Chapter 16 - Surface Area and Volume
Practice set 16.1
1. Find the volume of a box if its length, breadth and height are 20 cm, 10.5 cm and 8 cm respectively.
Given:
Length of the box = 20 cm
Breadth of the box = 10.5 cm
Height of the box = 8 cm
To find:
Volume of the box
Solution:
Volume of the box = l × b × h
∴ Volume of the box = 20 × 10.5 × 8
∴ Volume of the box = 1680 cc
Ans: Volume of the box is 1680 cc
2. A cuboid shape soap bar has volume 150 cc. Find its thickness if its length is 10 cm and breadth is 5 cm.
Given:
Volume of the soap bar = 150 cc
Length of the box = 10 cm
Breadth of the box = 5 cm
To find:
Thickness or height of the soap bar
Solution:
Volume of the box = l × b × h
∴ 150 = 10 × 5 × h
∴ \(\large \frac {150}{10\, ×\, 5}\) = h
∴ h = 3 cm
Ans: Thickness of the soap bar is 3 cm
3. How many bricks of length 25 cm, breadth 15 cm and height 10 cm are required to build a wall of length 6 m, height 2.5 m and breadth 0.5 m?
Given:
Length of the brick = 25 cm
Breadth of the brick = 15 cm
Height of the brick = 10 cm
Length of the wall = 6 m = 6 × 100 cm = 600 cm
Breadth of the wall = 0.5 m = 0.5 × 100 cm = 50 cm
Height of the wall = 2.5 m = 2.5 × 100 cm = 250 cm
To find:
Number of bricks required
Solution:
Volume of the brick = l × b × h
∴ Volume of the brick = 25 × 15 × 10
∴ Volume of the brick = 3750 cc
Volume of the wall = l × b × h
∴ Volume of the wall = 600 × 50 × 250
∴ Volume of the wall = 75,00,000 cc
Number of bricks required = \(\large \frac {Volume \,of \, the \, wall}{Volume \,of \, the \, brick}\)
∴ Number of bricks required = \(\large \frac {75,00,000}{3750}\)
∴ Number of bricks required = 2000
Ans: 2000 bricks are required to build the wall.
4. For rain water harvesting a tank of length 10 m, breadth 6 m and depth 3m is built. What is the capacity of the tank ? How many litres of water can it hold?
Given:
Length of the tank = 10 m = 10 × 100 cm = 1000 cm
Breadth of the tank = 6 m = 6 × 100 cm = 600 cm
Height of the tank = 3 m = 3 × 100 cm = 300 cm
To find:
Capacity of the tank and litres of water the tank can hold
Solution:
Capacity of tank = Volume of the tank
Volume of the tank = l × b × h
∴ Volume of the tank = 1000 × 600 × 300
∴ Volume of the tank = 18,00,00,000 cc
∴ Capacity of tank = 18,00,00,000 cc
And,
Litres of water the tank can hold = \(\large \frac {18,00,00,000\, cc}{1000}\) …[∵ 1000cc = 1 litre]
∴ Litres of water the tank can hold = 1,80,000 litres
Ans: The capacity of the tank is 18,00,00,000 cc and it can hold 1,80,000 litres of water.
Practice set 17.2
1. In each example given below, the radius of base of a cylinder and its height are given. Then find the curved surface area and total surface area.
(1) r = 7 cm, h = 10 cm
Given:
Radius of the cylinder = 7 cm
Height of the cylinder = 10 cm
To find:
Curved surface area and Total surface area of the cylinder
Solution:
Curved surface area of the cylinder = 2πrh
∴ Curved surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 7 × 10
∴ Curved surface area of the cylinder = 44 × 10
∴ Curved surface area of the cylinder = 440 sq.cm
Total surface area of the cylinder = 2πr (h + r)
∴ Total surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 7 (10 + 7)
∴ Total surface area of the cylinder = 44 × 17
∴ Total surface area of the cylinder = 748 sq.cm
Ans: Curved surface area of the cylinder is 440 sq.cm and total surface area of the cylinder is 748 sq.cm.
