How Big, How Heavy — Class 5 Mathematics (CBSE)
Based on the NCERT Math Magic Grade 5 textbook. Understand volume, capacity, and packing, then solve the practice set without looking at the answers.
1. Why this chapter matters
How much water can a tank hold? How many boxes fit in a carton? How heavy is a brick? These are questions about volume and capacity. This chapter introduces students to the concept of volume — the amount of space an object takes up. They learn the relationship between cubic centimetres and millilitres (1 cm^3 = 1 mL), how to measure volume using the displacement method, and how to calculate packing efficiency. Volume and capacity are used in cooking, construction, shipping, and science experiments.
2. What is volume?
Volume is the amount of space an object occupies. It is measured in cubic units.
- 1 cubic centimetre (1 cm^3) = the volume of a cube with side 1 cm.
- Volume of a cuboid = length x breadth x height (all in the same unit).
- Volume of a cube = side x side x side (or side^3).
Finding volume by counting unit cubes
If a shape is made of 1 cm cubes, the volume is simply the number of cubes.
Example: A cuboid made of 3 layers, each layer having 4 rows of 5 cubes: Volume = 3 x 4 x 5 = 60 cm^3
Formula for cuboid and cube
| Shape | Dimensions | Volume formula | Example |
|---|---|---|---|
| Cube | side = 4 cm | side x side x side | 4 x 4 x 4 = 64 cm^3 |
| Cuboid | L = 5 cm, B = 3 cm, H = 2 cm | L x B x H | 5 x 3 x 2 = 30 cm^3 |
| Cuboid | L = 10 cm, B = 4 cm, H = 6 cm | L x B x H | 10 x 4 x 6 = 240 cm^3 |
3. Capacity
Capacity is the amount a container can hold. Capacity is often measured in litres (L) or millilitres (mL).
Relationship between volume and capacity
1 cm^3 = 1 mL 1,000 cm^3 = 1,000 mL = 1 L
Example: A rectangular tank is 20 cm long, 15 cm wide, and 10 cm high.
Volume = 20 x 15 x 10 = 3,000 cm^3 Capacity = 3,000 mL = 3 L
Common conversions
| Volume | Capacity |
|---|---|
| 1 cm^3 | 1 mL |
| 1,000 cm^3 | 1 L |
| 500 cm^3 | 500 mL (half a litre) |
| 250 cm^3 | 250 mL (quarter of a litre) |
| 2,500 cm^3 | 2.5 L |
Word problems with capacity
Problem 1: A water tank is 50 cm long, 30 cm wide, and 40 cm high. How many litres can it hold? Volume = 50 x 30 x 40 = 60,000 cm^3 Capacity = 60,000 mL = 60 L
Problem 2: A bottle holds 1.5 L of juice. How many glasses of 200 mL can be filled? 1.5 L = 1,500 mL Number of glasses = 1,500 / 200 = 7.5 glasses (7 full glasses + a little left)
4. Displacement method
The displacement method measures the volume of irregular objects (objects that are not cubes or cuboids).
Steps
- Take a measuring cylinder with water.
- Note the initial water level (let us say 50 mL).
- Gently lower the object into the water (tied to a thread).
- Note the new water level (let us say 73 mL).
- The volume of the object = rise in water level = 73 — 50 = 23 mL = 23 cm^3.
Why this works
An object displaces its own volume of water when submerged. The amount the water level rises is equal to the object's volume.
| Object | Initial water level | Final water level | Volume of object |
|---|---|---|---|
| Stone | 40 mL | 58 mL | 18 mL = 18 cm^3 |
| Key | 60 mL | 67 mL | 7 mL = 7 cm^3 |
| Marble | 30 mL | 35 mL | 5 mL = 5 cm^3 |
| Small potato | 50 mL | 72 mL | 22 mL = 22 cm^3 |
5. Packing
How many smaller boxes can fit into a larger box? This is a packing problem.
Method
- Calculate the volume of the large box (container).
- Calculate the volume of one small box (item).
- Divide: Volume of container / Volume of one item.
Important: In real packing, shape matters. You cannot always fill all space due to gaps. The calculation gives the maximum possible number.
Example: A carton is 60 cm long, 40 cm wide, and 30 cm high. How many boxes of 10 cm x 8 cm x 5 cm can fit?
Volume of carton = 60 x 40 x 30 = 72,000 cm^3 Volume of one box = 10 x 8 x 5 = 400 cm^3 Maximum boxes = 72,000 / 400 = 180 boxes
Along length: 60 / 10 = 6 boxes Along width: 40 / 8 = 5 boxes Along height: 30 / 5 = 6 boxes Total: 6 x 5 x 6 = 180 boxes. They fit exactly.
When shape causes gaps
If the carton is 62 cm long and the box is 10 cm long: 62 / 10 = 6 boxes (with 2 cm wasted space along that dimension). So the actual number is less than the volume-based calculation.
6. Comparing volumes
| Object | Approximate volume |
|---|---|
| Matchbox | 25 cm^3 |
| Brick | 1,500 cm^3 |
| Maths textbook | 500 cm^3 |
| Drinking glass | 250 cm^3 |
| School bag | 15,000 cm^3 |
| School desk | 50,000 cm^3 |
7. Activity corner
Activity 1: Take 5 small objects (eraser, marble, key, stone, pencil stub). Use the displacement method to find the volume of each. Record your readings in a table.
Activity 2: Find a small cardboard box. Measure its length, breadth, and height. Calculate its volume in cm^3. How many 1 cm cubes can you fit inside?
Activity 3: A water bottle has a capacity of 1 L. Mark the level for 250 mL, 500 mL, and 750 mL using a measuring cylinder.
8. Common mistakes
- Mistake: Confusing volume with capacity Fix: Volume is space occupied (cm^3). Capacity is what a container can hold (L, mL). For a container, volume and capacity are numerically equal (1 cm^3 = 1 mL).
- Mistake: Using different units for length, width, and height Fix: Convert all dimensions to the same unit (usually cm) before multiplying.
- Mistake: Forgetting that the object must be fully submerged in the displacement method Fix: The object should be completely under water. If it floats, push it down with a thin pin (or use a sinker).
- Mistake: Assuming volume-based packing count is always achievable Fix: Check each dimension separately. Gaps may reduce the actual number.
9. Key facts
- Volume of cuboid = length x breadth x height.
- Volume of cube = side x side x side.
- 1 cm^3 = 1 mL.
- 1,000 cm^3 = 1 L.
- Displacement method measures volume of irregular objects.
- Packing: check along each dimension, not just by volume.
- Always use the same unit for all measurements.
10. Self-test
- Find the volume of a cuboid with length 8 cm, breadth 5 cm, height 3 cm.
- A cube has side 7 cm. What is its volume?
- Convert 2,500 cm^3 to litres.
- A stone raises water level from 45 mL to 72 mL. What is the stone's volume?
- A carton is 40 cm x 30 cm x 20 cm. How many boxes of 10 cm x 6 cm x 5 cm can fit?
11. Answer key
-
Find the volume of a cuboid with length 8 cm, breadth 5 cm, height 3 cm. Answer: Volume = 8 x 5 x 3 = 120 cm^3.
-
A cube has side 7 cm. What is its volume? Answer: Volume = 7 x 7 x 7 = 343 cm^3.
-
Convert 2,500 cm^3 to litres. Answer: 2,500 cm^3 = 2,500 mL = 2.5 L.
-
A stone raises water level from 45 mL to 72 mL. What is the stone's volume? Answer: Volume = 72 — 45 = 27 mL = 27 cm^3.
-
A carton is 40 cm x 30 cm x 20 cm. How many boxes of 10 cm x 6 cm x 5 cm can fit? Answer: Along length: 40/10 = 4. Along width: 30/6 = 5. Along height: 20/5 = 4. Total = 4 x 5 x 4 = 80 boxes.
12. Quick revision
- Volume = L x B x H for cuboids, side^3 for cubes.
- 1 cm^3 = 1 mL, 1,000 cm^3 = 1 L.
- Use displacement method for irregular objects.
- For packing, check fit along each dimension.
- Always use consistent units.
- Practise by measuring real objects around you.
- Volume and capacity are connected through cm^3 = mL.
