Dimensional Analysis

Dimensional analysis is a method of converting between units by multiplying by conversion factors, arranged so that the units you don't want cancel out and the units you do want remain.

Dimensional analysis is a systematic technique for unit conversion in which a quantity is multiplied by one or more conversion factors — fractions equal to 1 — such that undesired units appear in both the numerator and denominator and therefore cancel. The process relies on the principle that multiplying by a fraction whose numerator and denominator are equivalent (e.g., $\frac{1 \text{ km}}{1000 \text{ m}}$ ) does not change the value of the quantity, only its units.

Key Formula

\text{Starting value} \times \frac{\text{Desired unit}}{\text{Starting unit}} = \text{Converted value}

Where:

$Starting value$ = the number with its original unit
$Desired unit / Starting unit$ = a conversion factor equal to 1, oriented so the starting unit cancels
$Converted value$ = the result in the desired unit

Worked Example

Problem: A car is travelling at 90 kilometres per hour. Convert this speed to metres per second.

Step 1: Write the starting value with its units.

90 \;\frac{\text{km}}{\text{hr}}

Step 2: Convert kilometres to metres. Since 1 km = 1000 m, multiply by a fraction that puts km in the denominator so it cancels.

90 \;\frac{\cancel{\text{km}}}{\text{hr}} \times \frac{1000 \;\text{m}}{1 \;\cancel{\text{km}}}

Step 3: Convert hours to seconds. Since 1 hr = 3600 s, multiply by a fraction that puts hr in the numerator so it cancels.

90 \times \frac{1000 \;\text{m}}{\cancel{\text{hr}}} \times \frac{1 \;\cancel{\text{hr}}}{3600 \;\text{s}}

Step 4: Multiply the numbers and confirm only the desired units remain.

\frac{90 \times 1000}{3600} \;\frac{\text{m}}{\text{s}} = 25 \;\frac{\text{m}}{\text{s}}

Answer: 90 km/hr equals 25 m/s.

Why It Matters

Dimensional analysis is one of the most widely used skills in science, engineering, and medicine. Pharmacists use it to calculate drug dosages, chemists use it to convert between moles and grams, and engineers use it whenever quantities involve mixed unit systems. Learning this technique now gives you a reliable, repeatable method for any unit conversion you'll encounter in physics, chemistry, or everyday life.

Common Mistakes

Mistake: Placing the conversion factor upside down, so the unwanted unit does not cancel.

Correction: Always check that the unit you want to eliminate appears once in the numerator and once in the denominator. If the starting unit is in the numerator, your conversion factor must have that same unit in the denominator.

Mistake: Forgetting to convert all units when a quantity has more than one (e.g., converting km/hr to m/s but only changing km to m and leaving hr unchanged).

Correction: Count every unit in your starting expression and set up a separate conversion factor for each one that needs to change.

Related Terms

Ratio — Conversion factors are ratios equal to 1