Year 6: Volume of cubes and cuboids
Part of the Year 6 area, volume and measures chapter. Chapter overview · whole chapter on one page
Volume of cubes and cuboids
The volume is the amount of space a solid takes up. We measure it by counting cubes. A cuboid with length l, width w and height h has volume l × w × h. A cube has all its edges the same, so its volume is side × side × side.
A layer of 1 cm cubes that is 5 cubes long and 4 cubes wide has 5 × 4 = 20 cubes. If the cuboid is 3 layers high, the volume is 20 × 3 = 60 cm³.
The formula can be used for a cuboid or a cube. For a solid made of two or more cuboids, use the formula for each cuboid and add the volumes. The formula cannot be used on the whole solid if it is not a cuboid.
| Length | Volume |
|---|---|
| 1 cm = 10 mm | 1 cm³ = 1000 mm³ |
| 1 m = 100 cm | 1 m³ = 1 000 000 cm³ |
| 1 km = 1000 m | 1 km³ = 1 000 000 000 m³ |
Key ideas
- Cuboid: volume = length × width × height.
- Cube: volume = side × side × side.
- Volume is measured in cubic units, such as cm³ or m³.
- 1 m³ = 1 000 000 cm³ (100 × 100 × 100). 1 km³ = 1 000 000 000 m³.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Multiply the length, the width and the height.
- Write the formula. Volume = length times width times height.V = l × w × h
- Multiply the length by the width. This gives the area of the base.6 × 4 = 24
- Multiply by the height. This gives the number of cubes that fill the box.24 × 3 = 72
- Write the answer. The volume is in cubic units.72
Check. Order does not matter: 3 × 4 × 6 = 72. ✓
Watch out. Volume is in cubic units, such as cm³. Multiply all three lengths in the same unit first.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Multiply the side by itself three times.
- Write the formula. A cube has three equal edges, so its volume is side times side times side.V = s × s × s
- Multiply the first two sides. This is the area of one face.0.5 × 0.5 = 0.25
- Multiply by the third side. This gives the volume.0.25 × 0.5 = 0.125
- Write the answer. The volume is in cubic units.0.125
Check. Each face has area 0.25, and the cube is 0.5 deep, so the volume is 0.25 × 0.5 = 0.125. ✓
Watch out. Do not multiply the side by 3. Volume is side times side times side, which is much more than 3 times the side.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Find the volume in m³, then change it to cm³.
- Multiply the length by the width. Multiply the decimals first.2.5 × 0.4 = 1.0
- Multiply by the height. This gives the volume in m³.1.0 × 1.2 = 1.2 m³
- Change 1.2 m³ to cm³. 1 m³ = 1 000 000 cm³, so multiply by 1 000 000.1.2 × 1 000 000 = 1200000
- Write the answer. The volume is in cm³.1200000
Check. 1200000 ÷ 1000000 = 1.2 m³, the volume we found first. ✓
Watch out. A cubic metre is a million cubic centimetres. Do not multiply by 100 only.
Watch out
Common mistake: 1 m³ is not 100 cm³. It is 100 × 100 × 100 = 1 000 000 cm³. Cubic units use the length change three times over.
Exam tip
If an answer is in cubic units, and the question asks for a bigger unit, the number must get smaller: divide.
Exercise 9F Volume of cubes and cuboids
Non-calculatorKey words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Multiply the length, the width and the height.
- Write the formula. Volume = length times width times height.V = l × w × h
- Multiply the length by the width. This gives the area of the base.5 × 2 = 10
- Multiply by the height. This gives the number of cubes that fill the box.10 × 4 = 40
- Write the answer. The volume is in cubic units.40
Check. Order does not matter: 4 × 2 × 5 = 40. ✓
Watch out. Volume is in cubic units, such as cm³. Multiply all three lengths in the same unit first.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Multiply the side by itself three times.
- Write the formula. A cube has three equal edges, so its volume is side times side times side.V = s × s × s
- Multiply the first two sides. This is the area of one face.3 × 3 = 9
- Multiply by the third side. This gives the volume.9 × 3 = 27
- Write the answer. The volume is in cubic units.27
Check. Each face has area 9, and the cube is 3 deep, so the volume is 9 × 3 = 27. ✓
Watch out. Do not multiply the side by 3. Volume is side times side times side, which is much more than 3 times the side.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Multiply the length, the width and the height.
- Write the formula. Volume = length times width times height.V = l × w × h
- Multiply the length by the width. This gives the area of the base.1.5 × 2 = 3
- Multiply by the height. This gives the number of cubes that fill the box.3 × 4 = 12
- Write the answer. The volume is in cubic units.12
Check. Order does not matter: 4 × 2 × 1.5 = 12. ✓
Watch out. Volume is in cubic units, such as cm³. Multiply all three lengths in the same unit first.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Use the cuboid formula for each part, then add the two volumes.
- Find the volume of the first cuboid. The formula can be used, because this part is a cuboid.4 × 3 × 2 = 24 cm³
- Find the volume of the second cuboid. This part is also a cuboid, so the same formula works.4 × 3 × 1 = 12 cm³
- Add the two volumes. The total volume is the sum of the two parts.24 + 12 = 36
- Write the answer. The volume is in cubic centimetres.36
Check. Check each part again: 4 × 3 = 12, so 12 × 2 = 24 and 12 × 1 = 12. Total 24 + 12 = 36 cm³. ✓
Watch out. A formula works for one cuboid at a time. Split a solid into cuboids first, then add.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Work out how many small cubes fit in one big cube, then multiply the volume by that number.
