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1. Find the area, circumference, and diameter of the circles whose radius is

(a) 10 cm

(b) 6.5 cm

(c) 8 cm (Take π = 22/7)

## Solution:

(a) 10 cm

Given radius r = 10 cm

The circle formulas are

The diameter of the circle d = 2r

d = 2 x 10

d = 20 cm

Circumference of the circle C = 2πr

C = 2 x (22/7) x 10

C = 440/7

C = 62.85 cm

Area of the circle A = πr²

A = π x (10)² = π x 10 x 10

A = 100 x (22/7) = 2200 / 7

A = 314.285 cm²

∴ Area of circle is 314.285 cm², circumference is 62.85 cm, and diameter is 20 cm.

(b) 6.5 cm

Given radius r = 6.5 cm

The diameter of the circle d = 2r

d = 2 x 6.5

d = 13 cm

Circumference of the circle C = 2πr

C = 2 x (22/7) x 6.5

C = 286/7 = 40.85 cm

Area of the circle A = πr²

A = (22/7) x (6.5)²

= (22/7) x 42.25

= 132.78 cm²

∴ Area of circle is 314.285 cm², circumference is 40.85 cm, and diameter is 13 cm.

(c) 8 cm

Given radius r = 8 cm

The diameter of the circle d = 2r

d = 2 x 8

d = 16 cm

Circumference of the circle C = 2πr

C = 2 x (22/7) x 8

= 352/7 = 50.28 cm

Area of the circle A = πr²

A = (22/7) x (8)²

= (22/7) x 64

= 1408/7

= 201.14 cm²

∴ Area of circle is 201.14 cm², the circumference is 50.28 cm, and diameter is 16 cm.

2. Find the area, circumference of the circle whose diameter is

(a) 12 cm

(b) 18 cm

(c) 15 cm

## Solution:

(a) 12 cm

Given diameter d = 12 cm

The Radius of the circle r = d/2

r = 12/2

r = 6 cm

Circumference of the circle C = 2πr

C = 2 x 3.14 x 6

C = 37.68 cm

Area of the circle A = πr²

A = 3.14 x (6)²

A = 3.14 x 36

A = 113.04 cm²

∴ Radius, circumference, and area of the circle is 6 cm, 37.68 cm, 113.04 cm².

(b) 18 cm

Given diameter d = 18 cm

The Radius of the circle r = d/2

r = 18/2

r = 9 cm

Circumference of the circle C = 2πr

C = 2 x 3.14 x 9

C = 56.52 cm

Area of the circle A = πr²

A = 3.14 x 9²

A = 3.14 x 81

A = 254.34 cm²

∴ Radius, circumference, and area of the circle is 9 cm, 56.52 cm, 254.34 cm².

(c) 15 cm

Given diameter d = 15 cm

The Radius of the circle r = d/2

r = 15/2

r = 7.5 cm

Circumference of the circle C = 2πr

C = 2 x 3.14 x 7.5

C = 47.1 cm

Area of the circle A = πr²

A = 3.14 x (7.5)²

= 3.14 x 56.25

= 176.62

∴ The radius, circumference, and area of the circle is 7.5 cm, 47.1 cm, 176.62 cm².

3. Calculate the circumference of the circle whose area is 212 cm².

## Solution:

Given that,

Area of the circle A = 212 cm²

πr² = 212

r² = 212/π

r² = (212/22) x 7

r² = 9.636 x 7

r = √(67.45)

r = 8.21 cm

Circumference of the Circle C = 2πr

C = 2 x (22/7) x 8.21

C = 361.24/7

C = 51.6 cm

∴ Circle circumference is 51.6 cm.

4. If the circumference of a circular sheet is 47 cm, find its area.

## Solution:

Given that,

Circle circumference C = 47 cm

2πr = 47

r = 47/2π

r = 47/(2 x 3.14)

r = 47/6.28

= 7.48 cm

Area of the circle A = πr²

A = 3.14 x (7.48)²

A = 3.14 x 56 = 175.8 cm²

∴ The circle area is 175.8 cm².

