Dimensional Analysis: Converting Units by Chaining Conversion Factors
October 21, 2026
Dimensional analysis, also called the factor-label method, is a systematic way to convert between units. You multiply the quantity by one or more conversion factors, each written as a fraction equal to 1 (such as 5,280 feet per 1 mile), arranged so that unwanted units cancel and desired units remain. Because each factor equals 1, you do not change the quantity, only how it is expressed.
The method works for any conversion, however complicated, and it protects against the most common error: multiplying when you should divide.
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Write the starting quantity with its units. Then write a conversion factor as a fraction with the unit you want to cancel in the denominator (opposite position to where it is in your quantity) and the new unit in the numerator. Repeat until you reach the desired unit. Cancel units as you go, then multiply the numbers on top and divide by the numbers on the bottom.
Example: convert 60 miles per hour to feet per second. 60 mi/h × (5,280 ft / 1 mi) × (1 h / 3,600 s) = 60 × 5,280 ÷ 3,600 ft/s = 88 ft/s. Miles cancel, hours cancel, and feet per second remain.
More Worked Examples
Flow rate: a hose delivers 5 gallons per minute. How many liters per hour? 5 gal/min × (3.78541 L / 1 gal) × (60 min / 1 h) = 1,135.6 L/h. Density: a material weighs 2,400 kg per cubic meter. What is that in pounds per cubic foot? 2,400 kg/m³ × (2.20462 lb / 1 kg) × (1 m³ / 35.3147 ft³) = 149.8 lb/ft³. Fuel use: 35 miles per gallon to kilometers per liter: 35 mi/gal × (1.609344 km / 1 mi) × (1 gal / 3.78541 L) = 14.88 km/L.
Cooking: a recipe uses 250 grams of flour and your scale reads in ounces. 250 g × (1 oz / 28.3495 g) = 8.82 oz.
| Conversion | Chain | Result |
|---|---|---|
| 60 mph to ft/s | 60 x 5,280 / 3,600 | 88 ft/s |
| 5 gal/min to L/h | 5 x 3.78541 x 60 | 1,135.6 L/h |
| 2,400 kg/m3 to lb/ft3 | 2,400 x 2.20462 / 35.3147 | 149.8 lb/ft3 |
| 35 mpg to km/L | 35 x 1.609344 / 3.78541 | 14.88 km/L |
| 250 g to oz | 250 / 28.3495 | 8.82 oz |
Squared and Cubed Units
When units are squared or cubed, the conversion factor must be too. One meter is 3.28084 feet, so one square meter is 3.28084² = 10.7639 square feet, and one cubic meter is 3.28084³ = 35.3147 cubic feet. To convert 120 square meters to square feet: 120 m² × (3.28084 ft / 1 m)² = 120 × 10.7639 = 1,291.7 ft². Forgetting to square the factor is a major source of error.
Sanity Checks
After converting, ask whether the answer makes sense. Feet are smaller than meters, so a number of feet should be larger than the same length in meters. If you converted 100 meters and got 30.5, you divided instead of multiplied. Use rough benchmarks: 1 mile ≈ 1.6 km, 1 kg ≈ 2.2 lb, 1 gallon ≈ 3.8 L, 1 inch = 2.54 cm.
Keep extra digits during the calculation and round at the end to avoid compounding rounding errors.
Where It Matters
Medicine (dose calculations in mg per kg per hour), engineering, chemistry, aviation, and cooking all rely on correct unit conversion. Notable disasters, like the loss of the Mars Climate Orbiter in 1999 because of a pound-force versus newton mismatch, show the cost of getting it wrong. Writing units on every number is a simple habit that prevents mistakes.
Frequently Asked Questions
What is dimensional analysis?
A method of converting units by multiplying by conversion factors arranged so that unwanted units cancel.
How do I convert square units?
Square the linear conversion factor, such as 3.28084² for square meters to square feet.
How do I know which way to write the conversion factor?
Put the unit you want to cancel in the opposite position (numerator or denominator) from where it appears in your quantity.
What is 60 mph in feet per second?
88 ft/s.
How can I check my conversion?
Make sure the units cancel correctly and the size of the answer makes sense.