Areas Related to Circles

Solution of NCERT Exercise 12.1

Unless stated otherwise, use π = 22/7

Question (1) The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

Solution:

[Strategy to solve the question: (a) First calculate the circumferences of each of the given circle. (b) Add their circumferences to find the circumference of new circle radius of which is to be calculated. (c) Using new circumference calculated, calculate the radius of the required circle.]

Given, Radius of circle (1) = 19 cm

Radius of circle (2) = 9 cm

Radius of the circle which has circumference equal to the sum of the circumferences of the two circles = ?

We know that, circumference of circle = 2 π r

Thus, Circumference of bigger circle (1) = 2 × π 19 cm

= 38 π cm

Circumference of smaller circle (2) = 2 × π 9 cm

= 18 π cm

Circumference of new circle = circumference of circle (1) + circumference of circle (2)

= 38 π cm + 18 π cm

= 56 π cm

Let, radius of new circle = r_n

Thus, circumference of new circle = 2 π r_n

⇒ 56 π cm = 2 π r_n

⇒ 2 π r_n = 56 π cm

⇒ r_n = 56/2 = 28 cm

Thus, Radius of the required circle = 28 cm Answer

Question (2) The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

Solution:

[Strategy to solve the question (a) First calculate the areas of each of the given circle. (b) Add their circumferences to find the area of new circle radius of which is to be calculated. (c) Using new area calculated, calculate the radius of the required circle.]

Given,

Radius of circle (1) = 8 cm

Radius of circle (2) = 6 cm

Radius of the circle having area equal to the sum of the areas of the two circles = ?

We know that Area of a Circle = π r²

Therefore, Area of circle (1) = π (8 cm)²

= π 64 cm²

And, Area of circle (2) = π (6 cm)²

= π 36 cm²

Now, Area of new circle = Area of circle (1) + Area of circle (2)

= π 64 cm² + π 36 cm²

= π (64 + 36) cm²

= π 100 cm²

Now, let radius of new circle = r

Now, Area of new circle = π r²

⇒ π 100 cm² = π r²

⇒ r = 10 cm

Thus, radius of required circle = 10 cm Answer

Question (3) Figure depicts an archery target marked with its five scoring regions from the centre outwards as Gold, Red, Blue, Black and White. The diameter of the region representing Gold score is 21 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.

[Strategy to solve the question: (a) First find the area of gold region. (b) Find the area of red circle after finding the radius of red circle. Radius of red circle is equal to sum of radius of gold circle and 10.5 cm, because each of the other bands is 10.5 cm wider. (c) Now, subtract the area gold circle from area of red circle to find the area of red scoring region. (d) In similar way find the areas of rest of the scoring regions.]

Solution

Given, Diameter of Gold region = 21 cm

Therefore, radius of gold region = 21/2 = 10.5 cm

The diameter of each of the other band is 10.5 cm wide.

Therefore, Diameter of Red circle = Diameter of gold circle + 10.5 cm + 10.5 cm

= 21 cm + 21 cm = 42 cm

Thus, radius of red circle = 42/2 = 21 cm

This means radius of red circle = radius of gold circle + 10.5 cm

= 10.5 cm + 10.5 cm = 21 cm

Thus, areas of each of the five scoring region = ?

(a) Calculation of Area of gold scoring region

We know that area of a circle = π r²

Radius of gold scoring region = 10.5 cm

Thus, Area of gold scoring region =

(b) Calculation of Area of red scoring region

Radius of gold circle = 10.5 cm

Radius of red circle = 21 cm

Therefore, Area of red scoring region = Area of Red circle – Area of Gold circle

= 1039.5 cm²

(c) Calculation of Area of Blue scoring zone

As given in the question diameter of the next circle is 10.5 cm wide than the diameter of previous circle.

This means, radius of the next circle is 10.5 cm wider than previous circle,

Here, Diameter of Red circle = 42 cm

And thus, Radius of red circle = 21 cm

Therefore, Radius of blue circle = Radius of red circle + 10.5 cm

= 21 cm + 10.5 cm = 31.5 cm

⇒ Radius of blue circle = 31.5 cm

Now, Area of blue scoring region = Area of blue circle – Area of red circle

= π (31.5 cm)² – π (21 cm)²

= 1732.5 cm²

Thus, Area of blue scoring region = 1732.5 cm²

(d) Calculation of Area of Black scoring region

As given in the question diameter of the next circle is 10.5 cm wide than the diameter of previous circle.

Radius of black circle = Radius of blue circle + 10.5 cm

= 31.5 cm + 10.5 cm = 42 cm

Thus, Radius of black circle = 42 cm

Now, Area of black scoring region

= Area of black circle – Area of blue circle

= 2425.5 cm²

Thus, Area of black scoring region = 2425.5 cm²

(e) Calculation of white scoring region

As given in the question diameter of the next circle is 10.5 cm wide than the diameter of previous circle.

Radius of white circle = Radius of black circle + 10.5 cm

= 42 cm + 10.5 cm = 52.5 cm

Now, Area of white scoring region

= Area of white circle – Area of black circle

[Using formula a² – b² = (a + b)(a – b)]

= 3118.5 cm²

Thus, Area of white scoring region = 3118.5 cm²

Question (4) The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?

[Strategy to solve the question (a) Calculate the circumference of one wheel. (b) Calculate the distance covered by car in 10 minute (c) Find the number of rotation of wheel by dividing distance covered by car in 10 minute by circumference of wheel.]

Solution

Given, Diameter of each wheel = 80 cm

Therefore, Radius of wheel = 80/2 = 40 cm

Speed of car = 66 km / hour

Number of complete rotation in 10 minute by each of wheel = ?

We know that, circumference of a circle = 2 π r

Therefore, circumference of wheel of car = 2 π 40 cm

= 40 π cm

Now, distance covered by car in 1 hour = 60 minute = 66 km

Therefore, distance covered by car in 10 minute

And 11 km = 11 × 1000 × 100 cm

Now,

∵ to cover 40 π cm number of revolution taken by wheel = 1

∴ To cover 1 cm number of revolution taken by wheel

∴ To cover 11 × 1000 × 100 cm number of revolution taken by wheel

= 4375 revolution

Thus, number of revolution by each of the wheel in 10 minute = 4375 Answer

Question (5) Tick the correct answer in the following and justify your choice : If the perimeter and the area of a circle are numerically equal, then the radius of the circle is

(A) 2 unit

(B) π units

(C) 4 units

(D) 7 units

Answer: (A) 2 unit

Explanation

Given, If, Perimeter = Area

The radius , r = ?

If Perimeter of circle = Area of circle

⇒ 2 π r = π r²

⇒ π r × r = 2 π r

Thus, Radius = 2 unit Answer

MCQs Test

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