Associated Topics || Dr. Math Home || Search Dr. Math

What is Dimensional Analysis?

```
Date: 11/26/2001 at 21:20:59
From: Danielle
Subject: Dimensional Analysis

What is dimensional analysis and how does it work?
```

```
Date: 11/28/2001 at 12:00:37
From: Doctor Greenie
Subject: Re: Dimensional Analysis

Hi, Danielle -

Dimensional analysis is a tool that can be used to determine how to
manipulate formulas by analyzing the dimensions in the formulas. In
the physical world, these units are often units of distance, mass (or
weight) and time; but many other units are possible (see, for example,
the "fun" problem in the Dr. Math archives for which I have provided a
link at the end of this response).

One elementary application of dimensional analysis is in the
memorization of simple geometric formulas. For a particular example,
many students, when beginning to learn formulas related to circles,
get confused between the formulas for the circumference and area of a
circle:

circumference = pi * diameter

area = pi * (radius squared)

Using dimensional analysis, it is impossible to confuse these two
formulas:

(1) "pi times diameter"

In this formula, "pi" is a pure number, and the diameter is a length.
If the diameter is in inches, then "pi times diameter" is in inches;
if the diameter is in miles, then "pi times diameter" is in miles.
The units on "pi times diameter" are always units of length. So "pi
times diameter" measures a length - so it can't be an area formula; in
particular, it can't be the formula for the area of a circle,  But it
can be a formula for a length; and in fact it is the formula for the
circumference of a circle.

In this formula, "pi" is again a pure number, and the radius is a
length. If the radius is in inches, then "pi * (radius squared)" is in
square inches; if the radius is in miles, then "pi * (radius squared)"
is in square miles. The units on "pi * (radius squared)" are always
units of (length squared), which are units of area. So "pi * (radius
squared)" measures an area - so it can't be the formula for any
length; in particular, it can't be the formula for the circumference
of the circle. But it can be the formula for an area; and in fact it
is the formula for the area of a circle.

Another elementary application of dimensional analysis is in
converting units of measure. For example, young students often get
confused over whether to divide by 12 or multiply by 12 when
converting a measurement in inches to a measurement in feet:

180 inches = ??? feet

You can analyze this problem using dimensional analysis to determine
whether to multiply or divide by 12. In this application of
dimensional analysis, you make fractions equivalent to "1" and
"multiply" your given measurement by those fractions; when you do
this, you can "cancel" like units in the numerator and denominator,
just as you can cancel common numerical factors in the numerator and
denominator of any fraction.

In this example, 1 foot = 12 inches, so the following two fractions
are equivalent to "1":

1 foot
(1)   --------- = 1
12 inches

and

12 inches
(2)   --------- = 1
1 foot

Now try multiplying the given measurement, "180 inches," by each of
these fractions:

(1)

1 foot     (180 * 1) (feet * inches)
180 inches * --------- = -------------------------
12 inches          12 inches

Here, the units "inches" cancel in numerator and denominator, leaving

1 foot     180
180 inches * --------- = --- feet = 15 feet
12 inches    12

or (2)

12 inches   (180 * 12) (inches * inches)
180 inches * --------- = ---------------------------- = ????!!!
1 foot               feet

Here, there are no common units in numerator and denominator that can
be cancelled - so we know this is not the right way to convert the
measurement "180 inches" to feet.

Finally, here is a link to a page in the Dr. Math archives where
dimensional analysis is used to solve a "fun" problem:

Dimensional Analysis
http://mathforum.com/library/drmath/view/56728.html

I hope this helps.  Write back if you have further questions.

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry
High School Geometry
Middle School Definitions
Middle School Geometry
Middle School Terms/Units of Measurement
Middle School Two-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search