5. A circular swimming pool is 25 feet in diameter. How many feet around is the pool?

## Solution:

Given that,

The diameter of the circular swimming pool = 25 feet

Radius = 25/2 = 12.5 feet

Circumference of the swimming pool = 2πr

= 2 x 3.14 x 12.5 = 78.5 feet

∴ 78.5 feet is around the circular swimming pool.

6. The diameter of a bangle is 26 cm. How many times the bangle will revolve in order to travel a distance of 500 cm.

## Solution:

Given that,

The diameter of the bangle d = 26 cm

Distance covered in 1 rotation = Circumference of the bangle

So, circumference C = πd

C = 3.14 x 26

C = 81.64 cm

Number of times bangle should revolve = 500/81.64

= 6.12

∴ 6 times the bangle should revolve to cover a distance of 500 cm.

7. From a circular sheet of a radius 7 cm, a circle of radius 3 cm is removed. Find the area of the remaining sheet.

## Solution:

Given that,

The radius of the outer circle R = 7 cm

Radius of inner circle r = 3 cm

Area of the Outer Circle A = πr²

A = 3.14 x 7²

= 3.14 x 49

= 153.86 cm²

Area of the inner circle a = πr²

a = 3.14 x 3²

= 3.14 x 9

= 28.26 cm²

Area of the remaining sheet = Area of the outer circle – Area of the inner circle

= 153.86 – 28.26 = 125.6 cm².

8. If the circumference of a circle is 86 cm, find its area, diameter, and radius.

## Solution:

Given that,

Circumference of the circle C = 86 cm

2πr = 86

r = 86/2π

r = 86/(2 x 3.14) = 86/6.28

r = 13.69 cm

The diameter of the circle d = 2r

d = 13.69 x 2 = 27.38 cm

Area of the circle A = πr²

A = 3.14 x (13.69)² = 3.14 x 13.69 x 13.69

A = 588.48 cm²

∴ Circle radius is 13.69 cm, the area is 588.48 cm², and diameter is 27.38 cm.

9. Find the perimeter of the semicircle whose diameter is 32 m.

## Solution:

Given that,

Semicircle diameter = 32 cm

Semicircle radius = diameter/2

= 32/2 = 16 cm

The perimeter of the semicircle = πr

= 3.14 x 16

= 50.24 cm

∴ The perimeter of the semicircle is 50.24 cm.

10. The ratio of the radii of two wheels is 10: 13. Find the ratio of their circumference and areas.

## Solution:

Given that,

The ratio of radii of two wheels is 10: 13.

The ratio of wheels circumferences = 2πr : 2πR

= r : R = 10 : 13

The ratio of wheels areas = πr² : πR²

= r² : R²

= 10² : 13²

= 100 : 169

∴ The ratio of wheels circumferences is 10: 13 and areas is 100 : 169.

11. The radius of a cycle wheel is 25 cm. Find the number of turns required to cover a distance of 1585 m.

## Solution:

Given that,

The radius of the cycle r = 25 cm

Distance covered in 1 turn = Circumference of the cycle wheel

So, circumference C = 2πr

C = 2 x 3.14 x 25

C = 157 cm

Number of turns cycle wheel should revolve = (1585 x 100)/157

= 1,009.55

∴ 1,009.55 number of turns required to cover a distance of 1585 m.

12. A girl wants to make a square-shaped figure from a circular wire of a radius 49 cm. Determine the sides of a square.

## Solution:

Given that,

radius r = 49 cm

Length of the wire = Circumference of the circle = 2πr

= 2 x (22/7) x 49 = 2156/7

= 308 cm

Let the side of the square be ‘s’.

The perimeter of the square = Length of the wire = 4s

s = 308/4

s = 77 cm

Therefore, the sides of the square is 77 cm.

