2.3 Dimensional Analysis

Dimensional Analysis

Dimensional analysis is a method for converting between measurement units that uses the fact that any number or expression can be multiplied by one without changing its value. It can handle simple and very complicated measurement unit conversions. Dimensional analysis uses fractions that have a numerator and denominator that are equal but in different units. Since the numerator and denominator of the fraction are equal, the fraction is equal to 1. In dimensional analysis, this ratio which converts one unit of measure into another without changing the quantity, is called a conversion factor and is formed by turning equal measures into fractions equal to 1. For example, since 1 ft = 12 in, we can form the conversion factors  [latex]\frac{1\text{ foot}}{12\text{ inches}}[/latex]  or [latex]\frac{12\text{ inches}}{1\text{ foot}}[/latex], where each fraction equals 1.

The main idea of performing unit conversions through dimensional analysis, is to multiply by a series of conversion factors equal to 1 where we introduce measurement units we want and cancel out measurement units we don’t want that appear on both the numerator and denominator of different fractions. Dimensional analysis is basically performed in the following way:

[latex]\text{starting unit}\times\text{conversion factors} = \text{ending unit}[/latex]

ExampleS

1. Ashley’s height is 5’6″. Express Ashley’s height in inches. (1 ft = 12 in)

Solution

Since 5’6″ = 5′ + 6″, we can convert 5′ into inches, then add the extra 6 in.

We start by writing 5′ as a fraction over 1:   [latex]5\;ft=\dfrac{5\;ft}{1}[/latex]. This just makes it obvious that the unit feet is on the numerator.

This is our starting fraction. Now we want to multiply this by a conversion factor that introduces inches to our numerator and feet to our denominator so that feet will cancel out:  [latex]\dfrac{12\;in}{1\;ft}[/latex]

So,

[latex]\begin{aligned}\dfrac{5\;ft}{1}&=\dfrac{5\;\cancel{ft}}{1} \times \dfrac{12\;in}{1\;\cancel{ft}} \\&= \dfrac{5 \times 12\;in}{1}\\&=60\;in \end{aligned}[/latex]

The unit feet cancel out since [latex]\dfrac{feet}{feet}=1[/latex].

Now we know that 5ft = 60 in, we can add on the remaining 6 in:   5’6″ = 60 in + 6 in = 66 in.

So, Ashley is 66 inches tall.

 

2. The area of a living room is 35 yd2. Express the area in square feet. (1 yd2 = 9 ft2)

Solution

We start with 35 yd2 written as our starting fraction:  [latex]35\;yd^2=\dfrac{35\;yd^2}{1}[/latex]

Now we want to multiply this fraction by a conversion factor that introduces square feet to the numerator and square yards to the denominator so that yd2 will cancel out:  [latex]\dfrac{9\;ft^2}{1\;yd^2}[/latex]

So,

[latex]\begin{aligned}35 yd^2&=\dfrac{35\;\cancel{yd^2}}{1} \times \dfrac{9\;ft^2}{1\;\cancel{yd^2}}\\&=\dfrac{35\times9\;ft^2}{1}\\&=315\;ft^2\end{aligned}[/latex]

Therefore, 35 yd2 = 315 ft2.

 

3. The diameter of a quarter coin is about 2.3 centimeters. Convert 2.3 centimeters to millimeters. (1 cm = 10 mm)

Solution

We start by writing 2.3 cm as a fraction over 1: [latex]\dfrac{2.3\;cm}{1}[/latex]

Since centimeters is on the numerator of the starting fraction, our conversion factor must have centimeters on the denominator so that centimeters cancel out: [latex]\dfrac{10mm}{1cm}[/latex]

[latex]\dfrac{2.3\;\cancel{cm}}{1} \times \dfrac{10\;mm}{1\;\cancel{cm}} = \dfrac{2.3\times 10\;mm}{1}=23\;mm[/latex]  

The diameter of a quarter is about 23 mm.

 

Try It

1. A medium sized apple weighs about 6 ounces. Express the weight in pounds. (1 lb = 16 oz)

2. A stick of butter weighs 113 grams. How many kilograms is the stick of butter? (1 kg = 1000 g)

3. Jenny’s height is 1.83 meters. Convert 1.83 meters to centimeters. (1m = 100 cm)

There may be more than one conversion factor needed in dimensional analysis.

Example

1. A standard telephone pole stands about 10 yards tall. Express 10 yards in inches.

Solution

To convert 10 yards to inches, two conversion equations are needed:  1 yd = 3 ft and 1 ft = 12 in.  In other words, we will convert from yards to feet, then from feet to inches.

The two conversion factors need to be written in a way that the units yard and feet will be cancelled out with only the ending unit inch left in the dimensional analysis. This means that we need to introduce feet with yards on the denominator so we can cancel out yards in the numerator of the starting fraction:  [latex]\dfrac{3\;ft}{1\;yd}[/latex].

We then need to introduce inches on the numerator so that we get to the required end unit, and feet on the denominator so feet can cancel out the feet introduced by the first conversion factor:  [latex]\dfrac{12\;in}{1\;ft}[/latex]

Then,

[latex]\begin{aligned}10\;yd&=\dfrac{10\;\cancel{yd}}{1} \times \dfrac{3\;\cancel{ft}}{1\;\cancel{yd}} \times \dfrac{12\;in}{1\;\cancel{ft}}\\& = \dfrac{10 \times 3 \times 12\;in}{1}\\&=360\;in\end{aligned}[/latex]

Consequently, a standard telephone pole of 10 yards measures 360 inches tall.

 

2. Express the capacity of a 5-gallon water jug in cups. (1 gallon = 4 quarts, 1 quart = 2 pints, 1 pint = 2 cups)

Solution

We start with 5 gal written as a fraction:  [latex]\dfrac{5\;gal}{1}[/latex]. Then we can convert from gallons to quarts to pints to cups:

[latex]\begin{aligned}5\;gal&=\dfrac{5\;\cancel{gal}}{1}\times\dfrac{4\;\cancel{qt}}{1\;\cancel{gal}}\times\dfrac{2\;\cancel{pt}}{1\;\cancel{qt}}\times\dfrac{2\;c}{1\;\cancel{pt}}\\&=\dfrac{5\times4\times2\times2\;c}{1}\\&=80\;cups\end{aligned}[/latex]

Try It

Express 60 miles per hour in feet per second. (1 mile = 5280 feet, 1 hour = 60 minutes, 1 minute = 60 seconds)

 

Examples

  1. A bottle of apple juice has 48 fluid ounces. Express this capacity in cups. (1 cup = 8 fl oz)
  2. The volume of a suitcase is 7560 cubic inches. Express this volume in cubic feet. (1 ft3 = 1728 in3)
  3. Express 5 pounds of cinnamon in ounces. (1 lb = 16 oz)
  4. How many tablespoons are in 1 quart of ice cream? (1 fl. oz. = 2 tbsp, 1 cup = 8 fl. oz, 1 pint = 2 cups, 1 quart = 2 pints)
  5. You are planning a large party. You have invited 150 people and anticipate that 100 people will attend. In preparation, you anticipate that each person will drink around 2 cups of punch that you are providing. How many gallons do you need to purchase to make sure that no one goes thirsty?
  6. Ground beef costs $4.99 per pound. Express this price in cents per ounce.

 

Show/Hide Answers

1. [latex]\dfrac{48\;\cancel{fl\;oz}}{1} \times \dfrac{1\;cup}{8\;\cancel{fl\;oz}}=\dfrac{48\;cup}{8}=6\;cups[/latex]   Answer: 6 cups

2. [latex]\dfrac{7560\;\cancel{in^3}}{1} \times \dfrac{1\;ft^3}{1728\;\cancel{in^3}}=\dfrac{7560\;ft^3}{1728}=4.375\;ft^3[/latex]   Answer: 4.375 ft3

3. [latex]\dfrac{5\;\cancel{lb}}{1} \times \dfrac{16\;oz}{1\;\cancel{lb}}=\dfrac{5\times16\;oz}{1}=80\;oz[/latex]   Answer: 80 oz

4. [latex]\dfrac{1\;\cancel{quart}}{1}\times\dfrac{2\;\cancel{pints}}{1\;\cancel{quart}}\times\dfrac{2\;\cancel{cups}}{1\;\cancel{pint}}\times\dfrac{8\;\cancel{fl\;oz}}{1\;\cancel{cup}}\times\dfrac{2\;tbsp}{1\;\cancel{fl\;oz}}\\=\dfrac{1\times2\times2\times8\times2\;tbsp}{1}=64\;tbsp[/latex]   Answer: 64 tablespoons

5. 200 cups ([latex]2\times100[/latex]) are needed for the 100 anticipated participants.  

[latex]\dfrac{200\;\cancel{cups}}{1}\times\dfrac{1\;\cancel{pint}}{2\;\cancel{cups}}\times\dfrac{1\;\cancel{quart}}{2\;\cancel{pints}}\times\dfrac{1\;gallon}{4\;\cancel{quarts}}=\dfrac{200\;gallons}{1\times2\times2\times4}=12.5\;gallons[/latex]   Answer: 13 gallons

6. [latex]\dfrac{\cancel{$}4.99}{1\;\cancel{lb}}\times\dfrac{100\;¢}{\cancel{$}1}\times\dfrac{1\;\cancel{lb}}{16\;oz}=\dfrac{4.99\times100\times1\;¢}{1\times1\times16\;oz}=\dfrac{31.1875¢}{1\;oz}[/latex]   Answer: 31.1875 cents per pound

 

 

TRY IT

  1. Convert 3.5 pounds to ounces using dimensional analysis. (1 lb = 16 oz)
  2. Convert 7200 fluid ounces to gallons using dimensional analysis. (1 gal = 4 qt, 1 qt = 2 pt, 1 pt = 16 fl. oz.)
  3. Convert 10,000 steps to miles. (1 step = 2.5 feet, 1 mile = 5280 feet)
  4. A local steak house has an eating challenge. If you can finish their 72-ounce steak with 3 sides, and an entire pitcher of beer, you get the meal for free. In addition, your picture is put on their wall of fame, AND you get a free t-shirt! How many pounds of steak are 72 ounces? 
  5. Lisa is buying a new rug for her living room that measures 8 feet by 10 feet. Calculate the area of the rug in square inches. (1 ft2 = 144 in2)
  6. Maria made 5 gallons of soup at the soup kitchen. How many 1-cup ladles of soup is this? (1 gal = 4 qt, 1 qt = 2 pt, 1 pt = 2 cups)
  7. In February 2023, the average price of gas in Salt Lake City was $3.71 per gallon. What is the cost of gas in cents per pint?
Show/Hide Answers
  1. 56 oz.
  2. 56.25 gal
  3. 4.7 miles
  4. 4.5 lb
  5. 11,520 in2
  6. 80 cups
  7. 46.375 cents per pint

 

 

The origins of dimensional analysis have been disputed by historians with the first written application of dimensional analysis credited to an article by François Daviet at the Turin Academy of Science in 1799[1]. The modern use of dimensional analysis, which distinguishes mass, length, and time as fundamental units, while referring to other units as derived, was primarily established by James Clerk Maxwell, a Scottish mathematician and scientist, around 1873[2]. Commensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are expressed in different units of measure, e.g. yards and meters, pounds and kilograms, seconds and years. For example, the length of a dollar bill is the same whether it is measured as 6.14 inches or 156 mm.

Dimensional Analysis Across Systems

NASA has been notorious for having problems due to one team using SI units and the other team using U.S. Customary units. In 1998, NASA lost equipment worth millions of dollars thanks to shoddy conversion practices. SOHO, the Solar Heliospheric Observatory lost all communications with Earth for a week due to a mix up between U.S. and metric measures. The following year, NASA lost a Mars Orbiter worth $125 million for similar reasons. Nowadays, all scientific endeavors are calculated using the metric system. This includes all medications, after an order was misread and 0.5 grams of Phenobarbital (a sedative) were given to a patient instead of 0.5 grains (only 0.0325 grams)! Consequently, it is vital that if different systems are used conversions are accurate, and rounding errors are avoided.

To convert between measuring systems, equivalent measures need to be known across systems. Some equivalent measures are exact, like 1 inch = 2.54 cm, while others are approximate, like 1 yard ≅ 0.91 meters.

Examples

1. The longest tail ever recorded on a dog was 76.8 cm. It belongs to an Irish Wolfhound named Keon who lives in Belgium[3]. How long is Keon’s tail in inches? (1 inch = 2.54 cm)

Solution

We start with 76.8cm written as a fraction: [latex]\dfrac{76.8\;cm}{1}[/latex].

The conversion factor needs cm on the denominator so cm will cancel:  [latex]\dfrac{1\;in}{2.54\;cm}[/latex]

Then,

[latex]\begin{aligned}76.8\;cm&=\dfrac{76.8\;\cancel{cm}}{1} \times \dfrac{1\;in}{2.54\;\cancel{cm}} \\&= \dfrac{76.8\;in}{2.54}\\&=30.24\;in\end{aligned}[/latex]

Keon’s tail measures 30.24 inches.

 

2. How many liters can a 5-gallon jug hold? (1 gallon = 3.7854 liters)

Solution

We start with 5 gal written as a fraction: [latex]\dfrac{5\;gal}{1}[/latex].

The conversion factor needs gallons on the denominator so they will cancel:  [latex]\dfrac{3.7854\;liters}{1\;gallon}[/latex]

Then, [latex]\dfrac{5\;\cancel{gal}}{1} \times \dfrac{3.7854\;l}{1\;\cancel{gal}} = \dfrac{5\times 3.7854\;l}{1}=18.927\;l[/latex]

A 5-gallon jug can hold 18.927 liters.

 

Try It

1. How many gallons are equivalent to 2-liters of soda? (1 gallon = 3.7854 liters)

2. A typical can of soda holds 355 milliliters. How many cups can the can hold? (1 cup = 236.588 milliliters)

 

Often, multiple conversions factors are needed to complete a conversion.

Examples

1. The hotel room you booked in Paris is 21.2 m2, according to their website. How many square feet is the room? (1m2 = 10,000 cm2, 1 in2 = 6.4516 cm2, 1 ft2 = 144 in2)

Solution

We need to convert square meters to square feet. We start with the fraction [latex]\dfrac{21.2\;m^2}{1}[/latex].

[latex]\dfrac{21.2\;\cancel{m^2}}{1}\times\dfrac{10000\;\cancel{cm^2}}{1\;\cancel{m^2}}\times\dfrac{1\;\cancel{in^2}}{6.4516\;\cancel{cm^2}}\times\dfrac{1\;\;ft^2}{144\;\cancel{in^2}}\\=\dfrac{21\times10,000\;ft^2}{6.4516\times144}=226.04\;ft^2[/latex]

 

2. How many liters is equivalent to a quart of icecream? (1 gallon = 4 quarts, 1 gallon = 3.7854 liters)

Solution

We need to convert from quarts to liters. We start with the fraction [latex]\dfrac{1\;qt}{1}[/latex].

[latex]\dfrac{1\;\cancel{qt}}{1}\times\dfrac{1\;\cancel{gal}}{4\;\cancel{qt}}\times\dfrac{3.7854\;l}{1\;\cancel{gal}}=\dfrac{3.7854\;l}{4}=0.95\;l[/latex]

A quart of icecream s equivalent to 0.95 liters.

 

Try It

1. A medium pizza in Europe measures 30 cm. What is this measurement in inches? (1 inch = 2.54cm)

2. The average house size in the U.K. is 76 square meters[4]. What is this area in square feet? (1 m2 = 10,000 cm2, 1 in2 = 6.4516 cm2, 1 ft2 = 144 in2)

3. The height of a net stretched across a tennis court varies in height from 3 ft 6 in. at the posts to 3 feet at the center of the net. Convert these measurements to centimeters. (1 ft = 12 in, 1 in = 2.54 cm)

4. The tallest building in the world is the Burj Khalifa in the U.A.E. at a staggering 828 meters tall. This compares to the tallest building in the U.S. which is One World Trade Center at 1776 feet tall. How many feet taller than One World Trade Center is Burj Khalifa? [5] (1 m = 3.28084 ft)

5. On a European vacation you notice a speed limit of 150 km per hour. Convert this to miles per hour. (1 km = 0.62 mile)

Show/Hide Answers
  1. 11.8 inches
  2. 818 square feet
  3. 3 ft 6 in = 106.68 cm and 3 ft = 91.44 cm
  4. The Burj Khalifa is 2716.54 feet tall, which is 940.54 feet taller than One World Trade Center.
  5. 93 mph

 

  1. Martins, Roberto De A. (1981). The origin of dimensional analysis. Journal of the Franklin Institute. 311 (5): 331–337.ISSN 0016-0032, https://doi.org/10.1016/0016-0032(81)90475-0. ↵
  2. Roche, John J (1998). The Mathematics of Measurement: A Critical History, Springer, p. 203, ISBN 978-0-387-91581-4 ↵
  3. https://www.guinnessworldrecords.com/news/2017/1/video-meet-irish-wolfhound-who-boasts-worlds-longest-dog-tail-458596 ↵
  4. https://shrinkthatfootprint.com/how-big-is-a-house/ ↵
  5. https://www.guinnessworldrecords.com/records/hall-of-fame/burj-khalifa-tallest-building-in-the-world ↵