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Unit Conversions for Volume
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Introduction

In this lesson, you will find unit conversions for volume. Often, instead of using cubic inches or centimeters for volume, you use cups, quarts, or liters to measure volume. This lesson will first explain different conversion factors between measures of volume and then provide an example of a volume conversion.

These videos illustrate the lesson material below. Watching the videos is optional.

Conversion Factors Between Measures of Volume


Example 1
Define the conversion factor between centimeters cubed and liters.

A square is 10 centimeters in length (l) and 10 centimeters in width (w). The area is found as follows:

\begin{align*} Area = lw = 10 cm \times 10 cm =100 cm^2\end{align*}

A square that has 10 centimeters in length and 10 centimeters in width. 

Figure 1

If you look at just 1 centimeter of depth on this cube, it is comprised of 100 little cubes that are 1 centimeter by 1 centimeter by 1 centimeter.

The image shows one side of a cube that is comprised of 100 little cubes that are 1 centimeter by 1 centimeter by 1 centimeter. 

Figure 2

The entire large cube is made up of 10 layers, each containing 100 unit cubes. 10 layers of 100 unit cubes is 1,000 units cubed, or centimeters cubed:

\begin{align*} Volume = lwh = 10cm \times 10cm \times 10cm= 1000cm^3\end{align*}

\(1000 cm^3\) is defined as a liter.

The image shows a cube that has 10 centimeters for length, width, and height. 

Figure 3

The unit conversion between centimeters cubed and liters is \(1000\; cm^3 = 1\; liter\). This is using metric units.

Example 2
Define the conversion factor between yards (yd) cubed and feet (ft) cubed. Yards and feet are imperial units.

In imperial units, cubic inches, cubic feet, and cubic yards are calculated similarly based on their lengths cubed. For instance, let’s say you have a line that is 1 yard long. This is equal to 3 feet so,
\begin{align*} 1yd = 3ft\end{align*}

This image shows that 1 yard is equal to 3 feet. 

Figure 4

Consider a line that is 1 yard long. 1 yard is 3 feet.

If you make it a square yard, it is now 1 yard by 1 yard or 3 feet by 3 feet. Divide it into unit squares.
\begin{align*} 1yd &= 3ft &\color{navy}\small\text{Conversion factor}\\\\ (1yd)^2 &= (3ft)^2 &\color{navy}\small\text{Square both sides}\\\\ 1yd^2 &= 9ft^2 &\color{navy}\small\text{New conversion factor for area} \end{align*}

Each small square is 1 foot by 1 foot for a total of 9 feet squared (see Figure 5).

This image shows that a square that is 1 yard on each side is equal to a square that is 3 feet on each side. ! square yard equals 9 square feet. 

Figure 5

You can do the same thing using metric units and make the right side of Figure 5 into a square cube. Each square is a square cube with lengths of 1 yard by 1 yard by 1 yard. The dimensions of Figure 6 are 3 feet wide by 3 feet high by 1 foot depth.

The dimensions on this figure are 3 feet wide by 3 feet high by 1 foot depth. 

Figure 6

You can make this a cubic yard by adding 2 more layers of unit cubes to the first layer. Now you have 1 yard by 1 yard by 1 yard high or 3 feet by 3 feet by 3 feet high. The image below is now a 1 cubic yard or 1 yard cube. Each layer is made up of 9 units cubed and there are 3 layers for a total of 27 cubic feet.

Two cubes, the first is 1 cubic yard and the other is the same size but is made of 27 cubic feet (3 feet equal one yard, so there are three layers each made up of 9 units cubed, which totals 27 cubic feet). 

Figure 7

\begin{align*} 1yd &= 3ft &\color{navy}\small\text{Conversion factor you know}\\\\ (1yd)^3 &= (3ft)^3 &\color{navy}\small\text{Cube both sides}\\\\ 1yd^3 &=27ft^3 &\color{navy}\small\text{New conversion factor for volume} \end{align*}

The conversion factor between yards cubed and feet cubed is \(1 yd^3 = 27 ft^3\).

Imperial Units for Volume and their Conversion Factors

Some other imperial units for volume are not quite as straightforward, particularly the units used for cooking or for liquids. If a unit conversion is needed with some of the units below, then use a book or the internet to find a unit conversion:

  • Teaspoon
  • Cup
  • Tablespoon
  • Pint
  • Ounce
  • Quart
  • Gallon

Some common conversions are:

  • 3 teaspoons = 1 tablespoon
  • 2 tablespoons = 1 ounce
  • 8 ounces = 1 cup
  • 2 cups = 1 pint
  • 2 pint = 1 quart
  • 4 quarts= 1 gallon
  • 1 quart is approximately 0.946 liters

Again, if you ever need to find unit conversions, there are a lot of resources online to help you go from one unit to another.

Example 3
Mulch is decaying plant material that can be used to enrich the soil in a garden or a flowerbed. Figure 8 represents a yard cubed of mulch or \(1\;yard^3\). A store selling the mulch charges 20 dollars per 1 yard cubed. What is the cost per meter cubed?

This image shows 1 yard cubed. 

    Figure 8

    • Steps for Volume Unit Conversions:
      1. Identify the units you have.
      2. Identify the units you want.
      3. Find the conversion factors that will help you step-by-step get to the units you want.
      4. Arrange conversion factors so that unwanted units cancel out through cross-cancellation. 

    You know that \(1\;yard = 0.9144\;meters\). Now convert \(yd^2\) to \(m^2\):
    \begin{align*} 1 yd &= 0.9144 m &\color(navy}\small\text{Conversion factor you know}\\\\ 1 yd^2 &= (0.9144m)^2 &\color{navy}\small\text{Square both sides}\\\\ 1 yd^2 &= 0.8361m^2\ &\color{navy}\small\text{New conversion factor}\\\\ \end{align*}

    Similarly, if you make this a cube, you can find the number of \(m^3\) in this one \(yd^3\) by cubing the meters.

    \begin{align*} 1 yd &= 0.9144 m &\color{navy}\small\text{Conversion factor you know}\\\\ 1 yd^3 &= (0.9144m)^3 &\color{navy}\small\text{Cube both sides}\\\\ 1 yd^3 &= 0.765m^3\ &\color{navy}\small\text{New conversion factor}\\\\ \end{align*}

    Now you can use this conversion factor to help solve the problem.

    You need to figure out what you want the units to be at the end of the equation. You want to know how much it costs per meter cubed.

    • Starting Units: \(\$20 \;per\; 1 yd^3\) or \(\large\frac{\$20}{1yd^3}\)
    • Ending Units: \(\$ \;per\; 1m^3\) or \(\large\frac{\$}{1m^3}\)

    \begin{align*} &\frac{\$20}{1\; yd^3} = \frac{\$}{m^3} &\color{navy}\small\text{Convert \(\frac{\$}{yd^3}\) to \(\frac{\$}{m^3}\)}\\\\ & \frac{\$20}{1 \color{green}\cancel{yd^3}} \times \frac{1 \color{green}\cancel{yd^3}}{ 0.765\;m^3} &\color{navy}\small\text{Use conversion factor to cancel \(yd^3\)}\\\\ &\frac{\$20}{0.765\;m^3} &\color{navy}\small\text{Multiply across}\\\\ &\frac{\$26.14}{1\;m^3} &\color{navy}\small\text{Simplifying by dividing}\\\\ \end{align*}
    Simplify the numerator and the denominator to determine the answer, which is \($26.14 \;per\;1m^3\).

    Things to Remember


    • Steps for Volume Unit Conversions:
      1. Identify the units you have.
      2. Identify the units you want.
      3. Find the conversion factors that will help you step-by-step get to the units you want.
      4. Arrange conversion factors so that unwanted units cancel out through cross-cancellation. 

    Practice Problems

    1. A wooden block has a volume of 210 cubic centimeters (\(\text {cm}^3\)). Use the fact that 1 inch is approximately equal to 2.54 cm to convert this volume to cubic inches (\(\text {in}^3\)). Round your answer to the nearest tenth.
    \(\text{(1 in)}^{3}=\text{(2.54 cm)}^{3}\) (
    Solution
    x
    Solution: \(\displaystyle \frac{210\: \text{cm}^{3}}{1} \times \frac{1\:\text {in}^{3}}{16.387\:\text {cm}^{3}} = 12.8\:\text {in}^{3}\)
    )    2. The Great Pyramid of Giza has a total volume of 91,227,778 cubic feet (\( \text {ft}^3\)). Use the fact that 1 foot is approximately equal to 0.3048 meters to convert this volume to cubic meters (\( \text {m}^3\)). Round your answer to the nearest whole number.
    \(\text{(1 ft)}^3 = \text{(0.3048 m)}^3 \) (
    Solution
    x
    Solution: \(\displaystyle \frac{91,227,778\:\text{ft}^{3}}{1} \times \frac{0.02832\:\text{m}^{3}}{1\:\text {ft}^{3}} = 2,583,283\:\text{m}^{3}\)
    (Slight differences in rounding can cause the final answer to be off by a few numbers.)
    )    3. A cup of milk has a volume of 350 cubic centimeters (\( \text {cm}^3\)). Use the fact that 1 milliliter is equal to 1 cubic centimeter to convert this volume to milliliters. \(1\:\text{ml}=1\:\text{cm}^{3}\) (
    Solution
    x
    Solution: \(\displaystyle \frac{350\:\text {cm}^{3}}{1} \times \frac{1\:\text {ml}}{1\:\text {cm}^{3}} = 350\:\text {ml}\)
    )    4. Mary is landscaping her front yard and needs to fill an area with dirt that measures 2 m × 3 m × 5 m. Calculate the volume of dirt needed in cubic meters (\( \text {m}^3\)). Then convert the volume to cubic yards (\( \text {yd}^3\)) using the fact that 1 yard is approximately equal to 0.9144 meters. Round to the nearest hundredth.
    \(2\:\text{m}\times3\:\text{m}\times5\:\text{m}=30\:\text{m}^{3}\)
    \(\text{(1 yd)}^{3}=\text{(0.9144 m)}^{3}\) (
    Solution
    x
    Solution: \(\displaystyle \frac{30\:\text {m}^{3}}{1} \times \frac{1\:\text {yd}^{3}}{0.7646\:\text {m}^{3}} = 39.24\text {yd}^{3}\)

    Details:
    Step 1: Start with what you know.

    You need to fill an area with dirt that measures 2 m × 3 m × 5 m. To find the volume, multiply the three dimensions together: \(2\:\text{m}\times3\:\text{m}\times5\:\text{m}=30\:\text{m}^{3}\)

    So you need \(30\:\text{m}^{3}\) of dirt.

    Step 2: Determine what you want to get in the end. (Figure out what the end units should be.) = \(\text{yd}^{3}\)

    Step 3: Determine what conversion factor(s) to use. You may sometimes need more than one.

    Two equal cubes. The first is labeled with the dimensions 1 yard by 1 yard by 1 yard. The second is labeled with dimensions 0.9144 meter by 0.9144 meter by 0.9144 meter. Therefore, 1 yard cubed = 0.7646 meter cubed. 

    You know that 1 yd = 0.9144 m and you can use this information to find \( \text {yd}^3\):

    1 yd = 0.9144 m

    Cube both sides of the equation:

    \(\text{(1 yard)}^{3}\) = \(\text{(0.9144 m)}^{3}\)

    Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

    You have \( 30 \text {m}^3 \) so you need to multiply it by \(\dfrac{1\:\text{yd}^{3}}{0.7646\:\text{m}^{3}}\)

    Note that you chose the conversion factor that will enable you to cancel out the existing units and change them to the desired units (\( \text {m}^3\) is in the top of the original fraction, so you chose the conversion factor with \(\text {m}^3\) in the bottom so they will cancel out).

    Cancel out the \( \text {m}^3\).