13. To cover a distance of 5 km a wheel rotates 2500 times. Find the radius of the wheel?

## Solution:

Given that,

Number of rotations = 2500

The total distance covered = 5 km

Circumference of the wheel = Distance covered in 1 rotation = 2πr

In 2500 rotations, The distance covered = 5 km = 500000 cm

Hence, in 1 rotation, the distance covered = 500000/2500

= 250 cm

But this is equal to the circumference. Hence, 2πr = 250 cm

r = 250/2π

= 250/(2 x 3.14)

= 250/6.28 = 39.8 cm

Therefore, the radius of the wheel is 39.8 cm.

14. A well of diameter 160 cm has a stone parapet around it. If the length of the outer edge of the parapet is 516 cm, find the width of the parapet.

## Solution:

Given that,

The diameter of the well (d) = 160 cm

So, the radius of the well r = d/2 = 160/2 = 80 cm

The length of the outer edge of the parapet is 516 cm

2πR = 516 cm

R = 516/2π

R = 516/(2 x 3.14)

R = 82.16 cm

Now, the width of the parapet = (Radius of the parapet – Radius of the well)

= 82.16 – 80 = 2.16

Therefore, the width of the parapet is 2.16 cm.

15. Find the area of a circle whose circumference is the same as the perimeter of the square of side 15 cm.

## Solution:

Given that,

Square side s = 15 cm

The perimeter of the square P = s²

P = 15² = 15 x 15 = 225

Circumference of the circle = Perimeter of the square

2πr = 225

r = 225/(2π) = 225/(2 x 3.15)

r = 225/6.28 = 35.8 cm

Area of the circle A = πr²

A = 3.14 x 35.8²

A = 3.14 x 35.8 x 35.8 = 4,030.65 cm²

Therefore, the area of the circle is 4,030.65 cm².

16. From a rectangular metal sheet of size 60 by 40, a circular sheet as big as possible is cut. Find the area of the remaining sheet.

## Solution:

Given that,

The size of the rectangular metal sheet is 60 by 40.

The diameter of the largest circle = Length of the smallest side of the rectangle

Radius of the circle r = 40/2 = 20

Area of rectangle = l x b = 60 x 40 = 2400 cm²

Area of the circle = πr²

= 3.14 x (20)² = 3.14 x 400 = 1256 cm²

Remaining area = 2400 – 1256

= 1144 cm²

∴ The area of the remaining sheet is 1144 cm².

17. Two circles have areas in the ratio of 16 : 20. Find the ratio of their circumferences.

## Solution:

Given that,

The ratio of areas of two circles = 16 : 20

The ratio of circumferences = √(Ratio of area)

= √(16 : 20)

= √(4 : 5)

= 2 : 2.23

Therefore, the ratio of circles circumferences is 2: 2.23.

18. A square metallic frame has a perimeter of 289 cm. It is bent in the shape of a circle. Find the area of the circle, the side length of the square.

## Solution:

Given that,

The perimeter of the square = 289 cm

side² = 289

side = √(289)

side = 17

The perimeter of the square = Circumference of a circle

289 = 2πr

r = 289/2π

r = 289/(2 x 3.14)

= 289/6.28 = 46 cm

Area of the circle = πr²

= 3.14 x (46)²

= 6,644.24 cm²

Therefore, the circle area is 6,644.24 cm², side length of the square is 17 cm.

19. From a circular sheet of radius 18 cm, two circles of radii 4.5 cm and a rectangle of length 4 cm and breadth 2 cm are removed; find the area of the remaining sheet.

## Solution:

Given that,

The radius of the circle r = 18 cm

Radii of two small circles = 4.5 cm

Length of the rectangle = 4 cm

The breadth of the rectangle = 2 cm

Area of the circle = πr²

= 3.14 x 18²

= 1,017.36 cm²

Area of the smallest circle = 3.14 x 4.5²

= 63.585

As there are 2 small circles so the total area of the circles is 63.64 x 2 = 127.28 square cm.

Area of rectangle = l x w = 4 x 2 = 8 cm²

Total area of cutouts = 127.28 + 8 = 135.28 cm²

Area of sheet left = 1,017.36 – 135.28 = 882.08 cm²

Therefore, area of the remaining sheet is 882.08 cm².