Answer

Verified

448.5k+ 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

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

How many crores make 10 million class 7 maths CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE