Dimensional Analysis Calculator
Enter a starting quantity and a chain of conversion factors and see the factor-label method done for you: each factor is turned the right way up, units cancel with a line through them, and the units left over are checked against the ones you wanted. Free, no sign-up.
Write units with / and ·, and powers as ^2 or ²: mi/h, kg·m/s², J/(mol·K), cm3.
Fill this in to have the answer's units checked.
Conversion factors, in order
Each row is an equality such as 1 mi = 1609.344 m. The calculator turns it the right way up to cancel units.
Results
26.8 m/s
Answer
m/s
Units of the answer
Unrounded: 26.8224
The units match the target, m/s.
Exact conversion factors (definitions) do not limit the significant figures. Untick "Exact" for a measured factor such as a molar mass.
How to Use the Dimensional Analysis Calculator
- 1. Enter what you start with. A number and its units, for example 60.0 and mi/h. Units can be compound: kg·m/s², J/(mol·K), cm3.
- 2. Name the units you want (optional). The calculator checks the answer's units against them and says if something is left over.
- 3. Add the conversion factors. Each row is an equality — 1 mi = 1609.344 m — typed or picked from a menu of common exact conversions. Put them in the order you would write the chain; the calculator turns each one over as needed.
- 4. Mark measured factors. Factors are exact by default. Untick Exact for a measured quantity such as a molar mass, so it counts toward the significant figures.
The Factor-Label Method
Dimensional analysis treats units like algebraic quantities. Every conversion factor is a fraction equal to one, because its numerator and denominator are the same amount in different units — 1609.344 m / 1 mi. Multiplying by it changes the units without changing the quantity. Choose the orientation that puts the unit you want to remove on the opposite side from where it stands, and it cancels.
Chain
start × (new units / old units) × (newer units / new units) × …
Check
units left over = the units you wanted
Worked Examples
Speed (the default inputs). Convert 60.0 mi/h to m/s: 60.0 mi/h × (1609.344 m / 1 mi) × (1 h / 3600 s). Miles cancel, hours cancel, and metres per second remain: 60.0 × 1609.344 ÷ 3600 = 26.8224, which is 26.8 m/s to three significant figures. Both factors are exact definitions, so only the 60.0 limits the answer.
Grams to formula units. How many formula units are in 25.0 g of NaCl? 25.0 g × (1 mol / 58.44 g) × (6.02214076 × 10²³ particles / 1 mol) = 2.58 × 10²³ particles. Here 58.44 g/mol is a measured value, so untick Exact on that row; the Avogadro constant is exact. The Grams to Moles Converter specialises in this chain.
Powers of a unit. To convert 1.00 g/cm³ to kg/m³, a cubed unit needs its factor three times: (1 kg / 1000 g) × (100 cm / 1 m) × (100 cm / 1 m) × (100 cm / 1 m) gives 1.00 × 100³ ÷ 1000 = 1000 kg/m³ (1.00 × 10³). The calculator cancels one power of cm each time you apply the factor, and the same pattern converts m² to cm² with two.
Compound constants. The gas constant 8.314 J/(mol·K) times (1 cal / 4.184 J) is 1.987 cal/(mol·K), since the joules cancel.
Where the conversion factors come from
Use defined values where they exist: 2.54 cm per inch, 0.45359237 kg per pound and 4.184 J per thermochemical calorie are exact by definition, and so is the Avogadro constant since 2019. For anything else, take the value from a reference such as NIST Special Publication 811, whose appendices list conversion factors, and treat it as measured to its stated number of digits.
Common Mistakes
- Writing the factor upside down. The unit to be removed must sit in the opposite half of the fraction; the calculator picks the orientation that cancels, but when you do it by hand, check that the units cancel before multiplying numbers.
- Squaring the number but not the unit factor. 1 m² is 10,000 cm², not 100 cm²: the factor must be applied twice.
- Treating a unit conversion as a measured number. Defined factors such as 100 cm per metre have unlimited significant figures; only measured values limit the answer.
- Skipping the unit check. An answer in the wrong units is the quickest sign of an inverted factor; give the target units so the calculator checks for you.
Frequently Asked Questions
- What is dimensional analysis?
- Dimensional analysis, also called the factor-label or unit-factor method, is a way of converting quantities by multiplying by conversion factors written as fractions equal to one, arranged so that unwanted units cancel. It also checks the answer: if the units left are not the ones you wanted, the setup is wrong.
- How do you set up a conversion factor?
- Write the equality between the two units, such as 1 mi = 1609.344 m, then use it as a fraction with the unit you want to eliminate on the opposite side from where it sits in your quantity: in the denominator if it is in the numerator, and in the numerator if it is in the denominator. Both orientations equal one; only one cancels the unit you are removing.
- How do you convert squared or cubed units?
- Apply the linear conversion factor once for each power. To change 5.00 m² to cm², multiply by (100 cm / 1 m) twice, giving 5.00 × 10⁴ cm². A common error is to use 100 for an area and 1000 for a volume when the factors are 100² and 100³.
- Why won't my units cancel?
- Either a factor is inverted, or the chain skips a step: converting g to molecules, for example, needs both g to mol and mol to particles. The calculator reports a factor that shares no unit with the quantity so far, and the target-units check flags any unit left over.
- How many significant figures should the answer have?
- Count only the measured numbers. Defined conversion factors (100 cm per m, 60 s per minute) and counted quantities are exact and never limit the answer, but a measured factor such as a molar mass or a density does. Untick Exact on the row to include it.
- Is this the same as checking the dimensions of an equation?
- They use the same idea of units as algebraic objects. Checking an equation asks whether both sides reduce to the same units; dimensional analysis as used here converts a quantity from one set of units to another. Either way, a result in the wrong units is a sign of an error.