Kerala Syllabus Class 6 Mathematics - Unit 3 Volume - Questions and Answers | Teaching Manual 

Questions and Answers for Class 6 Mathematics - Unit 3 Volume - Study Notes | Text Books Solution STD 6 - Maths: Unit 3 Volume - Questions and Answers | ഗണിതം - വ്യാപ്തം - ചോദ്യോത്തരങ്ങൾ 

ആറാം ക്ലാസ്സ്‌  Mathematics - Unit 3 Volume എന്ന പാഠം ആസ്പദമാക്കി തയ്യാറാക്കിയ ചോദ്യോത്തരങ്ങള്‍. ഈ അധ്യായത്തിന്റെ Teachers Handbook, Teaching Manual എന്നിവ ഡൗൺലോഡ് ചെയ്യാനുള്ള ലിങ്ക് ചോദ്യോത്തരങ്ങളുടെ അവസാനം നൽകിയിട്ടുണ്ട്. പുതിയ അപ്‌ഡേറ്റുകൾക്കായി ഞങ്ങളുടെ Telegram Channel ൽ ജോയിൻ ചെയ്യുക.

Kerala Syllabus STD 6 Maths: Unit 3 Volume - Textbook Solutions

♦ Textbook Activities (Textbook Page No: 35)
♦ How many small cubes are used to make this large cube? If one small block is removed from each corner of the large block, how many would be left?
This rectangular block contains 4 × 4 × 4 = 64 small blocks. If one small block is removed from each corner of the large block, how many would be left? Let's sss.
The large rectangular block ha s8 corners.
So, subtract 8 from the total number of blocks.
Number of small blocks left = 64 − 8 = 56

♦ Textbook Activities (Textbook Page No: 36)
♦ All sides of the large cube are painted. How many small cubes would have no paint at all?
All the small cubes of the top layer will have paint on at least one face. In the middle layer, the small cube at the centre will not have paint on any of its faces. All the small cubes of the bottom layer will have paint on at least one face. Number of small cubes without paint = 1.

♦ Textbook Activities (Textbook Page No: 37, 38)
♦ All blocks shown below are made up of cubes of side 1 centimetre. Calculate the volume of each:
1. Number of small cubes = 4 × 4 × 4 = 64
Its volume = 64 cu.cm

2. Number of small cubes = 2 × 2 × 3 = 12
Its volume = 12 cu.cm

3. Number of small cubes = 4 × 3 × 3 = 36
Its volume = 36 cu.cm

4. Number of small cubes = 7 × 2 × 3 = 42
Its volume = 42 cu.cm

5. Number of small cubes = 3 × 2 × 2 = 12
Its volume = 12 cu.cm
The volume of a rectangular block is the product of its length, width and height.
♦ Textbook Activities (Textbook Page No: 40)
(1) The length, width and height of a brick are 21 centimetres, 15 centimetres and 7 centimetres. What is its volume?
Length = 21 cm
Width = 15 cm
Height = 7 cm
Volume of the brick = Length × width × height 
           = 21 × 15 × 7 = 2205 cu.cm

(2) An iron cube is of side 8 centimetres. What is its volume? 1 cubic centimetre of iron weighs 8 grams. What is the weight of this cube?
All sides of the cube are equal.
Volume = 8 × 8 × 8 = 512 cu.cm
Weight of 1 cubic centimetre of iron = 8 gms
The total weight of the cube = 512 × 8 = 4096 grams

♦ Textbook Activities (Textbook Page No: 41)
Volume and length

♦ A wooden block of length 9 centimetres and width 4 centimetres has a volume of 180 cubic centimetres. What is its height?
Volume is the product of length, width and height.
Length = 9 cm, width = 4cm
The volume of the block = 180 cubic centimetres
Length × width × height = volume
9 × 4 × height = 180, 36 × height = 180
Height = 180/36 = 5 cm.

