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#### Question 1:

Find the area of a circle whose radius is
(i) 7 cm
(ii) 2.1 m
(iii) 7 km

(i) We know that the area A of a circle of radius is given by A$\mathrm{\pi }{r}^{2}$.
Here, r  = 7 cm
.
.
(ii) ​We know that the area A of a circle of radius r is given by A$\mathrm{\pi }{r}^{2}$.
Here, r  = 2.1 m
∴ .
.
(iii) ​We know that the area A of a circle of radius r is given by A =  $\mathrm{\pi }{r}^{2}$.
Here, r  = 7 km
∴ .
.

#### Question 2:

Find the area of a circle whose diameter is
(i) 8.4 cm
(ii) 5.6 m
(iii) 7 km

(i) Let r be the radius of the circle. Then, r = 8.4 $÷$ 2 = 4.2 cm.
∴ Area of the circle  = ${\mathrm{\pi r}}^{2}$
.

(ii) Let r be the radius of the circle. Then, r = 5.6 $÷$ 2 = 2.8 m.
Area of the circle  = ${\mathrm{\pi r}}^{2}$

(iii) ​​Let r be the radius of the circle. Then, r = $÷$ 2 = 3.5 km.
Area of the circle  = ${\mathrm{\pi r}}^{2}$

#### Question 3:

The area of a circle is 154 cm2. Find the radius of the circle.

Let the radius of the circle be r cm.
Area of the circle (A) = 154 cm2

Hence, the radius of the circle is 7 cm.

#### Question 4:

Find the radius of a circle, if its area is
(i) 4 π cm2
(ii) 55.44 m2
(iii) 1.54 km2

(i) ​Let the radius of the circle be r cm.

∴ Area of the circle (A) = 4$\mathrm{\pi }$ cm2

(ii) ​Let the radius of the circle be r cm.

∴ Area of the circle (A) = 55.44 m2

(iii) ​Let the radius of the circle be r cm.

∴ Area of the circle (A) = 1.54 km2

#### Question 5:

The circumference of a circle is 3.14 m, find its area.

We have :

Circumference of the circle = 3.14  ​m = $2\mathrm{\pi r}$

Area of the circle (A) = ${\mathrm{\pi r}}^{2}$

#### Question 6:

If the area of a circle is 50.24 m2, find its circumference.

We have :
Area of the circle (A) = ​${\mathrm{\pi r}}^{2}$ = 50.24 m2

Circumference of circle (C) = $2\mathrm{\pi r}$

#### Question 7:

A horse is tied to a pole with 28 m long string. Find the area where the horse can graze. (Take π = 22/7).

We have :
Length of the string = ​28 m
The area over which the horse can graze is the same as the area of a circle of radius 28 m.
Hence, required area = π​r2.

#### Question 8:

A steel wire when bent in the form of a square encloses an area of 121 cm2. If the same wire is bent in the form of a circle, find the area of the circle.

We have :
Area of the square = 121 cm2
⇒ (side)2 = (11)2 cm2
⇒ side =  11 cm.
So, the perimeter of the square = 4(side) = (4 x 11) cm = 44 cm.

Let r be the radius of the circle. Then,
Circumference of the circle = Perimeter of the square
⇒     2πr =  44
⇒     2 x $\frac{22}{7}$x r =  44
⇒ r = 7 cm.
∴ Area of the circle = π​r2  =  .

#### Question 9:

A road which is 7 m wide surrounds a circular park whose circumference is 352 m. Find the area of of road.

We have :
Circumference of the circular park = 2πr = 352 m
​  ⇒     2 x $\frac{22}{7}$r =  352
⇒ r = 56 m.
Radius of the path including the 7m wide road = (r +7) =  56 +7 = 63 m.
∴ Area of the road :

#### Question 10:

Prove that the area of a circular path of uniform width h surrounding a circular region of radius r is π h (2r + h).

Radius  of the circular region = r
​Radius of the circular path of uniform width h surrounding the circular region of radius r = (rh).
​∴ Area of the path

#### Question 11:

The perimeter of a circle is 4πr cm. What is the area of the circle?

We have :
Given perimeter of the circle = 4$\mathrm{\pi }$r cm = 2​$\mathrm{\pi }$ (2r) cm

We know that, the perimeter of a circle = 2$\mathrm{\pi }$r

∴ Radius of the circle = 2r cm
Area of the circle= $\mathrm{\pi }$r2   = ​$\mathrm{\pi }$ (2r)2 = 4​$\mathrm{\pi }$r2 .

#### Question 12:

A wire of 5024 m length is in the form of a square. It is cut and made a circle. Find the ratio of the area of the square to that of the circle.

