Let the units do the work
Dimensional analysis is a method for converting units by multiplying by conversion factors: fractions equal to 1. Units cancel like numbers, which shows you whether you set things up correctly.
Conversion factors
Every equivalence makes two factors:
1 m = 100 cm → (100 cm / 1 m) or (1 m / 100 cm)
Both equal 1, so multiplying by either doesn’t change the quantity, only its units. Choose the one that cancels the unit you’re getting rid of.
One-step conversion
Example: Convert 3.2 km to m.
3.2 km × (1,000 m / 1 km) = 3,200 m
km cancels, leaving m ✓.
Multi-step conversion
Example: How many seconds are in 2 days?
2 days × (24 h / 1 day) × (60 min / 1 h) × (60 s / 1 min) = 172,800 s
Days, hours and minutes all cancel.
Converting compound units
Compound units (also called derived units) like km/h need two conversions: one for the top unit and one for the bottom.
Example: Convert 72 km/h to m/s.
72 km/h × (1,000 m / 1 km) × (1 h / 3,600 s) = 72,000/3,600 = 20 m/s
Area and volume
Square or cube the factor.
Example: 2.5 m² to cm²
2.5 m² × (100 cm / 1 m)² = 2.5 × 10,000 = 25,000 cm²
Unit rates and money
Example: Gas costs $1.62/L. A car uses 7.5 L/100 km. Cost to drive 360 km?
360 km × (7.5 L / 100 km) × ($1.62 / 1 L) = $43.74
The setup shows exactly which numbers multiply and which divide.
Imperial conversions
Example: 26.2 miles (a marathon) in km. 1 mile ≈ 1.609 km.
26.2 mi × (1.609 km / 1 mi) ≈ 42.2 km
Checking with units
If the units don’t cancel to what you want, the setup is wrong. That’s dimensional analysis’s best feature: it catches mistakes.
Worked examples
Example 1: 5,400 mL to L: 5,400 mL × (1 L / 1,000 mL) = 5.4 L.
Example 2: 15 m/s to km/h: 15 × 3,600/1,000 = 54 km/h.
Example 3: A heart beats 70 times per minute. Beats in a day? 70 × 60 × 24 = 100,800.
Common mistakes
- Flipping a conversion factor so units don’t cancel.
- Converting only the top unit of a compound unit.
- Forgetting to square or cube for area and volume.
Try it yourself
- 0.75 h to seconds
- 90 km/h to m/s
- 3 m³ to L (1 m³ = 1,000 L)
- $12/h to $ per minute
- 50 lb to kg (1 kg ≈ 2.2 lb)
Answers: 1) 2,700 s 2) 25 m/s 3) 3,000 L 4) $0.20/min 5) about 22.7 kg
Quick recap
- Dimensional analysis multiplies by conversion factors equal to 1.
- Units cancel like numbers; set up factors so the unwanted unit cancels.
- Convert both parts of compound units.