♦ Area and volume
♦ What is the area of a rectangle of length 8 centimetres and width 2 centimetres? 
Area of a rectangle of length 8 cm and width 2 cm 
Length × width = 8 × 2 = 16 sq.cm

♦ What about the volume of a rectangular block of length 8 centimetres, width 2 centimetres and height 1 centimetre?
Volume of a rectangular block of length 8 cm, width 2 cm and height 1 cm = 8 × 2 × 1 = 16 cu.cm
Here, the length and width of the rectangle and the rectangular block are equal. Also, the height of the rectangular block is 1 cm. Therefore, the area of rectangle and volume of rectangular block are equal.

♦ The table shows the measurements of some rectangular blocks. Calculate the missing measures.
1. Volume = 3 × 8 × 7 = 168 cu.cm
2. Volume = 6 × 4 × 5 = 120 cu.cm
3. Height = 48/6 × 4 = 2 cm
4. Width = 48/8 × 2 = 3 cm
5. Lentgh = 48/2 × 2 = 12 cm
6. Lentgh = 80/2 × 4 = 10 cm
7. Width = 210/14 × 5 = 3 cm

♦ New shapes
♦ We can make shapes other than rectangular block by stacking cubes. For example, see this:
It is made by stacking cubes of side 1 centimetre. Can you calculate its volume?
• How many cubes are there at the very bottom?
Number of cubes at the very bottom = 9 × 9 × 1 = 81

• How many cubes are there in the step just above it?
Number of cubes in the step just above it = 9 × 7 × 1 = 63

• How many cubes are there on the top-most step?
Number of cubes on the top-most step = 9 × 3 × 1 = 27

• How many cubes in all?
Total number of cubes = 81 + 63 + 45 + 27 = 216

• What is the volume of the stairs?
Volume of the stairs = 216 sq.cm

♦ Textbook Activities (Textbook Page No: 42)
♦ Now look at this figure:
It is made by stacking square blocks. The bottom block is of side 9 centimetres. As we move up, the sides decrease by 2 centimetres at each step. All blocks are of height 1 centimetre. What is the volume of this figure?
To find the volume of this figure, simply add up the volumes of each block.
Volume of the square block at the bottom = 9 × 9 × 1 = 81 cu.cm
Volume of the square block just above it = 7 × 7 × 1 = 49 cu.cm
Volume of the square block just above it = 5 × 5 × 1 = 25 cu.cm
Volume of the square block just above it = 3 × 3 × 1 = 9 cu.cm
Volume of the square block at the top = 1 × 1 × 1 = 1 cu.cm
Total volume = 81 + 49 + 25 + 9 + 1 = 165 cu.cm

♦ What is the volume of a rectangular block of length 4 centimetres, width 3 centimetres and height 1 centimetre?
Volume of a rectangular block of length 4 cm, width 3 cm and height 1 cm = 4 × 3 × 1 = 12 cu.cm

♦ If the length, width and height are doubled, what happens to the volume?
When the length, width and height are doubled, they become 8 cm, 6 cm and 2 cm, respectively.
Its volume = 8 × 6 × 2 = 96 cu.cm
12 × 8 = 96
The volume becomes 8 times the first. 

♦ Calculate the volumes of the shapes shown below. All lengths are in centimetres.
1. Divide the given shape into different parts.
Length of part (I) = 20 cm
Width = 12 − 8 = 4 cm
Height = 2 cm
Volume = 20 × 4 × 2 = 160 cu.cm

Length of part (II) = 8 cm
Width = 4 cm
Height = 2 cm
Volume = 8 × 4 × 2 = 64 cu.cm

Length of part (III) = 4 cm
Width = 4 cm
Height = 2 cm
Volume = 4 × 4 × 2 = 32 cu.cm

Length of part (IV) = 8 × 4 × 2 = 64 cu.cm
Total volume = 160 + 64 + 32 + 64 = 320 cu.cm

2. Divide the given shape into different parts.
Length of part (I) = 16 cm
Width =  4 cm
Height = 3 cm
Volume = 16 × 4 × 3 = 192 cu.cm

Length of part (II) = 4 cm
Width =  4 cm
Height = 3 cm
Volume = 4 × 4 × 3 = 48 cu.cm

Length of part (III) = 16 × 4 × 3 = 192 cu.cm
Total volume = 192 + 48 + 192 = 432 cu.cm

3. Divide the given shape into different parts.
Length of part (I) = 6 + 6 + 4 =16 cm
Width =  4 cm
Height = 3 cm
Volume = 16 × 4 × 3 = 192 cu.cm

Length of part (II) = 11 cm
Width =  4 cm
Height = 3 cm
Volume = 11 × 4 × 3 = 132 cu.cm
Total volume = 192 + 132 = 324 cu.cm

♦ Textbook Activities (Textbook Page No: 43)
(1) A truck is loaded with sand, 4 metres long, 2 metres wide and 1 metre high. The price of 1 cubic metre of sand is 1000 rupees. What is the price of this truckload?
Volume of the sand = Length × Width × Height = 4 × 2 × 1 = 8 cu.cm
The price of 1 cubic metre of sand = 1000 rupees
The total price of sand = 8 × 1000 = 8000 rupees

(2) What is the volume in cubic centimetres of a platform 6 metres long, 1 metre wide and 50 centimetres high?
Length = 6m = 600 centimetre
Width = 1m = 100 centimetre
Height = 50 cm
Volume = 600 × 100 × 50 = 3000000 cu.cm 

(3) What is the volume of a piece of wood which is 5 metres long, 1 metre wide and 25 centimetres high? The price of 1 cubic metre of wood is 60000 rupees. What is the price of this piece of wood?
Since the price per 1 cubic metre of wood is given, we need to convert all measurements to metres to calculate the volume.
Length of the piece of wood = 5 m, Width = 1m Height = 25 cm = 25/100 m
The volume of the piece of wood = length × width × height
     = 5 × 1 × 25/100 = 125/100 = 1.25 sq.m
The price of one cubic metre of wood = 60000 rupees
The cost of this piece of wood = 1.25 × 60000 = 75000 rupees

♦ Capacity
♦ Look at this hollow box (see textbook page 43):
It is made with thick wooden planks. Because of the thickness, its inner length, width and height are less than the outer measurements.
The inner length, width and height are 40 centimetres, 20 centimetres and 10 centimetres.
So, a rectangular block of these measurements can exactly fit into the space within this box.
The volume of this rectangular block is the volume within the box.
This volume is called the capacity of the box.
Thus the capacity of this box is: 40 × 20 × 10 = 8000 cubic centimetres.

♦ So, what is the capacity of a box whose inner length, width and height are 50 centimetres, 25 centimetres and 20 centimetres?
The capacity of the box = Length × Width × Height 
                   = 50 × 25 × 20 = 25000 cu.cm

♦ Textbook Activities (Textbook Page No: 44)
♦ Litre and cubic metre
1 litre = 1000 cubic centimetres 
1 cubic metre = 1000000 cubic centimetres
1 cubic metre = 1000 litres.

♦ Liquid measures
♦ What is the capacity of a cubical vessel of inner side 10 centimetres?
10 × 10 × 10 = 1000 cubic centimetres
1 litre is the amount of water that fills this vessel.

1 litre = 1000 cubic centimetres

If a cube of side 10 centimetres is completely immersed in a vessel filled with water, then the amount of water that overflows would be 1 litre.

♦ How many litres of water does a vessel of length 20 centimetres, width 15 centimetres and height 10 centimetres contain?
The capacity of the vessel = Length × Width × Height 
                   = 20 × 15 × 10 = 3000 cu.cm
The amount of water that the vessel can hold = 3000/1000 litre = 3 litres.

♦ In the water
♦ A vessel is filled with water. If a cube of side 1 centimetre is immersed in it, how many cubic centimetres of water would overflow? 
Amount of water that would overflow if a cube of side 1cm is immersed = 1 × 1 × 1 = 1 cu.cm

♦ What if 20 such cubes are immersed?
Amount of water that would overflow if 20 such cubes are immersed = 20 cu.cm 

♦ Textbook Activities (Textbook Page No: 45, 46)
♦ Raising water

♦ A swimming pool is 25 metres long, 10 metres wide and 2 metres deep. It is half filled. How many litres of water does it contain now?
20 × 10 × 1 = 250 cu.cm = 250000 litres

♦ Suppose the water level is increased by 1 centimetre. How many more litres of water does it contain now?
When the water level in the pool rises by 1cm, the capacity of water increases by 25 × 10 × 1/100
2.5 cu.m = 2500 litres