We have:
Perimeter of the square = 5024 m = Circumference of the circle
⇒​ 4 x Side of the square = 5024
∴ Side of the square = .
Let the area of the square be A1 and the area of the circle be A2.
Area of the square (A1)= side x side = .
Circumference of the circle = 5024 m
⇒ 2 $\mathrm{\pi }$r = 5024 m

Area of the circle (A2)= $\mathrm{\pi }$r2   = ​$\left[\frac{22}{7}×\frac{5024×7}{2×22}×\frac{5024×7}{2×22}\right]=\left[\frac{5024×5024××7}{2×2×22}\right]{\mathrm{m}}^{2}$.

Hence, the ratio of the area of the square to the area of the circle is 11:14.

#### Question 13:

The radius of a circle is 14 cm. Find the radius of the circle whose area is double of the area of the circle.

Let the area of the circle whose radius is 14 cm be A1.
Let the radius and area of the circle, whose area is twice the area of the circle A1 , be r2 and A2, respectively.

Thus,
A1.
A2 = 2 $×$ A1 = 2 $×$ 616 = 1232 cm2
A2 = $\mathrm{\pi }{\left({\mathrm{r}}_{2}\right)}^{2}$ = 1232 cm2

Hence, the radius of the circle A2 is 14$\sqrt{2}$ cm.

#### Question 14:

The radius of one circluar field is 20 m and that of another is 48 m. Find the radius of the third circular field whose area is equal to the sum of the areas of two fields.

Let the area of the circle whose radius is 20 m be A1 , and the area of the circle whose radius
is 48 m be A2. Let A3 be the area of a circle that is equal to the sum of the areas of the two fields, with the radius of its field being r cm.
A3 =  A1 + A2

#### Question 15:

The radius of one circular field is 5 m and that of the other is 13 m. Find the radius of the circular field whose area is the difference of the areas of first and second field.

Let the area of the circular field whose radius is 5 m be A1 , and the area of the circular field whose radius is 13 m be A2. Let A3 and r cm be the area and radius of the circular field, that is equal to the difference of the areas of the two fields.
∴ A3 = A2 - A1

Hence, the radius of the circular field is 12 m.

#### Question 16:

Two circles are drawn inside a big circle with diameters $\frac{2}{3}\mathrm{rd}$ and $\frac{1}{3}\mathrm{rd}$ of the diameter of the big circle as shown in Fig. 18. Find the area of the shaded portion, if the length of the diameter of the circle is 18 cm.

Let the left circle be denoted as the 1st circle and the right circle be denoted as the 2nd circle.
Diameter of the big circle = 18 cm
Radius of the big circle = 9 cm
Diameter of the 1st circle =
Radius of the 1st circle = 6 cm
Diameter of the 2nd circle =
Radius of the 2nd circle = 3 cm
Area of the 1st circle = $\mathrm{\pi }\left(6{\right)}^{2}=36{\mathrm{\pi cm}}^{2}$
Area of the 2nd circle =
Area of the big circle =
Area of the shaded portion = Area of the big circle - (Area of the Ist circle + Area of the IInd circle)
Area of the shaded portion = $81\mathrm{\pi }-\left(36\mathrm{\pi }+9\mathrm{\pi }\right)=36\mathrm{\pi }$ cm2.

#### Question 17:

In Fig. 19, the radius of quarter circular plot taken is 2 m and radius of the flower bed is 2 m. Find the area of the remaining field.

Radius of the quarter circular plot = 2 m
Area of the quarter circular plot =
Radius of each flower bed = 2 m
Area of four flower beds =
Area of the rectangular region = Length $×$ Breadth
Area of the rectangular region = 8 $×$ 6 = 48 m2
Area of the remaining field = Area of the rectangular region - (Area of the quarter circle + Area of the four flower beds)

Area of the remaining field = [48 - (12.57 + 12.57)] m2 = 22.86 m2 .

#### Question 18:

Four equal circles, each of radius 5 cm, touch each other as shown in Fig. 20. Find the area included between them. (Take π = 3.14).

Side of the square = 10 cm
Area of the square = Side $×$ Side
Area of the square =

Area of the four quarter circles =
Area included in them = Area of the square - Area of the four quarter circles
Area included in them = ( 100 - 78.57 ) cm2 = 21.43 cm2 .

#### Question 19:

The area of circle is 100 times the area of another circle. What is the ratio of their circumferences?

Let the area of the first circle be A1, the circumference be  C1 and the radius be r1.
Let the area of the second circle be A2 , the circumference be C2  and the radius be r2.

Thus,

We know that :

A1 =  100A2
∴

Substituting the values, we get:
∴​

Hence, the ratio of their circumferences is 10:1.

