Calculating the volume & surface area of cubes and cuboids
🎯 In this topic you will
- Derive and use the formula for the volume of cubes and cuboids
🧠 Key Words
- cubic centimetre (cm³)
- cubic millimetre (mm³)
- cubic metre (m³)
- volume
Show Definitions
- cubic centimetre (cm³): A unit of volume equal to the space occupied by a cube with 1 cm edges.
- cubic millimetre (mm³): A very small unit of volume equal to a cube with 1 mm edges.
- cubic metre (m³): A standard SI unit of volume equal to the space occupied by a cube with 1 m edges.
- volume: The amount of three-dimensional space an object occupies, measured in cubic units.
Look at this cube. It has a length, a width and a height of $1 \text{ cm}$.
It is called a centimetre cube. You say that it has a volume of 1 cubic centimetre (1 cm³).
This cuboid is $4 \text{ cm}$ long, $3 \text{ cm}$ wide and $2 \text{ cm}$ high.
If you divide the cuboid into centimetre cubes, it looks like this.
You can see that there are $12$ cubes in each layer and that there are two layers. This means that the total number of centimetre cubes in this cuboid is $24$.
You say that the volume of the cuboid is $24 \text{ cm}^3$.
You can use algebra to work out the volume of a cuboid, using the formula:
Volume = length × width × height or $V = l \times w \times h$
If the sides of a cuboid are measured in millimetres, the volume will be in cubic millimetres (mm³). If the sides of a cuboid are measured in metres, the volume will be in cubic metres (m³).
❓ EXERCISES
💡 Tip
Remember: in a cube the length, width and height are all the same.
1. Copy and complete the workings to find the volume of these cubes.
a.
Volume = length × width × height
= $3 \times 3 \times 3$
= $\_\_\_\_\ \text{cm}^3$
b.
Volume = length × width × height
= $5 \times \_\_\_\_ \times \_\_\_\_$
= $\_\_\_\_\ \text{m}^3$
👀 Show answer
b. $5 \times 5 \times 5 = 125\ \text{m}^3$
2. Copy and complete the workings to find the volume of these cuboids.
a.
Volume = length × width × height
= $10 \times 6 \times 4$
= $\_\_\_\_\ \text{cm}^3$
b.
Volume = length × width × height
= $12 \times \_\_\_\_ \times \_\_\_\_$
= $\_\_\_\_\ \text{mm}^3$
👀 Show answer
b. $12 \times 8 \times 5 = 480\ \text{mm}^3$
3. Work out the volume of each of these cuboids.
a.
b.
c.
👀 Show answer
b. $5 \times 6 \times 3 = 90\ \text{cm}^3$
c. $9 \times 6 \times 1 = 54\ \text{cm}^3$
❓ EXERCISES
4. This is part of Steph’s homework.
incorrect.
Explain the mistake that Steph has made and work out the correct answer.
👀 Show answer
Mistake: Steph multiplied lengths in mixed units (cm and mm). All three dimensions must be in the same unit before calculating volume.
Correct working (in cm): convert $35\ \text{mm}=3.5\ \text{cm}$.
$V = 12 \times 9 \times 3.5 = 378\ \text{cm}^3$.
Check (in mm):$120\ \text{mm} \times 90\ \text{mm} \times 35\ \text{mm} = 378{,}000\ \text{mm}^3$, and $378{,}000\ \text{mm}^3 \div 1000 = 378\ \text{cm}^3$.
💡 Tip
Make sure the length, width and height are all in the same units before you work out the volume.
5. The table shows the length, width and height of four cuboids.
Copy and complete the table.
| Length | Width | Height | Volume |
|---|---|---|---|
| $5\ \text{cm}$ | $12\ \text{mm}$ | $6\ \text{mm}$ | $\_\_\_\_\ \text{mm}^3$ |
| $12\ \text{cm}$ | $8\ \text{cm}$ | $4\ \text{cm}$ | $\_\_\_\_\ \text{cm}^3$ |
| $8\ \text{m}$ | $6\ \text{m}$ | $90\ \text{cm}$ | $\_\_\_\_\ \text{m}^3$ |
| $1.2\ \text{m}$ | $60\ \text{cm}$ | $25\ \text{cm}$ | $\_\_\_\_\ \text{cm}^3$ |
👀 Show answer
5 (table) answers
a. Convert $5\ \text{cm}=50\ \text{mm}$. Volume: $50 \times 12 \times 6 = 3600\ \text{mm}^3$.
b. $12 \times 8 \times 4 = 384\ \text{cm}^3$.
c. Convert $90\ \text{cm}=0.9\ \text{m}$. Volume: $8 \times 6 \times 0.9 = 43.2\ \text{m}^3$.
d. Convert $1.2\ \text{m}=120\ \text{cm}$. Volume: $120 \times 60 \times 25 = 180{,}000\ \text{cm}^3$.