(1) The inner sides of a cubical box are of length 4 centimetres. What is its capacity? How many cubes of side 2 centimetres can be stacked inside it?
Inner length of the box = 4 cm
Width = 4 cm
Height = 4 cm
Capacity of the box = 4 × 4 × 4 = 64 cu.cm
Volume of a cube of side 2 cm = 2 × 2 × 2 = 8 cu.cm
Number of small cubes that can be stacked in the big box = 64/8 = 8

(2) The inner sides of a rectangular tank are 70 centimetres, 80 centimetres and 90 centimetres. How many litres of water can it contain?
Capacity of the tank = 70 × 80 × 90 = 504000 cu.cm
 = 504000 / 1000 litre = 504 litres

(3) The length and width of a rectangular box are 90 centimetres and 40 centimetres. It contains 180 litres of water. How high is the water level?
180 litre = 180 × 1000 cu.cm 
  = 180000 cu.cm
Length × Width × Height = Capacity
90 × 40 × Height = 180000
Height = 180000 / 90 × 40 = 50 cm
There is water in a rectangular box at a height of 50 cm

(4) The inner length, width and height of a tank are 80 centimetres, 60 centimetres and 50 centimetres, and it contains water 15 centimetres high. How much more water is needed to fill it?
Capacity of the tank = Length × Width × Height 
  80 × 60 × 50 = 240000 cu.cm
Height of the water level = 15 cm
Amount of water = 80 × 60 × 15 = 72000 cu.cm
More water needed to fill the tank = 240000 − 72000 = 168000 cu.cm 
   = 168 litres

(5) The panchayat decided to make a rectangular pond. The length, width and depth were decided to be 20 metres, 15 metres and 2 metres. How many litres of water is needed to fill this pond to a height of one and a half metres?
Length = 20 m, Width = 15 m, Height = 2 m
The capacity of the rectangular pond = Length × Width × Height 
    = 20 × 15 × 2 = 600 cu.cm
    = 600 × 1000 litres
    = 600000 litres
Amount of water needed to fill the pond up to one and a half metres = 20 × 15 × 1½
 = 20 × 15 × ³⁄₂ = 450 cubic metres
 = 450 × 1000 litres = 450000 litres

(6) The inner length and width of an aquarium are 60 centimetres and 30 centimetres. It is half filled with water. When a stone is immersed in it, the water level rose by 10 centimetres. What is the volume of the stone?
When a stone was immersed, the water level rose by 10 cm. The capacity of water of length 60 cm, width 30 cm and height 10 cm is equal to the volume of the stone.
Volume of the stone = 60 × 30 × 10 = 18000 cu.cm

(7) A rectangular iron block has length 20 centimetres, width 10 centimetres and height 5 centimetres. It is melted and recast into a cube. What is the length of a side of this cube?
Volume of the iron block = 20 × 10 × 5 = 1000 cu.cm
Melting the rectangular block to form a cube doesn't change its volume.
Volume of the recast cube = 1000 cu. cm
Length, width and height of a cube are equal.
10 × 10 × 10 = 1000 
One side of the cube = 10 cm

(8) A tank 2 metres long and 1 metre wide is to contain 10000 litres of water. What should be the height of the tank?
Capacity = 10000 litres = 10000 / 1000 cu.cm = 10 cu.m
Length × Width × Height = capacity
2 × 1 × Height = 10 cu.m
Height = 10 / 2 × 1 = 10/2 = 5 m
∴ Height of tank = 5m

(9) From the four corners of a square piece of paper of side 12 centimetres, small squares of side 1 centimetre are cut off. The edges of this are bent up and joined to form a container of height 1 centimetre. What is the capacity of this container? If squares of side 2 centimetres are cut off, what would be the capacity?
Length of the remaining part when small squares of side 1 cm are cut off from the corners = 12 − 1 − 1 = 10 cm
Capacity of the container = 10 × 10 × 1 = 100 cu.cm
Length of the remaining part when small squares of side 2 cm are cut off from the corners = 12 − 2 − 2 = 8 cm
Present capacity = 8 × 8 × 2 = 128 cu.cm