Converting between units for area
🎯 In this topic you will
- Convert between metric units of area.
🧠 Key Words
- area
- centimetre (cm²)
- conversion factor
- compound shape
- dimensions
- square millimetre (mm²)
- square metre (m²)
Show Definitions
- area: The amount of space inside a 2D shape, measured in square units.
- centimetre (cm²): A unit of area equal to a square that is 1 centimetre on each side.
- conversion factor: A number used to change one unit of measurement into another.
- compound shape: A shape made up of two or more simple geometric shapes.
- dimensions: Measurements of length, width, or height that define the size of a shape.
- square millimetre (mm²): A unit of area equal to a square that is 1 millimetre on each side.
- square metre (m²): A unit of area equal to a square that is 1 metre on each side.
The diagram shows three squares.
- The first square has a side length of $1$ mm.
- The second square has a side length of $1$ cm.
- The third square has a side length of $1$ m.
- The first square has an area of 1 square millimetre (1 mm$^2$).
- The second square has an area of 1 square centimetre (1 cm$^2$).
- The third square has an area of 1 square metre (1 m$^2$).
To convert between units of area you need to know the conversion factors.
Look at the square with a side length of $1$ cm and area $1 \text{ cm}^2$.
If you divide the square into squares with side length $1$ mm, you get $10 \times 10 = 100$ of these smaller squares.
This shows that:
$1 \text{ cm}^2 = 100 \text{ mm}^2$
You can do the same with the square with a side length of $1$ m and area $1 \text{ m}^2$.
If you divide the square up into squares with side length $1$ cm, you get $100 \times 100 = 10000$ of these smaller squares.
This shows that:
$1 \text{ m}^2 = 10000 \text{ cm}^2$
When you measure the area of a shape, decide which units you would use to measure a length of the shape; for example, cm. You then measure the area of the shape in those units squared, so cm$^2$.
❓ EXERCISES
1. What units would you use to measure the area of:
a. a postage stamp? b. a bank note? c. a tennis court? d. a cinema screen?
👀 Show answer
b. $ \text{cm}^2 $ (square centimetres)
c. $ \text{m}^2 $ (square metres)
d. $ \text{m}^2 $ (square metres)
💡 Tip
Use the conversion factor $1 \text{ cm}^2 = 100 \text{ mm}^2$.
2. Copy and complete these area conversions between cm$^2$ and mm$^2$.
a. $8 \text{ cm}^2 = \_\_\_\_ \text{ mm}^2$ $8 \times 100 = \_\_\_\_ \text{ mm}^2$
b. $0.75 \text{ cm}^2 = \_\_\_\_ \text{ mm}^2$ $0.75 \times \_\_\_\_ = \_\_\_\_ \text{ mm}^2$
c. $600 \text{ mm}^2 = \_\_\_\_ \text{ cm}^2$ $600 \div 100 = \_\_\_\_ \text{ cm}^2$
d. $45 \text{ mm}^2 = \_\_\_\_ \text{ cm}^2$ $45 \div \_\_\_\_ = \_\_\_\_ \text{ cm}^2$
👀 Show answer
b. $0.75 \text{ cm}^2 = 75 \text{ mm}^2$
c. $600 \text{ mm}^2 = 6 \text{ cm}^2$
d. $45 \text{ mm}^2 = 0.45 \text{ cm}^2$
💡 Tip
Use the conversion factor $1 \text{ m}^2 = 10000 \text{ cm}^2$.
3. Copy and complete these area conversions between m$^2$ and cm$^2$.
a. $3 \text{ m}^2 = \_\_\_\_ \text{ cm}^2$ $3 \times 10000 = \_\_\_\_ \text{ cm}^2$
b. $8.1 \text{ m}^2 = \_\_\_\_ \text{ cm}^2$ $8.1 \times \_\_\_\_ = \_\_\_\_ \text{ cm}^2$
c. $70000 \text{ cm}^2 = \_\_\_\_ \text{ m}^2$ $70000 \div 10000 = \_\_\_\_ \text{ m}^2$
d. $780 \text{ cm}^2 = \_\_\_\_ \text{ m}^2$ $780 \div \_\_\_\_ = \_\_\_\_ \text{ m}^2$
👀 Show answer
b. $8.1 \text{ m}^2 = 81000 \text{ cm}^2$
c. $70000 \text{ cm}^2 = 7 \text{ m}^2$
d. $780 \text{ cm}^2 = 0.078 \text{ m}^2$
🧠 Think like a Mathematician
Marcus says:
“I never know when I need to multiply or to divide by the conversion factor.”
