Units and Conversion

Units of Measurement and Conversion

Length Units

$\$1 \text{ km} = 1000 \text{ m} = 100{,}000 \text{ cm} = 1{,}000{,}000 \text{ mm}$$

To convert: multiply when going to smaller units, divide when going to larger units.

Area Units

Area conversions use the square of the length conversion factor.

$\$1 \text{ m}^2 = 10{,}000 \text{ cm}^2 \quad (= 100^2)$$
$\$1 \text{ km}^2 = 1{,}000{,}000 \text{ m}^2 \quad (= 1000^2)$$
$\$1 \text{ hectare (ha)} = 10{,}000 \text{ m}^2$$

Volume Units

Volume conversions use the cube of the length factor.

$\$1 \text{ m}^3 = 1{,}000{,}000 \text{ cm}^3 \quad (= 100^3)$$
$\$1 \text{ L} = 1000 \text{ mL} = 1000 \text{ cm}^3$$
$\$1 \text{ m}^3 = 1000 \text{ L}$$

Worked Example — Area Conversion

A paddock measures \$2.5 \text{ km} \times 1.8 \text{ km}$.

Area $= 2.5 \times 1.8 = 4.5 \text{ km}^2$.

Convert to hectares: \$4.5 \text{ km}^2 \times 100 \text{ ha/km}^2 = 450 \text{ ha}$.

(Note: \$1 \text{ km}^2 = 100 \text{ ha}$ since \$1 \text{ km}^2 = 1{,}000{,}000 \text{ m}^2$ and \$1 \text{ ha} = 10{,}000 \text{ m}^2$, so $\frac{1{,}000{,}000}{10{,}000} = 100$.)

Worked Example — Volume to Capacity

A rectangular tank is $2$ m long, $1.5$ m wide, and $0.8$ m deep.

$$V = 2 \times 1.5 \times 0.8 = 2.4 \text{ m}^3$$

Convert to litres: \$2.4 \times 1000 = 2400 \text{ L}$.

Common Error: Forgetting to Square or Cube

Converting 3 m to cm: multiply by 100 → 300 cm. Correct.

Converting 3 m² to cm²: multiply by $100^2 = 10{,}000$ → 30,000 cm². NOT by 100.

COMMON MISTAKE: Applying the length conversion factor to area or volume. Always square the factor for area, cube it for volume.

EXAM TIP: Write out the conversion factor explicitly and check your units cancel correctly: \$3 \text{ m}^2 \times \frac{10{,}000 \text{ cm}^2}{1 \text{ m}^2} = 30{,}000 \text{ cm}^2$.