Pick one before you read on. It is fine to be wrong.
By the end you can
Count cubes to find the volume of a box
Find the volume of a cuboid as length × width × height
Change between cm³, ml and litres
A tank, a box and a brick are all solid shapes. A solid shape with six rectangle faces is a cuboid. How much space is inside one? That space is its volume.
We count volume in cubes. A cubic centimetre is a cube with edges of 1 cm. We write it 1 cm³. A box with a volume of 24 cm³ holds 24 of these small cubes.
One cubic centimetre, 1 cm³.
Draw it first
Fill a box layer by layer
1 Cover the bottom of the box with cubes. One layer holds 4 × 3 = 12 cubes.
2 Add a second layer. Now there are 2 × 12 = 24 cubes.
3 Add a third layer. The box is full, with 3 × 12 = 36 cubes. Its volume is 36 cm³.
Look at the bottom layer. Its cubes are length × width. That is the same as the area of the bottom. Then the height says how many layers there are. So the volume of a cuboid is length × width × height.
Worked example
A box is 5 cm long, 3 cm wide and 2 cm high. What is its volume?
One layer has 5 × 3 = 15 cubes.
The box is 2 cm high, so there are 2 layers.
The volume is 15 × 2 = 30 cm³.
Answer: 30 cm³
Two layers of 15 cubes make 30 cm³.
The unit is cm³ because you multiply three lengths in cm. A cube has the same length, width and height. A cube with edges of 5 cm has a volume of 5 × 5 × 5 = 125 cm³.
Liquids are measured in ml and litres. The two measures fit together. A cube with edges of 1 cm holds 1 ml of water. So 1 cm³ is 1 ml.
A bigger cube has edges of 10 cm. Its volume is 10 × 10 × 10 = 1 000 cm³, which is 1 000 ml. We call 1 000 ml one litre.
Worked example
A box is 10 cm long, 10 cm wide and 10 cm high. How much water does it hold?
The volume is length × width × height.
10 × 10 × 10 = 1 000 cm³.
1 cm³ holds 1 ml, so the box holds 1 000 ml.
And 1 000 ml is 1 litre.
Answer: 1 litre
This cube is 10 cm each way. It holds 1 000 cm³, which is 1 litre. Not to scale.
Worked example
A tank is 40 cm long, 25 cm wide and 20 cm high. How many litres of water fill it?
The base is 40 cm by 25 cm. Its area is 40 × 25 = 1 000 cm².
The tank is 20 cm high, so the volume is 1 000 × 20 = 20 000 cm³.
1 litre is 1 000 cm³, so divide by 1 000.
20 000 ÷ 1 000 = 20 litres.
Answer: 20 litres
The tank holds 20 000 cm³, which is 20 litres. Not to scale.
See it three ways
With things
Stack small cubes, such as dice, into a box shape. Make it 4 across, 3 deep and 2 high. Count them all.
You need: 12 small cubes of the same size, such as dice or sugar lumps, a flat table.
Build a box that is 3 cubes long, 2 cubes wide and 1 cube high. Count the cubes.
Put a second layer on top. Count again.
Work out 3 × 2 × 2 and compare it with your count.
What you will see: The count and the product are the same, 12. If each cube has edges of 1 cm, the volume of the box is 12 cm³.
If you cannot try it, this is what happens.
A wrong idea many people have
To find the volume of a box, add its length, width and height.
Why it feels right: You add lengths to find a perimeter, so adding seems natural.
The test that breaks it: Take a box 4 cm by 3 cm by 2 cm. Adding gives 4 + 3 + 2 = 9. But the bottom layer alone has 4 × 3 = 12 cubes, so the box holds more than 9.
What is true: Volume counts cubes in layers. Multiply the length, the width and the height.
Key words
cuboid
a solid shape with six rectangle faces, such as a brick.
volume
the space inside a solid shape, counted in cubes.
cubic centimetre
a cube with edges of 1 cm. Written cm³.
litre
a unit for liquids. 1 litre is 1 000 ml, which is 1 000 cm³.
