Dimensional analysis is a way to solve problems by tracking the units: you multiply by conversion factors, cancel any unit that appears on both the top and the bottom, and check that the units left over are the ones the answer needs. It is also called the factor-label method or unit analysis.
In Spanish
análisis dimensional
Also called
factor-label method, unit analysis, unit factor method
Choose each factor so an unwanted unit cancels: miles with miles, hours with hours. What is left, feet per second, is the unit you wanted.
Formal definition and notation
Dimensional analysis treats units as algebraic quantities that are multiplied, divided and cancelled exactly like the numbers they label. A given quantity is multiplied by a chain of conversion factors, each equal to 1, chosen so that every unwanted unit cancels; the product has the same size but the target units. The same bookkeeping checks whether a formula or answer has sensible units.
Unit conversion is the goal (change 2 km into meters). Dimensional analysis is one systematic method to do it, and it also handles rates and checks formulas.
Quick check: 3 questions
Question 1 of 3
Which chain converts 2 hours into seconds?
Show answer and explanation
Answer: B. B cancels hours, then minutes, leaving seconds: 2 × 60 × 60 = 7200 s. A has both factors upside down, C uses 100 instead of 60, and D stops at minutes.
Question 2 of 3
Convert 72 km/h to meters per second.
Show answer and explanation
Answer: A.72 × 1000 m1 km × 1 h3600 s = 20 m/s. 259.2 multiplies by 3.6 instead of dividing, 1.2 converts only hours to minutes, and 72,000 converts only km.
Question 3 of 3
A recipe needs 3 cups of milk. How many fluid ounces is that? (1 cup = 8 fl oz)
Show answer and explanation
Answer: C.3 cups × 8 fl oz1 cup = 24 fl oz. 0.375 uses the factor upside down, 11 adds, and 38 is 3 and 8 written side by side.
Dimensional Analysis: frequently asked questions
Why is it called dimensional analysis?
In science, a dimension is a kind of measurement, such as length, time or mass, and units like meters or seconds measure those dimensions. The method analyzes the units of every quantity in a calculation, keeping and cancelling them like numbers. Chemistry books also call it the factor-label method, because each factor carries a unit label.
How do you do dimensional analysis step by step?
Write the given amount with its unit. Multiply by a conversion factor that has that unit on the bottom and the next unit on top. Repeat until the only units left are the ones you want. Cancel units, then multiply the numbers on top and divide by the numbers on the bottom.
Can dimensional analysis check if an answer is wrong?
Yes. If the units left at the end are not the units the question asks for, something is wrong. A speed must come out in distance per time; an area must come out in square units. For example, multiplying a speed in miles per hour by hours must give miles, so the units confirm d = rt.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSN-Q.A.1
Cite this entry
Effortless Math Team. “Dimensional Analysis.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/dimensional-analysis/
Flashcard
TermDimensional Analysismih × ftmi × hs
DefinitionSolve by tracking units: multiply by conversion factors and cancel units that appear top and bottom. 60 mi/h becomes 88 ft/s.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Conversion FactorA conversion factor is a fraction whose top and bottom are the same amount written in different units, such as 12 in1 ft, so it equals 1. Multiplying a measurement by it changes the units without changing the amount: 5 ft × 12 in1 ft = 60 in. Pick the version that cancels the unit you start with.Measurement · Grade 6, Grade 7, Algebra 1Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3GramA gram is a metric unit of mass, about the mass of a paper clip. Its symbol is g. There are 1,000 grams in a kilogram and 1,000 milligrams in a gram. Grams measure light things, such as food portions, coins and letters; heavier things are measured in kilograms.Measurement · Grade 3, Grade 4, Grade 5FootA foot is a US customary unit of length equal to 12 inches, about the length of a standard school ruler. Its abbreviation is ft, and its plural is feet. There are 3 feet in a yard and 5,280 feet in a mile, and 1 foot is exactly 30.48 centimeters.Measurement · Grade 2, Grade 3, Grade 4SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5Unit ConversionUnit conversion is rewriting a measurement in a different unit without changing the amount, such as writing 3 feet as 36 inches. You multiply by a conversion factor, a fraction equal to 1 such as 12 in1 ft. Converting to a smaller unit gives a larger number; converting to a larger unit gives a smaller number.Measurement · Grade 4, Grade 5, Grade 6RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8DensityDensity is the amount of mass packed into each unit of volume, found by dividing mass by volume: D = mV. Its units combine the two, such as grams per cubic centimeter. A 40 g block with a volume of 16 cm³ has D = 40 ÷ 16 = 2.5 g/cm³. Population density works the same way: people per unit of area.Problem Solving · Grade 8, Algebra 1, GeometryScientific NotationScientific notation writes a number as a value from 1 to less than 10 times a power of 10, such as 5.4 × 103 for 5,400. The exponent records how far the decimal point moved: positive for large numbers and negative for numbers between 0 and 1, as in 0.0062 = 6.2 × 10−3.Number System · Grade 8, Algebra 1Average SpeedAverage speed is the total distance traveled divided by the total time taken. If a car goes 150 miles in 3 hours, its average speed is 150 ÷ 3 = 50 miles per hour, even if it sped up and slowed down along the way. It is not the average of the separate speeds unless each speed lasted equally long.Problem Solving · Grade 6, Grade 7, Algebra 1RateA rate is a ratio that compares two quantities measured in different units, such as 150 miles in 3 hours or 12 dollars for 4 pounds. The word “per” signals a rate. Dividing gives the unit rate, the amount for 1 unit: 150 miles in 3 hours is 50 miles per hour.Ratios and Proportions · Grade 6, Grade 7Unit RateA unit rate is a rate that compares a quantity to one unit of another quantity, such as 50 miles per hour or 4 dollars per notebook. To find it, divide the first quantity by the second: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. The word per means for each one.Ratios and Proportions · Grade 6, Grade 7, Grade 8Metric SystemThe metric system is a system of measurement in which units are related by powers of 10. Its base units are the meter for length, the gram for mass and the liter for capacity. Prefixes name larger and smaller units: kilo- means 1,000, centi- means one hundredth and milli- means one thousandth, so 1 km = 1,000 m.Measurement · Grade 4, Grade 5, Grade 6