Find the area of the shaded region in the figure shown

Last updated date: 25th Mar 2023
•
Total views: 307.5k
•
Views today: 2.84k
Answer
307.5k+ views
Hint- In order to solve such a question, find the relationship between the area of a square, the area of four sectors of the circle and the area of the shaded region. Also establish some relationship between the area of four sectors.
Complete step-by-step answer:
Given that: radius of four circles $ = 7cm$
So the side of the square $ = 2 \times 7cm = 14cm$
Since these four circles have the same radius and are equal in area so they must be subtending at the right angle at the square.
From the figure we can see that
Area of the shaded region = Area of square – sum of area of four sectors subtending right angle.
Since, all the circles are equal and sectors are subtending at the right angle. So, area of all the four sectors are also equal and sum of the area of these 4 equal sector is equal to the area of circle of same radius i.e. $7cm$
Sum of area of four sectors
\[
= 4 \times \left( {\dfrac{1}{4} \times \pi {r^2}} \right) \\
= \pi {r^2} \\
= \pi {\left( 7 \right)^2} \\
= \dfrac{{22}}{7} \times 7 \times 7 \\
= 154c{m^2} \\
\]
Area of the square
$
= {\left( {{\text{side}}} \right)^2} \\
= {\left( {14} \right)^2} \\
= 14 \times 14 \\
= 196c{m^2} \\
$
So the area of the shaded portion
$
= 196c{m^2} - 154c{m^2} \\
= 42c{m^2} \\
$
Hence, the area of the shaded portion is $42c{m^2}$ .
Note- In order to solve such questions related to area, figure plays a vital role in establishing the relationship between the area of the portion to be found out and the area of the given geometrical figures known. Formulas of area of sector and the area of squares must be remembered.
Complete step-by-step answer:
Given that: radius of four circles $ = 7cm$
So the side of the square $ = 2 \times 7cm = 14cm$
Since these four circles have the same radius and are equal in area so they must be subtending at the right angle at the square.
From the figure we can see that
Area of the shaded region = Area of square – sum of area of four sectors subtending right angle.
Since, all the circles are equal and sectors are subtending at the right angle. So, area of all the four sectors are also equal and sum of the area of these 4 equal sector is equal to the area of circle of same radius i.e. $7cm$
Sum of area of four sectors
\[
= 4 \times \left( {\dfrac{1}{4} \times \pi {r^2}} \right) \\
= \pi {r^2} \\
= \pi {\left( 7 \right)^2} \\
= \dfrac{{22}}{7} \times 7 \times 7 \\
= 154c{m^2} \\
\]
Area of the square
$
= {\left( {{\text{side}}} \right)^2} \\
= {\left( {14} \right)^2} \\
= 14 \times 14 \\
= 196c{m^2} \\
$
So the area of the shaded portion
$
= 196c{m^2} - 154c{m^2} \\
= 42c{m^2} \\
$
Hence, the area of the shaded portion is $42c{m^2}$ .
Note- In order to solve such questions related to area, figure plays a vital role in establishing the relationship between the area of the portion to be found out and the area of the given geometrical figures known. Formulas of area of sector and the area of squares must be remembered.
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

The coordinates of the points A and B are a0 and a0 class 11 maths JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?