❓ EXERCISES
6. Look at this compound shape.
a. Write down the value of $x$.
b. Copy and complete the workings to find the volume of the shape.
Top cuboid: $V = l \times w \times h = 5 \times \square \times 3$
= $\square\ \text{cm}^3$
Bottom cuboid: $V = l \times w \times h = 11 \times 4 \times 6$
= $\square\ \text{cm}^3$
Volume of shape: $\square + \square = \square\ \text{cm}^3$
👀 Show answer
6a. The missing depth is common to both parts and is shown as $4\ \text{cm}$ on the base, so $x = 4\ \text{cm}$.
6b. Top cuboid: $5 \times 4 \times 3 = 60\ \text{cm}^3$.
Bottom cuboid: $11 \times 4 \times 6 = 264\ \text{cm}^3$.
Total volume: $60 + 264 = 324\ \text{cm}^3$.
7. Work out the volume of these compound shapes.
a.
b.
👀 Show answer
7a. Split into base and top.
Base: $15 \times 6 \times 5 = 450\ \text{m}^3$ (length × depth × height).
Top block: height is $12 - 5 = 7\ \text{m}$, so $8 \times 6 \times 7 = 336\ \text{m}^3$.
Total: $450 + 336 = 786\ \text{m}^3$.
7b. Split into top bar and central stem (dimensions in mm).
Top bar: $20 \times 6 \times 7 = 840\ \text{mm}^3$.
Stem: width $6$, depth $6$, height $9$ ⇒ $6 \times 6 \times 9 = 324\ \text{mm}^3$.
Total: $840 + 324 = 1164\ \text{mm}^3$.
🧠 Think like a Mathematician
Amadeo tries different values for the height, until he gets the correct volume.
a. What are the disadvantages of this method?
b. Discuss different methods that Amadeo could use. What are the advantages of these methods? Which is the best method?
👀 Show Answer
- a. The trial-and-error method is inefficient and can take a long time if the numbers are large. It also relies on guessing and checking, which is not precise.
- b. Amadeo could solve the equation directly by dividing: $h = \dfrac{168}{21} = 8$. This is faster and more accurate. Using algebra or rearranging formulas is the best method because it always gives the exact answer without unnecessary steps.
❓ EXERCISES
9. A metal cuboid has a length of $75\ \text{mm}$, a height of $4\ \text{mm}$ and a volume of $1500\ \text{mm}^3$.
What is the width of the cuboid?
👀 Show answer
10. A cuboid has a length of $6\ \text{cm}$, a width of $4\ \text{cm}$ and a height of $5\ \text{cm}$.
a. What is the volume of the cuboid?
b. Write down the dimensions of three other cuboids that have the same volume.
👀 Show answer
a.$V=6\times4\times5=120\ \text{cm}^3$.
b. Examples (all give $120\ \text{cm}^3$):
$10\times3\times4$, $12\times5\times2$, $20\times3\times2$.
11. The diagram shows a shape made from gold. The shape is melted and made into gold cubes. The side length of each cube is $12\ \text{mm}$.
How many whole cubes can be made from this shape?
👀 Show answer
Split into two cuboids with common depth $20\ \text{mm}$:
Left block: $30\times20\times50=30{,}000\ \text{mm}^3$.
Right block: $30\times20\times25=15{,}000\ \text{mm}^3$.
Total volume: $45{,}000\ \text{mm}^3$.
Each cube volume: $12^3=1{,}728\ \text{mm}^3$.
Number of whole cubes: $\left\lfloor\dfrac{45{,}000}{1{,}728}\right\rfloor=26$.
❓ EXERCISES
12a. Copy and complete this table.
| Side length of cube | Volume of cube | Power notation | Value |
|---|---|---|---|
| $2\ \text{cm}$ | $2 \times 2 \times 2$ | $2^3$ | $8\ \text{cm}^3$ |
| $3\ \text{cm}$ | $3 \times 3 \times 3$ | $3^3$ | $27\ \text{cm}^3$ |
| $4\ \text{cm}$ | $4 \times 4 \times 4$ | $4^3$ | $\_\_\_\_\ \text{cm}^3$ |
| $5\ \text{cm}$ | $5 \times 5 \times 5$ | $5^3$ | $\_\_\_\_\ \text{cm}^3$ |
👀 Show answer
12b. Sofia says: “If you know the volume of a cube, you can find the side length of the cube by working out the cube root of the volume.”
Is Sofia correct? Explain your answer. Use your table in part a to help with your explanation.
👀 Show answer
🧠 Reasoning Tip
If you know the volume of a cube, you can find the side length by working out the cube root of the volume.
12c. Work out the side length of a cube with volume:
i. $1000\ \text{cm}^3$
ii. $216\ \text{cm}^3$
👀 Show answer
ii. $\sqrt[3]{216} = 6\ \text{cm}$