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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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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 = ˆš(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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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
ˆ´ 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 = ˆš(Ratio of area)
= ˆš(16 : 20)
= ˆš(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 = ˆš(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.