(2) r = 1.4 cm, h = 2.1 cm
Given:
Radius of the cylinder = 1.4 cm
Height of the cylinder = 2.1 cm
To find:
Curved surface area and Total surface area of the cylinder
Solution:
Curved surface area of the cylinder = 2πrh
∴ Curved surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 1.4 × 2.1
∴ Curved surface area of the cylinder = 44 × 0.2 × 2.1
∴ Curved surface area of the cylinder = 18.48 sq.cm
Total surface area of the cylinder = 2πr (h + r)
∴ Total surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 1.4 (2.1 + 1.4)
∴ Total surface area of the cylinder = 44 × 0.2 × 3.5
∴ Total surface area of the cylinder = 44 × 0.7
∴ Total surface area of the cylinder = 30.8 sq.cm
Ans: Curved surface area of the cylinder is 18.48 sq.cm and total surface area of the cylinder is 30.8 sq.cm.
(3) r = 2.5 cm, h = 7 cm
Given:
Radius of the cylinder = 2.5 cm
Height of the cylinder = 7 cm
To find:
Curved surface area and Total surface area of the cylinder
Solution:
Curved surface area of the cylinder = 2πrh
∴ Curved surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 2.5 × 7
∴ Curved surface area of the cylinder = 44 × 2.5
∴ Curved surface area of the cylinder = 110 sq.cm
Total surface area of the cylinder = 2πr (h + r)
∴ Total surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 2.5 (7 + 2.5)
∴ Total surface area of the cylinder = \(\large \frac {44}{7}\) × 2.5 × 9.5
∴ Total surface area of the cylinder = \(\large \frac {44\, ×\,23.75}{7}\)
∴ Total surface area of the cylinder = \(\large \frac {1045}{7}\)
∴ Total surface area of the cylinder = 149.28 sq.cm
Ans: Curved surface area of the cylinder is 110 sq.cm and total surface area of the cylinder is 149.28 sq.cm.
(4) r = 70 cm, h = 1.4 cm
Given:
Radius of the cylinder = 70 cm
Height of the cylinder = 1.4 cm
To find:
Curved surface area and Total surface area of the cylinder
Solution:
Curved surface area of the cylinder = 2πrh
∴ Curved surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 70 × 1.4
∴ Curved surface area of the cylinder = 44 × 10 × 1.4
∴ Curved surface area of the cylinder = 44 × 14
∴ Curved surface area of the cylinder = 616 sq.cm
Total surface area of the cylinder = 2πr(h + r)
∴ Total surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 70 (1.4 + 70)
∴ Total surface area of the cylinder = 44 × 10 × 71.4
∴ Total surface area of the cylinder = 440 × 71.4
∴ Total surface area of the cylinder = 31,416 sq.cm
Ans: Curved surface area of the cylinder is 616 sq.cm and total surface area of the cylinder is 31,416 sq.cm.
(5) r = 4.2 cm, h = 14 cm
Given:
Radius of the cylinder = 4.2 cm
Height of the cylinder = 14 cm
To find:
Curved surface area and Total surface area of the cylinder
Solution:
Curved surface area of the cylinder = 2πrh
∴ Curved surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 4.2 × 14
∴ Curved surface area of the cylinder = 44 × 4.2 × 2
∴ Curved surface area of the cylinder = 369.6 sq.cm
Total surface area of the cylinder = 2πr(h + r)
∴ Total surface area of the cylinder = 2 × \(\large \frac {22}{7}\) × 4.2 (14 + 4.2)
∴ Total surface area of the cylinder = 44 × 0.6 × 18.2
∴ Total surface area of the cylinder = 480.48 sq.cm
Ans: Curved surface area of the cylinder is 369.6 sq.cm and total surface area of the cylinder is 480.48 sq.cm.