#### Question 1:

Find the circumference of a circle whose radius is
(i) 14 cm
(ii) 10 m
(iii) 4 km

(i) The circumference C of a circle of radius 'r' is given by:
C = 2$\mathrm{\pi }$r
Here, r = 14 cm
∴ C = .
(ii) The circumference C of a circle of radius 'r' is given by:
C = 2$\mathrm{\pi }$r
Here, r = 10 m
∴ C = .
(iii) ​The circumference C of a circle of radius 'r' is given by:
C = 2$\mathrm{\pi }$r
Here, r = 4 km
∴ C = .

#### Question 2:

Find the circumference of a circle whose diameter is
(i) 7 cm
(ii) 4.2 cm
(iii) 11.2 km

(i) We have:
Diameter = 7 cm
∴ Radius = $\frac{7}{2}$ cm
Let C be the circumference of a circle. Then,
C = 2$\mathrm{\pi }$r
∴ C = .
(ii)   We have:
Diameter = 4.2 cm
Let C be the circumference of a circle. Then,
C = 2$\mathrm{\pi }$r
∴ C = .
(iii) ​​We have:
Diameter = 11.2 km
Let C be the circumference of a circle. Then,
C = 2$\mathrm{\pi }$r
∴ C = .

#### Question 3:

Find the radius of a circle whose circumference is
(i) 52.8 cm
(ii) 42 cm
(iii) 6.6 km

(i) We know that the circumference C of a circle of  radius 'r' is given by:
C = 2$\mathrm{\pi }$r
.

​(ii) We know that the circumference C of a circle of radius 'r' is given by:
C = 2$\mathrm{\pi }$r

(iii) We know that the circumference C of a circle of radius 'r' is given by:
C = 2$\mathrm{\pi }$r

#### Question 4:

Find the diameter of a circle whose circumference is
(i) 12.56 cm
(ii) 88 m
(iii) 11.0 km

(i) ​We know that the circumference C of a circle of  diameter 'd' is given by:
C = $\mathrm{\pi }$d

(ii) ​We know that the circumference C of a circle of  diameter 'd' is given by:
C = $\mathrm{\pi }$d

(iii) ​We know that the circumference C of a circle of diameter 'd' is given by:
C = $\mathrm{\pi }$d

#### Question 5:

The ratio of the radii of two circles is 3 : 2. What is the ratio of their circumferences?

We have, the ratio of the radii = 3:2
So, let the radii of the two circles be 3r and 2r, respectively.
Let C1 and C2 be the circumferences of the two circles of radii 3r and 2r, respectively. Then,
C1= 2$\mathrm{\pi }$x 3r = 6$\mathrm{\pi }$r,  and C2 = ​2$\mathrm{\pi }$x2r = 4$\mathrm{\pi }$r

, or

C1: C2 = 3:2.

#### Question 6:

A wire in the form of a rectangle 18.7 cm long and 14.3 cm wide is reshaped and bent into the form of a circle. Find the radius of the circle so formed.

Length of the wire = Perimeter of the rectangle
= 2 ( l +) = 2 x (18.7 + 14.3) = 66 cm

Let the wire be bent in the form of a circle of radius r cm. Then,

circumference = 66 cm

#### Question 7:

A piece of wire is bent in the shape of an equilateral triangle of each side 6.6 cm. it is re-bent to form a circular ring. What is the diameter of the ring?

We have:
Length of the wire = The perimeter of  the equilateral triangle

=  3 x side = 3 x 6.6 =   19.8 cm.

Let the wire be bent to form a circular ring of radius r cm.Then,

Circumference =  19.8 cm

So, the diameter of the ring = 2 x 3.15 = 6.30 cm.

#### Question 8:

The diameter of a wheel of a car is 63 cm. Find the distance travelled by the car during the period, the wheel makes 1000 revolutions.

It may be noted that in one revolution, the cycle covers a distance equal to the circumference of the wheel.
Now, the diameter of the wheel = 63 cm
∴ Circumference of the wheel = .
Thus, the cycle covers 198 cm in one revolution.
∴ The distance covered by the cycle in 1000 revolutions = (198 x 1000 ) cm = 198000 cm = 1980 m.

#### Question 9:

The diameter of a wheel of a car is 98 cm. How many revolutions will it make to travel 6160 metres.

We have:
Diameter of the wheel of the car = 98 cm
∴ Circumference of the wheel of the car = .
Note that, in one revolution of the wheel, the car travels a distance equal to the circumference of the wheel.

∴ The distance travelled by the car in one revolution of the wheel = 308 cm.
Total distance travelled by the car = 6160 m = 616000 cm.
​​∴ Number of revolutions = .