    \(\displaystyle \frac{30\:\cancel{\text{m}^{3}}}{1}\times\frac{1\:\text{yd}^{3}}{0.7646\:\cancel{\text{m}^{3}}}\)

    Next, multiply straight across:

    \(\dfrac{30\times 1\:\text{yd}^{3}}{1\times0.7646}\)

    Which equals:

    \(\dfrac{30\:\text{yd}^{3}}{0.7646}\)

    Then divide 30 by 0.7646, which gives you the following:

    \(39.24\:\text{yd}^{3}\)

    You need \(39.24\:\text{yards}^{3}\) of dirt to fill that volume.
    )    5. The volume of a gumball is about 340 cubic millimeters (\( \text {mm}^3\)). Use the fact that 10 millimeters equal 1 centimeter to convert this volume to cubic centimeters (\( \text {cm}^3\)). Round to the nearest hundredth. \(\text{(10 mm)}^{3}\) = \(\text{(1 cm)}^{3}\) (
    Solution
    x
    Solution: \(\displaystyle \frac{340\:\text {mm}^{3}}{1} \times \frac{1\:\text{cm}^{3}}{1000\:\text{mm}^{3}} = .34\:\text{cm}^{3}\)
    )    6. A storage facility has a large open room with a total volume of 300,000 cubic feet (\( \text {ft}^3\)). Use the fact that 1 foot is approximately equal to 0.3048 meters to convert this volume to cubic meters (\( \text {m}^3\)). Round your answer to the nearest whole number.
    \(\text{(1 ft)}^{3}\) = \(\text{(0.3048 m)}^{3}\) (
    Solution
    x
    Solution: \(\displaystyle \frac{300,000\:\text {ft}^{3}}{1}\times\frac{0.0283\:\text{m}^{3}}{1\:\text{ft}^{3}}=8,490\:\text{m}^{3}\)
    (Slight differences in rounding can cause the final answer to be off by a few numbers.)

    Details:
    Step 1: Start with what you know: The room is 300,000 cubic feet.

    Step 2: Determine what you want to get in the end (figure out what the end units should be): cubic meters.

    Step 3: Determine what conversion factor(s) to use:

    You know the following:
    \(\text{1 ft} = 0.3048\; m\)

    This is a picture of two cubes that are equal in size. The dimensions of the first are 1 ft by 1ft by 1ft. The dimensions of the second cube are 0.3048 meter by 0.3048 meter by 0.3048 meter. By cubing both sides of the equation, 1 ft = 0.3048 meter, you get 1 ft cubed = 0.0283 meters cubed. 

    To determine how many cubic meters are in one cubic foot, start with the equation:
    \(\text{1 ft} = 0.3048\; m\)

    Then cube both sides of the equation:
    \(\text{(1 ft)}^{3}\) = \(\text{(0.3840 m)}^{3}\)

    Which gives you the following:
    \(\text{1 ft}^{3}\) = \(\text{0.0283 m}^{3}\)

    Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

    Since you have \(300,000\:\text{ft}^{3}\) you will multiply it by \(\dfrac{0.0283\:\text{m}^{3}}{1\:\text{ft}^{3}}\):

    \(\displaystyle \frac{300,000\:\text{ft}^{3}}{1}\times\dfrac{0.0283\:\text{m}^{3}}{1\:\text{ft}^{3}}\)

    Note that you chose the conversion factor that will enable you to cancel out the existing units and change them to the desired units (\(\text{ft}^{3}\) is on the top of the original fraction, so you chose the conversion factor with \(\text{ft}^{3}\) in the bottom so they will cancel out).

    Then cancel out \(\text{ft}^{3}\):
    \(\displaystyle \frac{300,000\:\cancel{\text{ft}^{3}}}{1}\times\frac{0.0283\:\text{m}^{3}}{1\:\cancel{\text{ft}^{3}}}\)

    Then multiply straight across:
    \(\displaystyle \frac{300,000\times0.0283\:\text{m}^{3}}{1\times1}\)

    Which equals:
    \(8490\:\text{m}^{3}\)
    )
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