Units and Conversions

Being able to convert between different units of measurement is a fundamental skill tested throughout the AQA GCSE Mathematics course. You need to convert between metric units, understand common imperial units, and convert between compound units. This lesson provides all the key conversions and methods you need.

---

Metric Units of Length

Unit	Abbreviation	Relationship
Kilometre	km	1 km = 1,000 m
Metre	m	1 m = 100 cm
Centimetre	cm	1 cm = 10 mm
Millimetre	mm	Base unit for small measurements

Converting Metric Lengths

graph LR
    A[km] -->|x 1000| B[m]
    B -->|x 100| C[cm]
    C -->|x 10| D[mm]
    D -->|/ 10| C
    C -->|/ 100| B
    B -->|/ 1000| A

• To convert from a larger unit to a smaller unit, multiply.
• To convert from a smaller unit to a larger unit, divide.

Worked Example 1: Convert 3.7 km to metres.

• 3.7 x 1,000 = 3,700 m

Worked Example 2: Convert 450 cm to metres.

• 450 / 100 = 4.5 m

Worked Example 3: Convert 2.85 m to mm.

• 2.85 x 100 = 285 cm
• 285 x 10 = 2,850 mm
• Or in one step: 2.85 x 1,000 = 2,850 mm

Exam Tip: A common error is converting the wrong way — multiplying when you should divide or vice versa. Ask yourself: "Am I going to a bigger or smaller unit?" Smaller unit = more of them = multiply.

---

Metric Units of Mass

Unit	Abbreviation	Relationship
Tonne	t	1 tonne = 1,000 kg
Kilogram	kg	1 kg = 1,000 g
Gram	g	1 g = 1,000 mg
Milligram	mg	Base unit for tiny measurements

Worked Example 4: Convert 5.2 kg to grams.

• 5.2 x 1,000 = 5,200 g

Worked Example 5: Convert 3,750 g to kg.

• 3,750 / 1,000 = 3.75 kg

---

Metric Units of Capacity and Volume

Unit	Abbreviation	Relationship
Litre	l	1 l = 1,000 ml
Millilitre	ml	1 ml = 1 cm cubed
Centilitre	cl	1 cl = 10 ml

Key fact: 1 litre = 1,000 cm cubed = 1,000 ml

Worked Example 6: A fish tank holds 45,000 cm cubed of water. How many litres is this?

• 45,000 / 1,000 = 45 litres

---

Metric Units of Area and Volume

Converting area and volume units requires squaring or cubing the conversion factor.

Conversion	Calculation
1 m squared = ? cm squared	100 x 100 = 10,000 cm squared
1 km squared = ? m squared	1,000 x 1,000 = 1,000,000 m squared
1 m cubed = ? cm cubed	100 x 100 x 100 = 1,000,000 cm cubed
1 cm squared = ? mm squared	10 x 10 = 100 mm squared

Worked Example 7: Convert 3.5 m squared to cm squared.

• 3.5 x 10,000 = 35,000 cm squared

Worked Example 8: Convert 0.002 m cubed to cm cubed.

• 0.002 x 1,000,000 = 2,000 cm cubed

Exam Tip: When converting area, square the conversion factor. When converting volume, cube it. This is one of the most common sources of errors on the exam — students often just multiply by 100 instead of 10,000 for area.

---

Common Imperial Units

You do not need to memorise all imperial conversions, but you must know the approximate metric equivalents given in the table below (AQA provides these in the exam when needed).

Imperial Unit	Approximate Metric Equivalent
1 inch	2.5 cm
1 foot	30 cm
1 mile	1.6 km
1 pound (lb)	450 g
1 ounce (oz)	28 g
1 pint	568 ml (approximately 0.57 litres)
1 gallon	4.5 litres

Worked Example 9: A road sign says "London 80 miles". How far is this in kilometres?

• 80 x 1.6 = 128 km

Worked Example 10: A recipe calls for 2 pounds of flour. How many grams is this approximately?

• 2 x 450 = 900 g

---

Converting Compound Units

Compound units involve two different measures. Converting them requires you to apply the conversion to each part of the unit.

Speed Conversions

Worked Example 11: Convert 72 km/h to m/s.

Step 1: Convert km to m: 72 km = 72,000 m Step 2: Convert hours to seconds: 1 hour = 3,600 seconds Step 3: 72,000 / 3,600 = 20 m/s

Short cut: Divide km/h by 3.6 to get m/s.

Worked Example 12: Convert 25 m/s to km/h.

• 25 x 3.6 = 90 km/h

Density Conversions

Worked Example 13: Convert a density of 2.5 g/cm cubed to kg/m cubed.

Step 1: 1 g = 0.001 kg, so multiply mass by 1,000 to convert g to kg per m cubed Step 2: 1 cm cubed = 0.000001 m cubed, or equivalently, 1 m cubed = 1,000,000 cm cubed Step 3: 2.5 g/cm cubed = 2.5 x 1,000,000 g/m cubed = 2,500,000 g/m cubed = 2,500 kg/m cubed

Short cut: Multiply g/cm cubed by 1,000 to get kg/m cubed.

Exam Tip: The shortcut for density conversion is: g/cm cubed x 1,000 = kg/m cubed. This appears frequently in exam questions that combine shapes with density calculations.

---

Time Conversions

Conversion	Value
1 minute	60 seconds
1 hour	60 minutes = 3,600 seconds
1 day	24 hours
1 week	7 days
1 year	365 days (approximately)