#### Question 10:

The moon is about 384400 km from the earth and its path around the earth is nearly circular. Find the circumference of the path described by the moon in lunar month.

We have:
The radius of the path described by the moon around the earth = 384400 km
∴ The circumference of the path described by the moon C = .

#### Question 11:

How long will John take to make a round of a circular field of radius 21 m cycling at the speed of 8 km/hr?

We have:
The radius of the circular field = 21 m
∴ Circumference of the circular field = .
If John cycles at the speed of 8 km/hr (In 1 hour John covers 8 km = 8000 m ), then,
John covers 8000 m in 1 hour.

​​∴ Time required to cover 132 m  =
1 hour = 3600 seconds
∴ 0.0615 hours = 0.0615 x 3600 = 59.4 seconds.

#### Question 12:

The hour and minute hands of a clock are 4 cm and 6 cm long respectively. Find the sum of the distances travelled by their tips in 2 days.

The radius of the path inscribed by the hour hand = Length of the hour hand = 4 cm
The radius of the path inscribed by the minute hand = Length of the minute hand = 6 cm

The circumference of the path inscribed by the hour hand = .
The hour hand makes 2 revolutions in one day.
∴ The distance covered by the hour hand in 2 days = .
The distance covered by the minute hand in 1 revolution =
The minute hand makes 1 revolution in one hour.
∴ In  1 day, it makes 24 revolutions.
In 2 days, it makes  2 x 24 revolutions.
∴ ​The distance covered by the minute hand in ​2 days =
The sum of the distances travelled by the hour and minute hands in 2 days = 1810.28 + 100.57 = 1910.85 cm.

#### Question 13:

A rhombus has the same perimeter as the circumference of a circle. If the side of the rhombus is 2.2 m, find the radius of the circle.

We have:

The side of a rhombus  = 2.2 m
Let C be the circumference of a circle having a radius r cm. Then,
the perimeter of the rhombus = 4$×\mathrm{side}$ = 4 x 2.2 = 8.8 m.

We know:
Perimeter of the rhombus = Circumference of the circle

The radius of the circle is 1.4 m.

#### Question 14:

A wire is looped in the form of a circle of radius 28 cm. It is re-bent into a square form. Determine the length of the side of the square.

We have:

The radius of the circle  = 28 cm
∴ Circumference of the circle  = .
Let a cm be the side of the square. Then,
the circumference of the circle  = the perimeter of the square

The side of the square is 44 cm.

#### Question 15:

A bicycle wheel makes 5000 revolutions in moving 11 km. Find the diameter of the wheel.

We have:

Total distance covered in 5000 revolutions =  11 km = 11,000 m
∴​ Distance covered in 1 revolution = .
Distance covered in 1 revolution =  Circumference of the wheel
​
Thus, the diameter of the wheel is 0.7 m = 70 cm.

#### Question 16:

A boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheel is 60 cm, calculate the speed per hour with which the boy is cycling.

​​We have:
The diameter of the wheel =  60 cm
Distance covered by the wheel in 1 revolution = Circumference of the wheel
∴ Distance covered by the wheel in 1 revolution =
∴ Distance covered in 140 revolutions = .

Thus, the wheel covers 26400 cm in 1 minute. Then,

The speed with which the boy is cycling is 15.84 km/hr.                                 ​

#### Question 17:

The diameter of the driving wheel of a bus is 140 cm. How many revolutions per minute must the wheel make in order to keep a speed of 66 km per hour?

We have:
Diameter of the wheel = 140 cm, desired speed of the bus = 66 km/hr
∴ Distance covered in 1 revolution = The circumference of the wheel = $\mathrm{\pi d}$ = .
Now, the desired speed of the bus = 66 km/hr  =.
∴ Number of revolutions per minute = .

Thus, the bus must make 250 revolutions per minute to keep the speed at 66 km/hr.

#### Question 18:

A water sprinkler in a lawn sprays water as far as 7 m in all directions. Find the length of the outer edge of wet grass.

The wet grass forms a circular region of radius 7 m.
∴ The length of the outer edge of the wet grass is .

#### Question 19:

A well of diameter 150 cm has a stone parapet around it. If the length of the outer edge of the parapet is 660 cm, then find the width of the parapet.

We have :
Diameter of the well  = 150 cm
Length of the outer edge of the parapet = 660 cm
Width of the parapet = ?
Radius of well = 150 $÷$ 2 = 75 cm.
Let the width of  the stone parapet be x cm.Clearly, the outer edge of the parapet forms a circular region of radius (x +75 cm). Therefore,

Thus, the width of the parapet is 30 cm.

#### Question 20:

An ox in a kolhu (an oil processing apparatus) is tethered to a rope 3 m long. How much distance does it cover in 14 rounds?