Back to your guess
The answer is A: Multiply the length, the width and the height.
A tank fills in layers. One layer has length × width cubes of water. The height tells you how many layers there are. So you multiply all three numbers.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
What is the volume of this box?
See the answer
40 cm³
One layer has 5 × 2 = 10 cubes. There are 4 layers. So the volume is 10 × 4 = 40 cm³.
If you chose A: You added 5 + 2 + 4. Volume counts cubes in layers, so you multiply. Go to M3.6.7.
If you chose C: That is 5 × 2, only the bottom layer. The box is 4 layers high, so multiply by 4 as well. Go to M3.6.7.
A cube has edges of 5 cm. What is its volume?
See the answer
125 cm³
A cube has the same length, width and height. 5 × 5 × 5 = 125 cm³.
If you chose A: You added 5 + 5 + 5. Multiply the three edges instead. Go to M3.6.7.
If you chose B: That is 5 × 5, only one layer. The cube is 5 layers high. Go to M3.6.7.
A box has a volume of 120 cm³. Its base is 8 cm by 5 cm. How high is it?
See the answer
3 cm
One layer is 8 × 5 = 40 cubes. The volume is 120 cubes, so the height is 120 ÷ 40 = 3 cm.
If you chose B: You divided 120 by 8 and stopped. The base is 8 × 5 = 40 cm², so divide 120 by 40. Go to M3.6.7.
If you chose C: That is the area of the base, 8 × 5. The height is how many layers of 40 cubes make 120. Go to M3.6.7.
A tank holds 18 000 cm³ of water. How many litres is that?
See the answer
18 litres
1 litre is 1 000 cm³. So 18 000 ÷ 1 000 = 18 litres.
If you chose A: You divided by 100. A litre is 1 000 cm³, so divide by 1 000. Go to M3.6.7.
If you chose B: You divided by 10 000. A litre is 1 000 cm³, which has three zeros. Go to M3.6.7.
A box is 20 cm long, 10 cm wide and 5 cm high. How many millilitres of water does it hold?
See the answer
1 000 ml
The volume is 20 × 10 × 5 = 1 000 cm³. Each cm³ holds 1 ml, so the box holds 1 000 ml.
If you chose A: You added 20 + 10 + 5. Volume counts cubes in layers, so you multiply. Go to M3.6.7.
If you chose C: That is 20 × 10, one layer of cubes. The box is 5 cm high, so there are 5 layers. Go to M3.6.7.
This box is 3 cm long, 3 cm wide and 4 cm high. Two layers are filled with cubes. How many cubes fill the whole box?
See the answer
36 cubes
One layer has 3 × 3 = 9 cubes. The full box has 4 layers. So 9 × 4 = 36 cubes.
If you chose B: That is one layer, 3 × 3. The box has 4 layers when it is full. Go to M3.6.7.
If you chose C: You added 3 + 3 + 4. Multiply the layer, 9 cubes, by the 4 layers. Go to M3.6.7.
A box is 6 cm long, 4 cm wide and 3 cm high. What is its volume?
See the answer
72 cm³
The volume is length × width × height: 6 × 4 × 3 = 72 cm³.
A tank holds 7 000 cm³ of water. How many litres is that?
See the answer
7 litres
1 litre is 1 000 cm³. So 7 000 ÷ 1 000 = 7 litres.
From an earlier chapter
The base of a box is 6 cm by 5 cm. What is the area of the base?
See the answer
30 cm²
The area of a rectangle is length × width: 6 × 5 = 30 cm².
Keep it
Copy this into your exercise book.
Volume = length × width × height, in cm³. 1 cm³ = 1 ml. 1 litre = 1 000 cm³ = 1 000 ml. A cube of 10 cm holds 1 litre.
Draw: A box of 1 cm cubes, 4 by 3 by 2, with its edges labelled. Write 4 × 3 × 2 = 24 cm³.