Task: Discuss a strategy that Marcus could use to help him decide when he should multiply and when he should divide by the conversion factor.
👀 Show Answer
- Strategy: If you are converting from a larger unit to a smaller unit (e.g., cm$^2$ to mm$^2$), you will always need more of the smaller units, so you multiply by the conversion factor.
- If you are converting from a smaller unit to a larger unit (e.g., mm$^2$ to cm$^2$), you will always need fewer of the larger units, so you divide by the conversion factor.
- A useful check: ask “Will the number get bigger or smaller?” If it should get bigger, multiply. If it should get smaller, divide.
❓ EXERCISES
5. Copy and complete the following area conversions. Show your working.
a. $6 \text{ cm}^2 = \_\_\_\_ \text{ mm}^2$
b. $7.2 \text{ cm}^2 = \_\_\_\_ \text{ mm}^2$
c. $3 \text{ m}^2 = \_\_\_\_ \text{ cm}^2$
d. $5.4 \text{ m}^2 = \_\_\_\_ \text{ cm}^2$
e. $900 \text{ mm}^2 = \_\_\_\_ \text{ cm}^2$
f. $865 \text{ mm}^2 = \_\_\_\_ \text{ cm}^2$
g. $20000 \text{ cm}^2 = \_\_\_\_ \text{ m}^2$
h. $48000 \text{ cm}^2 = \_\_\_\_ \text{ m}^2$
i. $125000 \text{ cm}^2 = \_\_\_\_ \text{ m}^2$
👀 Show answer
b. $720 \text{ mm}^2$
c. $30000 \text{ cm}^2$
d. $54000 \text{ cm}^2$
e. $9 \text{ cm}^2$
f. $8.65 \text{ cm}^2$
g. $2 \text{ m}^2$
h. $4.8 \text{ m}^2$
i. $12.5 \text{ m}^2$
6. Suyin and Tam use algebra to work out the area of this rectangle, in cm$^2$.
a. Who has the correct answer, Suyin or Tam?
b. Explain the mistake that the other person has made.
c. Which method do you prefer: the method used by Suyin or the method used by Tam? Explain why.
👀 Show answer
b. Tam’s mistake was not converting both dimensions to the same unit before multiplying.
c. Suyin’s method is preferable because it consistently converts both measurements into the same unit (cm).
7. Work out the area of the rectangle shown. Give your answer in:
a. mm$^2$ b. cm$^2$
👀 Show answer
💡 Tip
Start by dividing the compound shape into two rectangles.
8. Work out the area of this compound shape. Give your answer in:
a. mm$^2$ b. cm$^2$
👀 Show answer
Top rectangle: $10 \text{ cm} \times 2 \text{ cm} = 20 \text{ cm}^2$
Bottom rectangle: $2 \text{ cm} \times 3.2 \text{ cm} = 6.4 \text{ cm}^2$
Total = $26.4 \text{ cm}^2 = 2640 \text{ mm}^2$.
❓ EXERCISES
9. Zara says: “An area of $0.25 \text{ m}^2$ is the same as $25000 \text{ mm}^2$.”
Is Zara correct? Explain your answer.
👀 Show answer
$1 \text{ m}^2 = 100 \text{ cm} \times 100 \text{ cm} = 10000 \text{ cm}^2$
$1 \text{ cm}^2 = 100 \text{ mm}^2$ so $1 \text{ m}^2 = 10000 \times 100 = 1,000,000 \text{ mm}^2$.
Therefore $0.25 \text{ m}^2 = 250,000 \text{ mm}^2$, not $25,000 \text{ mm}^2$.
10. Sven is going to lay tiles on the floor of his bathroom.
The diagram shows the dimensions of the floor and the dimension of one tile.
a. Show that Sven needs 80 tiles to cover the bathroom floor.
b. Discuss the method you used to show part a.
c. Did all of you use the same method? Which method do you think is the best?
👀 Show answer
Each tile area: $20 \text{ cm} \times 20 \text{ cm} = 400 \text{ cm}^2 = 0.04 \text{ m}^2$.
Number of tiles: $3.2 \div 0.04 = 80$.
b. Method: Convert both areas into the same unit, then divide floor area by tile area.
c. Different students might use different unit conversions (all to cm$^2$ or all to m$^2$), but the method of dividing total area by tile area is the best approach.