Dimensional Analysis

Pronunciation: /dɪˈmɛn.ʃə.nl əˈnæl.ə.sɪs/ Explain

Mathematical equations used in science contain dimensions such as meters or seconds. Dimensional analysis is a tool for verifying these equations.[3] The dimensions, or units of measure, on both sides of an equation must agree for an equation to be valid.

A dimension is not the same as a unit of measure. A unit of measure may have multiple dimensions such as m/s (velocity), or m/s2 (acceleration).

Units of Measure and Their Dimensions
Unit of MeasureSymbolDimension
velocitym/smeters per second
mi/hmiles per hour
km/hkilometers per hour
accelerationm/s2meters per second squared
mi/h2miles per hour squared
km/h2kilometers per hour squared
energy (joules)kg·m2/s2kilogram meters squared per second squared
force (newtons)kg·m/s2kilogram meters per second squared
Table 1. Underlined dimensions in red are the SI (Système Internationale) units. Use of SI System units are preferred over other, non-SI units.


  1. a = 3d / t2 where a is acceleration, d is distance, and t is time. Is this equation valid? To validate this equation, change the variables to units of measure. Use the table of units of measure as a guide. Constants that are not exponents can be ignored.
    • The dimensions of acceleration are m/s2.
    • The dimension of distance is m.
    • The dimension of time is s.
    So a = 3d / t2 becomes m/s2 = m/s2. Dimensional analysis does not show any problem with this equation.
  2. v = k·d / (3t) where v is velocity, k is mass, d is distance, and t is time. Check the validity of this equation using dimensional analysis.
    • The dimensions of velocity are m/s.
    • The dimension of mass is kg.
    • The dimension of distance is m.
    • The dimension of time is s.
    So v = k·d / (3t) becomes m/s = kg·m / s. The dimensions on both sides of the equation are not the same. This equation is invalid.

Check Mark Understanding Check

The variables in this understanding check have the following meanings:

  • d: distance
  • t: time
  • v: velocity
  • a: acceleration
  • m: mass
  • g: energy
  • f: force

Check the validity of each equation using dimensional analysis. Then click the 'Click for answer' button to see the correct answer.

blank space1. d = t·a Click for Answer
blank space2. g = f·d Click for Answer
blank space3. v = f·t / d Click for Answer
blank space4. m·f = g Click for Answer.


  1. McAdams, David E.. All Math Words Dictionary, dimensional analysis. 2nd Classroom edition 20150108-4799968. pg 61. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. dimensional analysis. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 7/3/2018. http://www.merriam-webster.com/dictionary/dimensional analysis. Buy the book
  3. Bridgman, P. W.. Dimensional Analysis. www.archive.org. Yale University Press. 1922. Last Accessed 7/3/2018. http://www.archive.org/stream/dimensionalanal00bridgoog. Buy the book
  4. Gloria P Craig. Quick Guide to Solving Problems Using Dimensional Analysis. 1st edition. Lippincott Williams & Wilkins. January 1, 2003. Last Accessed 7/3/2018. Buy the book

More Information

  • Dimensional Analysis. Department of Physics, University of Guelph. 3/12/2009. http://www.physics.uoguelph.ca/tutorials/dimanaly/.

Cite this article as:

McAdams, David E. Dimensional Analysis. 4/19/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/d/dimensionalanalysis.html.

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Revision History

4/19/2019: Updated equations and expressions to the new format (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/4/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
1/23/2010: Added "References". (McAdams, David E.)
6/7/2008: Corrected spelling. (McAdams, David E.)
5/2/2008: Initial version. (McAdams, David E.)

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