Area, volume and surface area
Converting between units for area
A square of side 1 cm holds one hundred 1 mm squares — not ten. When you convert an area, the conversion factor gets squared too.
Squares of each size
A square with sides of 1 mm has an area of 1 SQUARE MILLIMETRE (1 mm²). A square with sides of 1 cm has an area of 1 SQUARE CENTIMETRE (1 cm²). A square with sides of 1 m has an area of 1 SQUARE METRE (1 m²). The little ² means the unit is a square, not a length.
Why the factor is squared
Take the 1 cm square and cut it into 1 mm squares. 1 cm = 10 mm, so it is 10 squares across AND 10 squares down. 10 × 10 = 100 1 cm² = 100 mm² The length factor is 10, but the area factor is 10 × 10 = 100. This is the step everyone forgets.
The two conversion factors
1 cm² = 100 mm² 1 m² = 10 000 cm² The second comes from 1 m = 100 cm, so 100 × 100 = 10 000. Going to SMALLER units, multiply. Going to BIGGER units, divide. 5 cm² = 5 × 100 = 500 mm² 600 mm² = 600 ÷ 100 = 6 cm²
Choosing sensible units
Pick the unit you would use for a LENGTH of the object, then square it. • A postage stamp — you'd measure it in cm, so its area is in cm². • A football pitch — you'd measure it in m, so its area is in m². • A pinhead — mm, so mm². An answer of 4 500 000 mm² for a football pitch is not wrong, but it is not sensible.
Worked examples
What units for the area of a football pitch?
- You would measure its length in metres.
- So the area is in metres squared.
Answer: Square metres, m²
5 cm² in square millimetres
- 1 cm² = 100 mm².
- Going to smaller units, so multiply.
- 5 × 100 = 500.
Answer: 500 mm²
600 mm² in square centimetres
- Going to bigger units, so divide.
- 600 ÷ 100 = 6.
Answer: 6 cm²
3 m² in square centimetres
- 1 m² = 10 000 cm².
- 3 × 10 000 = 30 000.
Answer: 30 000 cm²
70 000 cm² in square metres
- Divide by 10 000.
- 70 000 ÷ 10 000 = 7.
Answer: 7 m²
Why isn't 1 cm² equal to 10 mm²?
- 1 cm = 10 mm is a LENGTH conversion.
- An area is 10 across by 10 down.
- 10 × 10 = 100.
Answer: Because the factor is squared: 1 cm² = 100 mm²
Where you see this in real life
- Flooring is priced per square metre, so a room measured in centimetres has to be converted before you can order.
- Microchip features are quoted in square millimetres, because square centimetres would all round to zero.
Key words
- Area
- the amount of surface a shape covers
- Square centimetre
- the area of a square with 1 cm sides, written cm²
- Conversion factor
- the number you multiply or divide by to change units
- Square millimetre
- the area of a square with 1 mm sides, written mm²
- Dimensions
- the measurements of a shape, such as length and width