2. Find the total surface area of a closed cylindrical drum if its diameter is 50 cm and height is 45 cm. (π = 3.14)
Given:
Diameter of the cylindrical drum = 50 cm
Height of the cylindrical drum = 45 cm
π = 3.14
To find:
Total surface area of the closed cylindrical drum
Solution:
Radius of the cylindrical drum = \(\large \frac {Diameter\,of\,the\,cylindrical\,drum}{2}\)
∴ Radius of the cylindrical drum = \(\large \frac {50}{2}\)
∴ Radius of the cylindrical drum = 25 cm
Total surface area of the cylindrical drum = 2πr (h + r)
∴ Total surface area of the cylindrical drum = 2 × 3.14 × 25 (45 + 25)
∴ Total surface area of the cylindrical drum = 6.28 × 25 × 70
∴ Total surface area of the cylindrical drum = 6.28 × 1750
∴ Total surface area of the cylindrical drum = 10,990 sq.cm
Ans: Total surface area of the cylindrical drum is 10,990 sq.cm
3. Find the area of base and radius of a cylinder if its curved surface area is 660 sq.cm and height is 21 cm.
Given:
Curved surface area of the cylinder = 660 sq.cm
Height of the cylinder = 21 cm
To find:
Area of base of the cylinder
Radius of the cylinder
Solution:
Curved surface area of the cylinder = 2πrh
∴ 660 = 2 × \(\large \frac {22}{7}\) × r × 21
∴ 660 = 44 × r × 3
∴ \(\large \frac {660}{44\, ×\,3}\) = r
∴ r = 5 cm
Area of base of the cylinder is a circle
∴ Area of base of the cylinder = πr²
∴ Area of base of the cylinder = 3.14 × (5)²
∴ Area of base of the cylinder = 3.14 × 25
∴ Area of base of the cylinder = 78.75 sq.cm
Ans: Area of base of the cylinder is 78.75 sq.cm and radius of the cylinder is 5 cm.
4. Find the area of the sheet required to make a cylindrical container which is open at one side and whose diameter is 28 cm and height is 20 cm. Find the approximate area of the sheet required to make a lid of height 2 cm for this container.
Given:
Diameter of the cylindrical container = 28 cm
Height of the cylindrical container = 20 cm
Height of the lid for the container = 2 cm
To find:
Area of sheet required to make the cylindrical container
Area of sheet required to make a lid for the cylindrical container
Solution:
Diameter of the cylindrical container = 28 cm
∴ Radius of the cylindrical container = \(\large \frac {28}{2}\)
∴ Radius of the cylindrical container = 14 cm
Area of the sheet required to make the cylindrical container = Curved Surface area of the cylinder + Area of base of the cylinder
∴ Area of the sheet required to make the cylindrical container = 2πrh + πr²
∴ Area of the sheet required to make the cylindrical container = \((\)2 × \(\large \frac {22}{7}\) × 14 × 20\()\) + \((\large \frac {22}{7}\) × 14 × 14 \()\)
∴ Area of the sheet required to make the cylindrical container = (44 × 2 × 20) + (22 × 2 × 14)
∴ Area of the sheet required to make the cylindrical container = 1760 + 616
∴ Area of the sheet required to make the cylindrical container = 2376 sq.cm
Radius of the lid = Radius of the cylinder, as the lid has to fit the cylindrical container
Area of the sheet required to make a lid for the cylindrical container = Curved Surface area of the lid + Area of base of the lid
∴ Area of the sheet required to make a lid for the cylindrical container = 2πrh + πr²
∴ Area of the sheet required to make a lid for the cylindrical container = \((\)2 × \(\large \frac {22}{7}\) × 14 × 2\()\) + \((\large \frac {22}{7}\) × 14 × 14 \()\)
∴ Area of the sheet required to make a lid for the cylindrical container = (44 × 2 × 2) + (22 × 2 × 14)
∴ Area of the sheet required to make a lid for the cylindrical container = 176 + 616
∴ Area of the sheet required to make a lid for the cylindrical container = 792 sq.cm
Ans: Area of the sheet required to make the cylindrical container is 2376 sq.cm and area of the sheet required to make a lid for the cylindrical container is 792 sq.cm
