9.3 Measuring Volume

Goods packed in cardboard boxes. — **Figure 9.9** Packing cartons sit on a loading dock ready to be filled.Packing cartons sit on a loading dock ready to be filled. (credit: “boxing day” by Erich Ferdinand/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

Identify reasonable values for volume applications.
Convert between like units of measures of volume.
Convert between different unit values.
Solve application problems involving volume.

Volume is a measure of the space contained within or occupied by three-dimensional objects. It could be a box, a pool, a storage unit, or any other three-dimensional object with attributes that can be measured in the metric unit for distance–meters. For example, when purchasing an SUV, you may want to compare how many cubic units of cargo the SUV can hold.

Cubic units indicate that three measures in the same units have been multiplied together. For example, to find the volume of a rectangular prism, you would multiply the length units by the width units and the height units to determine the volume in square units:

$1 cm \times 1 cm \times 1 cm = 1 {cm}^{3}$

Note that to accurately calculate volume, each of the measures being multiplied must be of the same units. For example, to find a volume in cubic centimeters, each of the measures must be in centimeters.

Reasonable Values for Volume

Because volume is determined by multiplying three lengths, the magnitude of difference between different cubic units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a cubic meter, m³, is $100 \times 100 \times 100, or 1,000,000$ times greater in area than a cubic centimeter, cm³. This relationship between benchmark metric volume units is shown in the following table.

Units	Relationship	Conversion Rate
km³ to m³	$km \times km \times km = {km}^{3}$ $1 km equals 1,000 m$ $1,000 m \times 1,000 m \times 1,000 m = 1,000,000,000 m^{3}$	1 km³ = 1,000,000,000 m³
m³ to dm³	$m \times m \times m = m^{3}$ $1 ⁢ m equals 10 ⁢ dm$ $10 dm \times 10 dm \times 10 dm = 1,000 {dm}^{3}$	1 m³ = 1,000 dm³
dm³ to cm³	$dm \times dm \times dm = {dm}^{3}$ $1 ⁢ dm equals 10 ⁢ cm$ $10 cm \times 10 cm \times 10 cm = 1,000,000 {cm}^{3}$	1 cm³ = 1,000 dm³
cm³ to mm³	$cm \times cm \times cm = {cm}^{3}$ $1 ⁢ cm equals 10 ⁢ mm$ $10 mm \times 10 mm \times 10 mm = 1,000 {mm}^{3}$	1 cm³ = 1,000 mm³

To have an essential understanding of metric volume, you must be able to identify reasonable values for volume. When testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you determine whether the given volume is reasonable for the situation.

Converting Like Units of Measures for Volume

Just like converting units of measure for distance, you can convert units of measure for volume. However, the conversion factor, the number used to multiply or divide to convert from one volume unit to another, is different from the conversion factor for metric distance units. Recall that the conversion factor for volume is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are illustrated in Figure 9.11.

An illustration shows four units: cubic millimeter, cubic centimeter, cubic decimeter, and cubic meter. — **Figure 9.11** Common Conversion Factors for Metric Volume Units

Understanding Other Metric Units of Volume

When was the last time you purchased a bottle of soda? Was the volume of the bottle expressed in cubic centimeters or liters? The liter (L) is a metric unit of capacity often used to express the volume of liquids. A liter is equal in volume to 1 cubic decimeter. A milliliter is equal in volume to 1 cubic centimeter. So, when a doctor orders 10 cc (cubic centimeters) of saline to be administered to a patient, they are referring to 10 mL of saline.

The most frequently used factors for converting from cubic meters to liters are listed in Table 9.2.

**Table 9.2**
m³ to L	m³ to mL
$1 ⁢ dm cubed equals 1 ⁢ L$	$1 ⁢ dm cubed equals 1,000 ⁢ mL$
$1,000 ⁢ cm cubed equals 1 ⁢ L$	$1 ⁢ cm cubed equals 1 ⁢ mL$
$1,000,000 ⁢ mm cubed equals 1 ⁢ L$	$1 ⁢ mm cubed equals 0.001 ⁢ mL$

Solving Application Problems Involving Volume

Knowing the volume of an object lets you know just how much that object can hold. When making a bowl of punch you might want to know the total amount of liquid a punch bowl can hold. Knowing how many liters of gasoline a car’s tank can hold helps determine how many miles a car can drive on a full tank. Regardless of the application, understanding volume is essential to many every day and professional tasks.

Example 10 Using Volume to Solve Problems

A cubic shipping carton’s dimensions measure $2 m \times 2 m \times 2 m$ . A company wants to fill the carton with smaller cubic boxes that measure $10 cm \times 10 cm \times 10 cm$ . How many of the smaller boxes will fit in each shipping carton?

Step 1: Determine the volume of the shipping carton.

$2 m \times 2 m \times 2 m = 8 m^{3}$

Step 2: Use the appropriate conversion factor to convert the volume of the shipping carton from m³ to cm³.

$\begin{matrix} 8 m^{3} \times 1,000 = 8,000 {dm}^{3} \\ 8,000 {dm}^{3} \times 1,000 = 8,000,000 {cm}^{3} \end{matrix}$

Step 3: Determine the volume of the smaller boxes.

$10 cm \times 10 cm \times 10 cm = 1,000 {cm}^{3}$

Step 4: Divide the volume of the shipping carton, in cm³, by the volume of the smaller box, in cm³.

$\frac{8,000,000 {cm}^{3}}{1,000 {cm}^{3}} = 8,000$

The shipping carton will hold 8,000 smaller boxes.

Example 12 Solving Complex Volume Problems

A fish tank measures 60 cm long, 15 cm wide and 34 cm tall (Figure 9.12). The tank is 25 percent full. How many liters of water are needed to completely fill the tank?

Figure 9.12

Step 1: Determine the volume of the fish tank in cubic centimeters.

$60 cm \times 15 cm \times 34 cm = 30,600 {cm}^{3}$

Step 2: Convert the volume in cm³ to volume in liters.

$\begin{matrix} 1 {cm}^{3} & = & 1 mL \\ 30,600 {cm}^{3} & = & 30,600 mL \\ \frac{30,600 mL}{1,000} & = & 30.6 L \end{matrix}$

Step 3: Since the tank is 25 percent full, the tank is 75 percent empty. Convert 75 percent to its decimal equivalent. Multiply the total volume by 75 percent expressed in decimal form to determine how many liters of water are required to fill the tank.

$\begin{matrix} 75 % & = & 0.75 \\ 30.6 \times 0.75 & = & 22.95 \end{matrix}$

So, 22.95 liters of water are needed to fill the tank.

Key Terms

volume
cubic units

Key Concepts

Volume is a measure of the space contained within or occupied by three-dimensional objects.
Volume is measured in cubic units; the base unit for volume in the metric system is cubic meters (m³)
The liter (L) is a metric unit of capacity but is often used to express the volume of liquids. One liter is equivalent to one cubic decimeter, which is the volume of a cube measuring $10 cm \times 10 cm \times 10 cm$ .

Formulas

To determine the volume of a rectangular prism:

$Volume = length (l) \times width (w) \times height (h)$

$\begin{matrix} or \\ V = l w h \end{matrix}$

A rectangular prism with its length, width, and height marked l, w, and h. — **Figure 9.13** Rectangular Prism with Height $(h)$ , Length $(l)$ , and Width $(w)$ Labeled

You can convert between metric volume units and metric capacity units using the relationships shown in Table 9.2.

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Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.