Perimeter and Area

Exercise 11.3

1). Find the circumference of the circles with the following radius: (Take p =^{22}/_{7})

(a) 14 cm (b) 28 mm (c) 21 cm

Solution:

(a) 14 cm

Given: radius = 14 cm

Circumference = 2pr

= 2 X ^{22}/_{7} X 14

= 2 X 22 X 2

= 88 cm

Ans: circumference = 88 cm

(b) 28 mm

Given: radius = 28 mm

Circumference = 2pr

= 2 X ^{22}/_{7} X 28

= 2 X 22 X 4

= 196 mm

Ans: circumference = 196 mm

(c) 21 cm

Given: radius = 21 cm

Circumference = 2pr

= 2 X ^{22}/_{7} X 21

= 2 X 22 X 3

= 132 cm

Ans: circumference = 132 cm

2). Find the area of the following circles, given that:

(a) radius = 14 mm (Take p =^{22}/_{7})

(b) diameter = 49 m

(c) radius = 5 cm

Solution:

(a) radius = 14 mm (Take p =^{22}/_{7})

Given: radius = 14 mm

Area of circle = pr^{2}

= ^{22}/_{7 }X (14)^{2}

= ^{22}/_{7} X 14 X 14

= 616 sq mm

(b) diameter = 49 m

radius = diameter/_{2}

= ^{49}/_{2}

Area of circle = pr^{2}

= 22/7 X (^{49}/_{2})^{2}

= (22 X 49 X 49)/(7 X 2 X 2)

= (11 X 7 X 49)/2

= 1886.50 sq m

(c) radius = 5 cm

Area of circle = pr^{2}

= ^{22}/_{7 }X (5)2

= ^{22}/_{7} X 5 X 5

= 78.57 sq cm

3). If the circumference of a circular sheet is 154 m, find its radius. Also find the area of the sheet. (Take p =^{22}/_{7})

Given: Circumference of a sheet = 154 m

To Find: radius of the sheet = ?

Area of the sheet = ?

Circumference = 2pr

54 = 2 X ^{22}/_{7} X r

2 X ^{22}/_{7} X r = 154

r = ^{(154 X 7)}/ _{(22 X 2)}

r = 24.5 m

area of the sheet = pr^{2}

= ^{22}/_{7} X 24.5 X 24.5

= 22 X 3.5 X 24.5

= 1886.5 sq m

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 p = ^{22}/_{7})

Given: diameter of circular garden = 21 m

rounds of fence = 2

rate of fencing = ₹ 4 per meter

To Find: length of the rope to fence = ?

1 round of fence = circumference of garden

= pd

= ^{22}/_{7} X 21

= 22 X 3

= 66 m

Length of the rope = circumference X number of rounds

= 66 X 2

= 132 m

Cost of fencing = Rate X Length of the rope

= 4 X 132

=₹ 528

Ans: length of the rope = 132 m

Cost of fencing = ₹ 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 p = 3.14)

Given: Radius of larger circle = R = 4 cm

Radius of smaller circle = r = 3 cm

To Find: Area of remaining sheet = ?

Area of remaining sheet = pR^{2 }– pr^{2}

= p(R^{2 }– r^{2})

= ^{22}/_{7} (4^{2} – 3^{2})

=^{22}/_{7} (16-9)

= ^{22}/_{7} X 7

= 22 sq m

Ans: Area of the remaining sheet = 22 sq m.

^{ }

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 cost₹ s 15. (Take p = 3.14)

Given: diameter of circular table = 1.5 m

Rate of the lace = ₹ 15

To Find: length of the lace

Length of the lace = circumference of table cover

= pd

= 3.14 X 1.5

= 4.71 m

Cost of the lace = Rate X circumference

= 15 X 4.71

= ₹ 70.65

Ans: cost of the lace = ₹ 70.65

7). Find the perimeter of the adjoining figure, which is a semicircle including its diameter.

Given: diameter of semicircle = 10 cm

To Find: perimeter of semi circle = ?

Radius = diameter / 2

= 10/2 = 5 cm

Perimeter of semi circle = ½ X circumference + diameter

= ½ X 2pr + d

= pr + d.

= ^{22}/_{7} X 5 + 10

= 3.14 X 5 + 10

= 15.7 + 10

= 25.7 cm

Ans: perimeter of semicircle = 25.7 cm

8). Find the cost of polishing a circular table-top of diameter 1.6 m, if the rate of polishing is ₹ 15/m^{2}. (Take p = 3.14)

Given: diameter of table-top = 1.6 m

rate of polishing = ₹ 15/m^{2}

To Find: cost of polishing = ?

