**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.

- Unit Conversions for Volume (06:46 mins) | Transcript
- Example of Using Unit Conversions for Volume (05:46 mins) | Transcript

**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*}

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.

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 unit 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.

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*}

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{red}\small\text{Conversion factor}\\\\ (1yd)^2 &= (3ft)^2 &\color{red}\small\text{Square both sides}\\\\ 1yd^2 &= 9ft^2 &\color{red}\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 Figures 5 and 6).

Figure 5

Figure 6

You can do the same thing using metric units and make the right side of Figure 6 into a square cube. Each square is a square cube with lengths of 1 foot by 1 foot by 1 foot. The dimensions of Figure 7 are 3 feet wide by 3 feet high by 1 foot depth.

Figure 7

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.

Figure 8

\begin{align*} 1yd &= 3ft &\color{red}\small\text{Conversion factor you know}\\\\ (1yd)^3 &= (3ft)^3 &\color{red}\small\text{Cube both sides}\\\\ 1yd^3 &=27ft^3 &\color{red}\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:

- 2 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 9 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?

Figure 9

**Steps for Volume Unit Conversions:**- Start with what units you know.
- Determine what units you want to get in the end.
- Determine what conversion factor(s) to use. You may sometimes need more than one.
- Multiply by 1 in the form of the conversion factor that cancels out unwanted units.

You know that \(1\;yard = 0.9144\;meters\). Now convert \(yd^2\) to \(m^2\):

\begin{align*} 1 yd &= 0.9144 m &\color{red}\small\text{Conversion factor you know}\\\\ 1 yd^2 &= (0.9144m)^2 &\color{red}\small\text{Square both sides}\\\\ 1 yd^2 &= 0.8361m^2\ &\color{red}\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{red}\small\text{Conversion factor you know}\\\\ 1 yd^3 &= (0.9144m)^3 &\color{red}\small\text{Cube both sides}\\\\ 1 yd^3 &= 0.765m^3\ &\color{red}\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{red}\small\text{Convert \(\frac{\$}{yd^3}\) to \(\frac{\$}{m^3}\)}\\\\ & \frac{\$20}{1 \color{red}\cancel{yd^3}} \times \frac{1 \color{red}\cancel{yd^3}}{ 0.765\;m^3} &\color{red}\small\text{Use conversion factor to cancel \(yd^3\)}\\\\ &\frac{\$20}{0.765\;m^3} &\color{red}\small\text{Multiply across}\\\\ &\frac{\$26.14}{1\;m^3} &\color{red}\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:**- Start with what units you know.
- Determine what units you want to get in the end.
- Determine what conversion factor(s) to use. You may sometimes need more than one.
- Multiply by 1 in the form of the conversion factor that cancels out unwanted units.

### 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.\(\text{(1 in)}^{3}=\text{(2.54 cm)}^{3}\) (

\(\text{(1 ft)}^3 = \text{(0.3048 m)}^3 \) (

\(2\:\text{m}\times3\:\text{m}\times5\:\text{m}=30\:\text{m}^{3}\)

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

\(\text{(1 ft)}^{3}\) = \(\text{(0.3048 m)}^{3}\) (

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