Practice set 17.3
1. Find the volume of the cylinder if height (h) and radius of the base (r) are as given below.
(1) r = 10.5 cm, h = 8 cm
Given:
r = 10.5 cm
h = 8 cm
To find:
Volume of the cylinder
Solution:
Volume of the cylinder = πr²h
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × (10.5)² × 8
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × 10.5 × 10.5 × 8
∴ Volume of the cylinder = 22 × 1.5 × 84
∴ Volume of the cylinder = 2772 cu.cm
Ans: Volume of the cylinder is 2772 cu.cm
(2) r = 2.5 m, h = 7 m
Given:
r = 2.5 cm
h = 7 cm
To find:
Volume of the cylinder
Solution:
Volume of the cylinder = πr²h
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × (2.5)² × 7
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × 2.5 × 2.5 × 7
∴ Volume of the cylinder = 22 × 6.25
∴ Volume of the cylinder = 137.5 cu.cm
Ans: Volume of the cylinder is 137.5 cu.cm
(3) r = 4.2 cm, h = 5 cm
Given:
r = 4.2 cm
h = 5 cm
To find:
Volume of the cylinder
Solution:
Volume of the cylinder = πr²h
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × (4.2)² × 5
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × 4.2 × 4.2 × 5
∴ Volume of the cylinder = 22 × 0.6 × 21
∴ Volume of the cylinder = 277.2 cu.cm
Ans: Volume of the cylinder is 277.2 cu.cm
(4) r = 5.6 cm, h = 5 cm
Given:
r = 5.6 cm
h = 5 cm
To find:
Volume of the cylinder
Solution:
Volume of the cylinder = πr²h
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × (5.6)² × 5
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × 5.6 × 5.6 × 5
∴ Volume of the cylinder = 22 × 0.8 × 28
∴ Volume of the cylinder = 492.8 cu.cm
Ans: Volume of the cylinder is 492.8 cu.cm
2. How much iron is needed to make a rod of length 90 cm and diameter 1.4 cm?
Given:
Length of the rod = 90 cm
Diameter of the rod = 1.4 cm
To find:
Amount of iron needed to make a rod
Solution:
Rods are cylindrical in shape
Radius of the rod = \(\large \frac {1.4}{2}\)
∴ Radius of the rod = 0.7 cm
And, Height of the rod = Length of the rod
Amount of iron needed to make a rod = Volume of the rod
∴ Volume of the rod = πr²h
∴ Volume of the rod = \(\large \frac {22}{7}\) × (0.7)² × 90
∴ Volume of the rod = \(\large \frac {22}{7}\) × 0.7 × 0.7 × 90
∴ Volume of the rod = 22 × 0.1 × 63
∴ Volume of the rod = 138.6 cu.cm
Ans: Volume of the rod is 138.6 cu.cm
3. How much water will a tank hold if the interior diameter of the tank is 1.6 m and its depth is 0.7 m ?
Given:
Internal diameter of the tank = 1.6 m
Depth of the tank = 0.7 m
To find:
Volume of water the tank holds
Solution:
Shape of the tank is cylindrical
Internal radius of the tank = \(\large \frac {1.6}{2}\)
∴ Internal radius of the tank = 0.8 cm
Height of the tank = Depth of the tank
∴ Volume of water the tank holds = Internal volume of the tank
∴ Volume of water the tank holds = πr²h
∴ Volume of water the tank holds = \(\large \frac {22}{7}\) × (0.8)² × 0.7
∴ Volume of water the tank holds = 22 × 0.8 × 0.8 × 0.1
∴ Volume of water the tank holds = 22 × 0.064
∴ Volume of water the tank holds = 1.408 cu.cm
Now,
1 cu.cm = 1000 l of water
∴ Volume of water the tank holds = 1.408 × 1000
∴ Volume of water the tank holds = 1408 l
Ans: The tank holds 1408 l of water.
4. Find the volume of the cylinder if the circumference of the cylinder is 132 cm and height is 25 cm.
Given:
Circumference of the cylinder = 132 cm
Height of the cylinder = 25 cm
To find:
Volume of the cylinder
Solution:
Circumference of the cylinder = 2πr
∴ 132 = 2 × \(\large \frac {22}{7}\) × r
∴ ∴ \(\large \frac {132\, ×\,7}{44}\) = r
∴ r = 3 × 7
∴ r = 21 cm
Volume of the cylinder = πr²h
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × (21)² × 25
∴ Volume of the cylinder = \(\large \frac {22}{7}\) × 21 × 21 × 25
∴ Volume of the cylinder = 22 × 3 × 525
∴ Volume of the cylinder = 34650 cu.cm
Ans: Volume of the cylinder is 34650 cu.cm