1. Find the circumference of the circle with the following radius: (Take π = 22/7)
(a) 14 cm
Solution:
Given,
Radius of circle = 14 cm
Now,
Circumference of the circle = 2πr
= 2 × (22/7) × 14
= 2 × 22 × 2
= 88 cm
(b) 28 cm
Solution:
Given,
Radius of circle = 28 cm
Now,
Circumference of the circle = 2πr
= 2 × (22/7) × 28
= 2 × 22 × 4
= 176 cm
(c) 21 cm
Solution:
Given,
Radius of circle = 21 cm
Now,
Circumference of the circle = 2πr
= 2 × (22/7) × 21
= 2 × 22 × 3
= 132 cm
2. Find the area of the following circles, given that:
(a) Radius = 14 mm (Take π = 22/7)
Solution:
Given,
Radius of circle = 14 mm
Now,
Area of the circle = πr2
= 22/7 × 142
= 22/7 × 196
= 22 × 28
= 616 mm2
(b) Diameter = 49 m
Solution:
Given,
Diameter of circle (d) = 49 m
We know that, radius (r) is half of diameter = d/2
= 49/2
= 24.5 m
Now,
Area of the circle = πr2
= 22/7 × (24.5) 2
= 22/7 × 600.25
= 22 × 85.75
= 1886.5 m2
(c) Radius = 5 cm
Solution:
Given,
Radius of circle = 5 cm
Now,
Area of the circle = πr2
= 22/7 × 52
= 22/7 × 25
= 550/7
= 78.57 cm2
3. If the circumference of a circular sheet is 154 m, find its radius. Also find the area of the sheet. (Take π = 22/7)
Solution:
Given,
Circumference of the circle = 154 m
Now,
Circumference of the circle = 2πr
So,
154 = 2 × (22/7) × r
154 = 44/7 × r
r = (154 × 7)/44
r = (14 × 7)/4
r = (7 × 7)/2
r = 49/2
r = 24.5 m
Now,
Area of the circle = πr2
22/7 × (24.5)2
= 22/7 × 600.25
= 22 × 85.75
= 1886.5 m2
Hence, the radius of the circle is 24.5 m, and the area of the circle is 1886.5 m2.
4. A gardener wants to fence a circular garden of diameter 21m. Find the length of the rope he needs to purchase, if he makes 2 rounds of fence. Also find the cost of the rope, if it costs ₹ 4 per meter. (Take π = 22/7).
Solution:
Given,
Diameter (d) of the circular garden = 21 m
We know that, radius (r) = d/2
= 21/2
= 10.5 m
Now,
Circumference of the circle = 2πr
= 2 × (22/7) × 10.5
= 462/7
= 66 m
The length of rope required = 2 × 66 = 132 m
Now,
Cost of 1 m rope = ₹ 4 (given)
Cost of 132 m rope = ₹ 4 × 132
= ₹ 528
5. From a circular sheet of radius 4 cm, a circle of radius 3 cm is removed. Find the area of the remaining sheet. (Take π = 3.14)
Solution:
Given,
Total radius of circular sheet (R) = 4 cm
A circle of radius to be removed (r) = 3 cm
Now,
The area of the remaining sheet = πR2 – πr2
= π (R2 – r2)
= 3.14 (42 – 32)
= 3.14 (16 – 9)
= 3.14 × 7
= 21.98 cm2
Hence, the area of the remaining sheet is 21.98 cm2.
6. Saima wants to put a lace on the edge of a circular table cover of diameter 1.5 m. Find the length of the lace required and also find its cost if one meter of the lace costs ₹ 15. (Take π = 3.14)
Solution:
Given,
Diameter of the circular table cover = 1.5 m
So, radius (r) = d/2
= 1.5/2
= 0.75 m
Now,
Circumference of the circular table cover = 2πr
= 2 × 3.14 × 0.75
= 4.71 m
Hence, the length of the lace = 4.71 m
Now,
Cost of 1 m lace = ₹ 15 [given]
Cost of 4.71 m lace = ₹ 15 × 4.71
= ₹ 70.65
7. Find the perimeter of the adjoining figure, which is a semicircle including its diameter.
Solution:
Given,
Diameter of semi-circle = 10 cm
So, radius (r) = d/2
= 10/2
= 5 cm
Now,
Circumference of the semi-circle = πr
= (22/7) × 5
= 110/7
= 15.71 cm
Hence,
Perimeter of the given figure = Circumference of the semi-circle + semi-circle diameter
= 15.71 + 10
= 25.71 cm
8. Find the cost of polishing a circular table-top of diameter 1.6 m, if the rate of polishing is ₹15/m2. (Take π = 3.14)
Solution:
Given,
Diameter of the circular table-top = 1.6 m
So, radius (r) = d/2
= 1.6/2
= 0.8 m
Now,
Area of the circular table-top = πr2
= 3.14 × (0.8) 2
= 2.0096 m2
Cost of polishing 1 m2 area = ₹ 15 [given]
Cost of polishing 2.0096 m2 area = ₹ 15 × 2.0096
= ₹ 30.144
Hence, the cost of polishing 2.0096 m2 area is ₹ 30.144.
9. Shazli took a wire of length 44 cm and bent it into the shape of a circle. Find the radius of that circle. Also find its area. If the same wire is bent into the shape of a square, what will be the length of each of its sides? Which figure encloses more area, the circle or the square? (Take π = 22/7)
Solution:
Given,
Length of wire =44 cm
If the wire is bent into a circle,
So, the circumference of the circle = 2πr
44 = 2 × (22/7) × r
44 = 44/7 × r
(44 × 7)/44 = r
r = 7 cm
Then,
Area of the circle = πr2
= 22/7 × 72
= 22 × 7
= 154 cm2
Now,
If the wire is bent into a square,
Length of wire = perimeter of the square
So, the perimeter of square = 4 x side
44 = 4 x side
44 = 4S
S = 44/4
S = 11cm
Area of square = (side)2 = 112
= 121 cm2
Hence, the circle has more area than square
10. From a circular card sheet of radius 14 cm, two circles of radius 3.5 cm and a rectangle of length 3 cm and breadth 1cm are removed. (as shown in the adjoining figure). Find the area of the remaining sheet. (Take π = 22/7).
