\
\Rightarrow s = 2(7) + 2(7) + 2(7) \\
\Rightarrow s = 14 + 14 + 14, \\
\Rightarrow s = 42 \\
$
Thus the side of the square is $42 cm$ . And we know the formula for the area of square which is ${s^2}$. So we have:
$
\Rightarrow {A_{square}} = {s^2}, \\
\Rightarrow {A_{square}} = {\left( {42} \right)^2}, \\
\Rightarrow {A_{square}} = 1764c{m^2} \\
$
Area of circle is $\pi {r^2}$. And there are $9$ circles in the square. So, the total area of all the circles is:
$
\Rightarrow {A_{circles}} = 9\pi {r^2}, \\
\Rightarrow {A_{circles}} = 9 \times \dfrac{{22}}{7} \times {\left( 7 \right)^2}, \\
\Rightarrow {A_{circles}} = 1386c{m^2}. \\
$
Therefore, the area of the remaining part of the handkerchief is:
$
\Rightarrow {A_{remaining}} = {A_{square}} - {A_{circles}}, \\
\Rightarrow {A_{remaining}} = 1764 - 1386, \\
\Rightarrow {A_{remaining}} = 378c{m^2}. \\
$
Thus, the area of the remaining portion of the handkerchief is $378c{m^2}$.
Note: In such cases, when one of the standard geometrical figures is inscribed in another standard figure, finding the relation between the sides of both the figures is the key point to solve the question.
Last updated date: 24th Sep 2023
•
Total views: 363.6k
•
Views today: 9.63k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

What is the basic unit of classification class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers
