Dimensional Analysis Calculator

What dimensional analysis does

Dimensional analysis, which chemistry classes call the factor label method, is a way of converting units by multiplying. Every conversion factor is a fraction with the same quantity on top and bottom written in different units, so the fraction equals one. Multiplying by one changes nothing about the amount, only about the way it is written.

NIST's page on unit conversion describes the method the same way: a conversion ratio always equals one, and the units cancel the way the numbers in a fraction do. Divide miles by hours and the hours vanish, leaving speed. That bookkeeping is the whole trick, and it doubles as a proof: if the units of your answer do not come out right, no amount of arithmetic fixes the setup.

The calculator below carries a table of factors for 10 quantity types: length, mass, volume, area, speed, pressure, energy, power, force and flow. It multiplies your value by the factor for the starting unit, divides by the factor for the target unit, and prints the chain so you can check each step.

The factor label method in four steps

Worked on paper, the method is short enough to memorize:

Converting 2.5 hours to minutes shows the canceling: 2.5 x 60 min = 150 min, and the hour disappears because it sits on the bottom of the factor and the top of the measurement. NIST's unit conversion page, updated in 2026, lays out the same steps with one warning worth keeping: use exact factors when they exist, and do not round anything until the final answer is in hand.

Write each factor as a fraction and the canceling becomes visible on the page. Converting 88 ft/s back to miles per hour runs the chain in reverse: 88 ft/s x 1 mi / 5,280 ft x 3,600 s / 1 h gives 60 mi/h. A chain you can run backward is a chain you can check, and the order you multiply in does not change the answer, only the units you finish with.

The factors this calculator uses

Factors differ in one respect: some are exact by definition, others are measured. The table shows both, drawn from NIST Special Publication 811, Appendix B (2008 edition), which marks the exact factors in bold and rounds the rest to at least 7 digits.

A short list of conversions comes up often enough to be worth keeping on one card:

The international definitions sit behind these numbers: the BIPM SI Brochure (9th edition, 2019) defines the meter, the kilogram and the second, and every factor above follows from those definitions. Factors for hundreds of further units are in the printed 2008 edition of SP 811.

Worked example: 60 miles per hour to feet per second

Speed is a compound unit, so the chain has two links: the miles become feet and the hours become seconds.

Both factors are exact: 5,280 feet to a mile and 3,600 seconds to an hour are definitions, so 88 ft/s is an exact result, not an approximation. The calculator reaches the same place by a different road, through SI: 60 x 0.44704 = 26.8224 m/s, then 26.8224 / 0.3048 = 88 ft/s. The intermediate unit differs and the answer does not, which is the point of the method.

Compound units are the normal case in daily life. Fuel economy in miles per gallon, speed in miles per hour, tire pressure in pounds per square inch and pump output in gallons per minute all combine two units, so each one needs a link of its own. The calculator handles the second part by dividing the SI value by the factor of the target unit, which is the same operation written from the other side.

Worked example: 32 psi to kilopascals

Tire pressure gives a single link, but a rounded factor, and that is where the significant figure rule earns its keep. NIST lists 1 psi as 6894.757 pascals; the stored value runs on, 6894.757293168. Set the quantity to pressure, the start to psi, the target to kilopascals and the value to 32. The chain is 32 x 6894.757 = 220,632.2 Pa, and 220,632.2 / 1,000 = 220.632 kPa.

Only the final value gets rounded, to 6 digits. Rounding the factor first, say to 6,895, would give 32 x 6,895 = 220,640 Pa, about 8 Pa high from one careless step. On a tire gauge nobody would notice. In a pressure vessel calculation somebody might.

Checking an equation with dimensions

The same canceling tests whether a formula itself makes sense. Force equals mass times acceleration: kilograms times meters per second squared is the definition of the newton, so F = ma is dimensionally sound. A 1,500 kg car accelerating at 2 m/s squared needs 1,500 x 2 = 3,000 N, and the units of the answer, kg x m/s squared, are the newton by definition.

The test also catches mistakes. Add a length to a time and no factor rescues you: meters and seconds are different dimensions, and any equation that adds them is wrong regardless of the numbers. That is why engineers convert everything into one system before computing; the NIST guide to the SI exists in large part so that everyone converts the same way.

Two equations with the same shape can hide a missing piece. Work equals force times distance, and both sides come out in newton meters, which the SI calls the joule. A result in newtons instead of joules means the distance went missing, and no amount of arithmetic will put it back.

Chemistry: grams to moles

Chemistry runs the same play with a factor that is not a unit conversion at all. Molar mass converts between grams and moles, because one mole of a compound weighs its molar mass in grams. Sodium chloride has a molar mass of 58.44 g/mol, from 22.990 for sodium plus 35.45 for chlorine. So 25.0 g of table salt is 25.0 / 58.44 = 0.4278 mol, and the grams cancel exactly the way the hours did in the speed example.

Chain the two steps and you get routine lab arithmetic: grams to moles by molar mass, then moles per liter of solution for concentration. Write the units at every step and a slip shows up as a nonsense unit long before it becomes a ruined batch.

Rounding and significant figures

NIST's guidance for converted values is short: carry every digit through the calculation and round once, at the end. The rounding section of SP 811 also suggests keeping roughly the same number of significant digits in the converted value as in the one you started with. Convert 60 mph and you get 26.8224 m/s, six digits in and six digits out.

The calculator rounds its output to 6 significant digits and uses exact factors wherever definitions supply them. Where a factor is measured, psi or horsepower, the stored factor carries more digits than the display shows, so the rounding happens once, on the way out, and not twice.

One habit makes that easier: write down the unit you expect before you start. If you expect 88 ft/s and the tool hands back 26.8224 m/s because you forgot to change the target unit, the mistake shows up in one second instead of three.