Solution:
Given,
The radius of the circular card sheet = 14 cm
The radius of the two small circles = 3.5 cm
Length of the rectangle = 3 cm
Breadth of the rectangle = 1 cm
First, we have to find out the area of the circular card sheet,
Area of the circular card sheet = πr2
= 22/7 × 142
= 22 × 2 × 14
= 616 cm2
Now, we find the area of 2 small circles,
Area of the 2 small circles = 2 × πr2
= 2 × (22/7 × 3.52)
= 2 × ((22/7) × 12.25)
= 2 × 38.5
= 77 cm2
Now, we find the area of the rectangle,
Area of the rectangle = Length × Breadth
= 3 × 1
= 3 cm2
Now,
Area of the remaining sheet = Area of circular card sheet – (Area of two small circles + Area of the rectangle)
= 616 – (77 + 3)
= 616 – 80
= 536 cm2
Hence, the area of the remaining sheet is 536 cm2
11. A circle of radius 2 cm is cut out from a square piece of an aluminium sheet of side 6 cm. What is the area of the left over aluminium sheet? (Take π = 3.14)
Solution:
Given,
Radius of circle = 2 cm
Side of square sheet = 6 cm
First, we have to find out the area of the square aluminium sheet,
Area of the square = side2
So, the area of the square aluminium sheet = 62 = 36 cm2
Now,
Area of the circle = πr2
= 3.14 × 22
= 3.14 × 4
= 12.56 cm2
Now,
Area of the aluminium sheet left = Area of the square aluminium sheet – Area of the circle
= 36 – 12.56
= 23.44 cm2
Hence, the area of the aluminium sheet left is 23.44 cm2
12. The circumference of a circle is 31.4 cm. Find the radius and the area of the circle? (Take π = 3.14)
Solution:
Given,
Circumference of a circle = 31.4 cm
So, the circumference of a circle = 2πr
31.4 = 2 × 3.14 × r
31.4 = 6.28 × r
31.4/6.28 = r
r = 5 cm
Then,
Area of the circle = πr2
= 3.14 × 5 2
= 3. 14 × 25
= 78.5 cm2
Hence, radius of the circle is 5 cm and area of the circle is 78.5 cm2.
13. A circular flower bed is surrounded by a path 4 m wide. The diameter of the flower bed is 66 m. What is the area of this path? (π = 3.14).
Solution:
Given,
Diameter of the flower bed = 66 m
Radius of the flower bed = d/2
= 66/2
= 33 m
Now,
Area of flower bed = πr2
= 3.14 × 332
= 3.14 × 1089
= 3419.46 m
We have to find area of the flower bed and path together
So, radius of flower bed and path together = 33 + 4 = 37 m
Area of the flower bed and path together = πr2
= 3.14 × 372
= 3.14 × 1369
= 4298.66 m2
Now,
Area of the path = Area of the flower bed and path together – Area of flower bed
= 4298.66 – 3419.46
= 879.2 m2
Hence, the area of the path is 879.2 m2
14. A circular flower garden has an area of 314 m2. A sprinkler at the centre of the garden can cover an area that has a radius of 12 m. Will the sprinkler water the entire garden? (Take π = 3.14)
Solution:
Given,
Area of the circular flower garden = 314 m2
Sprinkler at the centre of the garden can cover an area that has a radius = 12 m
Area of the circular flower garden = πr2
314 = 3.14 × r2
314/3.14 = r2
r2 = 100
r = √100
r = 10 m
Hence, the radius of the circular flower garden is 10 m.
Since the sprinkler can cover an area of a radius of 12 m
Yes, the sprinkler will water the whole garden.
15. Find the circumference of the inner and the outer circles, shown in the adjoining figure? (Take π = 3.14).
Solution:
According to the figure,
The radius of inner circle = outer circle radius – 10
= 19 – 10
= 9 m
Circumference of the inner circle = 2πr
= 2 × 3.14 × 9
= 56.52 m
Now,
The radius of outer circle = 19 m
Circumference of the outer circle = 2πr
= 2 × 3.14 × 19
= 119.32 m
Hence, the circumference of the inner circle is 56.52 m and the circumference of the outer circle is 119.32 m
16. How many times a wheel of radius 28 cm must rotate to go 352 m? (Take π = 22/7).
Solution:
Given,
Radius of the wheel = 28 cm
Total distance = 352 m = 35200 cm
Now,
Circumference of the wheel = 2πr
= 2 × 22/7 × 28
= 2 × 22 × 4
= 176 cm
We have to find the number of rotations of the wheel,
Total number of times the wheel should rotate = Total distance covered by wheel / Circumference of the wheel
= 352 m/176 cm
= 35200 cm/ 176 cm
= 200
Hence, the wheel rotates 200 times
17. The minute hand of a circular clock is 15 cm long. How far does the tip of the minute hand move in 1 hour. (Take π = 3.14)
Solution:
Given,
Length of the minute hand of the circular clock = 15 cm
So, distance travelled by the tip of a minute hand in 1 hour = circumference of the clock
= 2πr
= 2 × 3.14 × 15
= 94.2 cm
Hence, the minute hand moves 94.2 cm in 1 hour
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