- Find how many cm make one m along one edge. 1 m = 100 cm.1 m = 100 cm
- Cube the factor: 100 × 100 × 100 = 1000000. A cube has three edges, so the factor is used three times.100 × 100 × 100 = 1000000, so 1 m³ = 1000000 cm³
- Multiply the volume by 1000000. The answer is in cm³, which is a smaller unit, so there are more of them. Multiplying makes the number bigger.1 × 1000000 = 1000000
- Write the answer. 1000000
Check. 1000000 ÷ 1000000 = 1. ✓
Watch out. Cubic units change by the length factor cubed. For example, 1 cm³ = 10 × 10 × 10 = 1000 mm³, not 10 mm³.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Work out how many small cubes fit in one big cube, then multiply the volume by that number.
- Find how many m make one km along one edge. 1 km = 1000 m.1 km = 1000 m
- Cube the factor: 1000 × 1000 × 1000 = 1000000000. A cube has three edges, so the factor is used three times.1000 × 1000 × 1000 = 1000000000, so 1 km³ = 1000000000 m³
- Multiply the volume by 1000000000. The answer is in m³, which is a smaller unit, so there are more of them. Multiplying makes the number bigger.1 × 1000000000 = 1000000000
- Write the answer. 1000000000
Check. 1000000000 ÷ 1000000000 = 1. ✓
Watch out. Cubic units change by the length factor cubed. For example, 1 cm³ = 10 × 10 × 10 = 1000 mm³, not 10 mm³.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Work out how many small cubes fit in one big cube, then divide the volume by that number.
- Find how many cm make one m along one edge. 1 m = 100 cm.1 m = 100 cm
- Cube the factor: 100 × 100 × 100 = 1000000. A cube has three edges, so the factor is used three times.100 × 100 × 100 = 1000000, so 1 m³ = 1000000 cm³
- Divide the volume by 1000000. The answer is in m³, which is a bigger unit, so there are fewer of them. Dividing makes the number smaller.3000000 ÷ 1000000 = 3
- Write the answer. 3
Check. 3 × 1000000 = 3000000. ✓
Watch out. Cubic units change by the length factor cubed. Do not divide by the length factor once.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Work out how many small cubes fit in one big cube, then multiply the volume by that number.
- Find how many mm make one cm along one edge. 1 cm = 10 mm.1 cm = 10 mm
- Cube the factor: 10 × 10 × 10 = 1000. A cube has three edges, so the factor is used three times.10 × 10 × 10 = 1000, so 1 cm³ = 1000 mm³
- Multiply the volume by 1000. The answer is in mm³, which is a smaller unit, so there are more of them. Multiplying makes the number bigger.4 × 1000 = 4000
- Write the answer. 4000
Check. 4000 ÷ 1000 = 4. ✓
Watch out. Cubic units change by the length factor cubed. For example, 1 cm³ = 10 × 10 × 10 = 1000 mm³, not 10 mm³.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Multiply the length, the width and the height.
- Write the formula. Volume = length times width times height.V = l × w × h
- Multiply the length by the width. This gives the area of the base.12 × 5 = 60
- Multiply by the height. This gives the number of cubes that fill the box.60 × 4 = 240
- Write the answer. The volume is in cubic units.240
Check. Order does not matter: 4 × 5 × 12 = 240. ✓
Watch out. Volume is in cubic units, such as cm³. Multiply all three lengths in the same unit first.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Work out the volume of each cuboid, then compare them.
- Find the volume of cuboid A. Multiply the three lengths.6 × 3 × 2 = 36 cm³
- Find the volume of cuboid B. Multiply the three lengths.4 × 4 × 3 = 48 cm³
- Find the difference. Take the smaller volume away from the bigger one.48 - 36 = 12
- Write the answer. The difference is in cubic centimetres.12
Check. 36 + 12 = 48 cm³. ✓
Watch out. Compare volumes only after you have worked out each one.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Multiply the three lengths in metres, then change the volume to litres.
- Multiply the three lengths. The tank is measured in metres.1.5 × 1 × 0.8 = 1.2 m³
- Change m³ to litres. There are 1000 litres in 1 m³, so multiply by 1000.1.2 × 1000 = 1200
- Write the answer. The answer is in litres.1200
Check. 1200 ÷ 1000 = 1.2 m³. ✓
Watch out. Multiply the three measurements first, then change the volume to litres.
Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.
Method. Work out the volume of each cube, then compare them.
- Find the volume of the cube with sides of 3 cm. Side times side times side.3 × 3 × 3 = 27 cm³
- Find the volume of the cube with sides of 6 cm. Double each edge, then cube the new side.6 × 6 × 6 = 216 cm³
- Compare the two volumes. Divide the new volume by the old volume.216 ÷ 27 = 8
- Write the answer. Sam is not right. The volume is 8 times bigger, not 2 times bigger.No
Check. 27 × 8 = 216, so the volume is 8 times bigger. ✓
Watch out. Doubling every edge multiplies the volume by 2 × 2 × 2 = 8.