Radius = diameter/2

= 1.6/2 = 0.8 m

Area of table-top = pr^{2}

= 3.14 X 0.8 X 0.8

= 2.0096

= 2.01

Cost of polishing = Rate X Area

= 15 X 2.01

= ₹ 30.15

Ans: Cost of polishing = ₹ 30.15

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 p =^{22}/_{7})

Given: length of the wire = 44 cm

To find: radius of the circle = ?

area of the circle = ?

side of the square = ?

area of the square = ?

circumference of the circle = length of the wire

2pr = 44

2 X ^{22}/_{7} X r = 44

r = (44 X 7)/(2 X 22)

r = 7 cm

area of the circle = pr^{2}

= ^{22}/_{7} X 7 X 7

= 22 X 7

= 154 Sq cm

Same wire is bent to shape of square

Perimeter of square = length of the wire

4 X side = 44

side = 44/4

side = 11 cm

area of square = side X side

= 11 X 11

= 121 sq cm

121 < 154

circle encloses more area

Ans: radius of the circle = 7 cm

area of the circle = 154 sq cm

side of the square = 11 cm

area of the square = 121 sq cm

circle encloses more area.

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 p =^{22}/_{7})

Given: radius of card sheet = R = 14 cm

radius of smaller circle = r = 3.5 cm

length of the rectangle = 3 cm

breadth of the rectangle = 1 cm

To Find: Area of the remaining sheet

Area of the remaining sheet = Area of circular card – (area of

two circles + area of rectangle)

Area of circular card = pr^{2}

= ^{22}/_{7} X 14 X 14

= 22 X 2 X 14

= 616 sq cm

Area of small circle = pr^{2}

= ^{22}/_{7} X 3.5 X 3.5

= 22 X 0.5 X 3.5

= 11 X 3.5

= 38.5 sq cm

Area of two small circles = 2 X pr^{2}

^{ }= 2 X 38.5

= 77 sq cm

Area of rectangle = length X breadth

= 3 X 1

= 3 sq cm

Area of the remaining sheet = Area of circular card – (area of

two circles + area of rectangle)

= 616 – ( 77 + 3)

= 616 – 80

= 536 sq cm

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 p = 3.14)

Given: radius of circle = r = 2 cm

side of square = s = 6 cm

To Find: area of left over aluminium sheet = ?

Area of circle = pr^{2}

= 3.14 X 2 X 2

= 12.56 sq cm

Area of square = side X side

= 6 X 6

= 36 sq cm

Area of the left over sheet = area of square – area of circle

= 36 – 12.56

= 23.44 sq cm

Ans: Area of the left over sheet = 23.44 sq cm

12). The circumference of a circle is 31.4 cm. Find the radius and the area of the circle? (Take p = 3.14)

Given: circumference of circle = 31.4 cm

To Find: radius = ?

Area = ?

circumference = 31.4

2pr = 31.4

2 X 3.14 X r = 31.4

r = 31.4 X ½ X ^{1}/_{3.14}

r = ^{(31.4 X 1})/_{(3.14 X 2)}

r = 5 cm

r = 5 cm

Area of circle = pr^{2}

= 3.14 X 5 X 5

= 78.5 sq cm

Ans: radius = 5 cm

Area of circle = 78.5 sq cm

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?

(p = 3.14)

Given: diameter of the flower bed = 66 m

width of the path = 4 m

To Find: area of the path = ?

Diameter of the flower bed = 66

radius of the flower bed r = Diameter/2

= 66/2

= 33 m

radius of the outer circle formed = R = 33 + 4

= 37 m

Area of remaining sheet = pR^{2 }– pr^{2}

= p(R^{2 }– r^{2})

= 3.14 (37^{2} – 33^{2})

= 3.14 (1369 – 1089)

= 3.14 X 280

= 879.20 sq m

14). A circular flower garden has an area of 314 m^{2}. 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 p = 3.14)

Given: Area of flower garden = 314 m^{2}

Radius of area that sprinkle can cover = 12 m

Area of flower garden = 314

pr^{2 }= 314

3.14 X r^{2} = 314

r^{2} = 314 / 3.14

r^{2} = 100

taking square root on both the sides

r = 10 m

since 10 < 12

sprinkler can water the entire garden

15). Find the circumference of the inner and the outer circles, shown in the adjoining figure? (Take p = 3.14)

Given: outer radius = R = 19 m

width of the road = 10 m

To Find: circumference of inner and the outer circles

radius of inner circle = outer radius – width of the road

= 19 – 10

r = 9 m

circumference of outer circle = 2pR

= 2 X 3.14 X 19

= 119.32 m

circumference of inner circle = 2pr

= 2 X 3.14 X 9

= 56.52 m

Ans: circumference of outer circle = 119.32 m

circumference of inner circle = 56.52 m

16). How many times a wheel of radius 28 cm must rotate to go 352 m? (Take p = ^{22}/_{7})

Given: radius of wheel = 28 cm

distance = 352 m = 352 X 100 = 35200 cm

To Find: times the wheel rotate to cover 35200 cm

one rotation of the wheel = circumference of circle

= 2pr

= 2 X ^{22}/_{7} X 28

= 2 X 22 X 4

= 176 cm

No of times wheel rotates = distatnce / circumference

= 35200 / 176

= 200

Ans: number of rotation of wheel = 200

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 p = 3.14)

Given: length of the minute hand = radius = 15 cm

Time = 1 hour

To Find = distance covered in 1 hour

Distance covered by minute hand = circumference

= 2pr

= 2 X 3.14 X 15

= 94.2 cm

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