### 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:

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.

### FORMULA

The formula used to determine volume depends on the shape of the three-dimensional object. Here we will limit our discussions to the area to rectangular prisms like the one in Figure 9.9 Given this limitation, the basic formula for volume is:

### 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, m^{3}, is $100\times 100\times 100,\mathrm{or}\mathrm{1,000,000}$ times greater in area than a cubic centimeter, cm^{3}. This relationship between benchmark metric volume units is shown in the following table.

Units | Relationship | Conversion Rate |
---|---|---|

km^{3} to m^{3} |
$\mathrm{km}\times \mathrm{km}\times \mathrm{km}={\mathrm{km}}^{3}$ $1\mathrm{km}=\mathrm{1,000}\mathrm{m}$ $\mathrm{1,000}\mathrm{m}\times \mathrm{1,000}\mathrm{m}\times \mathrm{1,000}\mathrm{m}=\mathrm{1,000,000,000}{\mathrm{m}}^{3}$ |
1 km^{3} = 1,000,000,000 m^{3} |

m^{3} to dm^{3} |
$\mathrm{m}\times \mathrm{m}\times \mathrm{m}={\mathrm{m}}^{3}$ $1\mathrm{m}=10\mathrm{dm}$ $10\mathrm{dm}\times 10\mathrm{dm}\times 10\mathrm{dm}=\mathrm{1,000}{\mathrm{dm}}^{3}$ |
1 m^{3} = 1,000 dm^{3} |

dm^{3} to cm^{3} |
$\mathrm{dm}\times \mathrm{dm}\times \mathrm{dm}={\mathrm{dm}}^{3}$ $1\mathrm{dm}=10\mathrm{cm}$ $10\mathrm{cm}\times 10\mathrm{cm}\times 10\mathrm{cm}=\mathrm{1,000,000}{\mathrm{cm}}^{3}$ |
1 cm^{3} = 1,000 dm^{3} |

cm^{3} to mm^{3} |
$\mathrm{cm}\times \mathrm{cm}\times \mathrm{cm}={\mathrm{cm}}^{3}$ $1\mathrm{cm}=10\mathrm{mm}$ $10\mathrm{mm}\times 10\mathrm{mm}\times 10\mathrm{mm}=\mathrm{1,000}{\mathrm{mm}}^{3}$ |
1 cm^{3} = 1,000 mm^{3} |

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.

### Example 9.19

#### Determining Reasonable Values for Volume

A grandparent wants to send cookies to their grandchild away at college. Which represents a reasonable value for the volume of a box to ship the cookies:

- 3,375 km
^{3}, - 3,375 m
^{3}, or - 3,375 cm
^{3}?

#### Solution

A volume of 3,375 km^{3} is equivalent to a rectangular prism with dimensions of $15\mathrm{km}\times 15\mathrm{km}\times 15\mathrm{km},$ which is far too large for a shipping box. An area of 3,375 m^{3} is equivalent to a surface of $15\mathrm{m}\times 15\mathrm{m}\times 15\mathrm{m},$ which is also too large. A reasonable value for the volume of the box is 3,375 cm^{3}.

### Your Turn 9.19

### Example 9.20

#### Identifying Reasonable Values for Volume

A food manufacturer is prototyping new packaging for one of its most popular products. Which represents a reasonable value for the volume of the box:

- 2 dm
^{3}, - 2 cm
^{3}, or - 2 mm
^{3}?

#### Solution

A decimeter is equal to 10 centimeters. A box with a volume of 2 dm^{3} might have the dimensions $1\mathrm{dm}\times 1\mathrm{dm}\times 2\mathrm{dm}$, or $10\mathrm{cm}\times 10\mathrm{cm}\times 20\mathrm{cm}$, which is reasonable. A box with a volume of 2 cm^{3} or 2 mm^{3} would be too small.

### Your Turn 9.20

### Example 9.21

#### Explaining Reasonable Values for Volume

A farmer has a hay loft. They calculate the volume of the hayloft as 64 cm^{3}. Does the calculation make sense? Explain your answer.

#### Solution

No. Centimeters are used to determine smaller distances, such as the length of a pencil. A hayloft is more than 64 centimeters long, so a more reasonable unit of value would be m^{2}. A volume of 64 m^{3} can be calculated using the dimensions 4 meters by 4 meters by 4 meters, which are reasonable dimensions for a hayloft. So, a more reasonable value for the volume of the hayloft is 64 m^{3}.

### Your Turn 9.21

### 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 9.10.

### Example 9.22

#### Converting Like Units of Measure for Volume Using Multiplication

A pencil case has a volume of 1,700 cm^{3}. What is the volume in cubic millimeters?

#### Solution

Use multiplication to convert from a larger metric volume unit to a smaller metric volume unit. To convert from cm^{3} to mm^{3}, multiply the value of the volume by 1,000.

The pencil case has a volume of 1,700,000 mm^{3}.

### Your Turn 9.22

### Example 9.23

#### Converting Like Units of Measure for Volume Using Multi-Step Multiplication

A shipping container has a volume of 33.2 m^{3}. What is the volume in cubic centimeters?

#### Solution

Use multiplication to convert a larger metric volume unit to a smaller metric volume unit. To convert from m^{3} to cm^{3}, first multiply the value of the volume by 1,000 to convert from m^{3} to dm^{3}, and then multiply again by 1,000 to convert from dm^{3} to cm^{3}.

The shipping container has a volume of 32,200,000 cm^{3}.

### Your Turn 9.23

### Example 9.24

#### Converting Like Units of Measure for Volume Using Multi-Step Division

A holding tank at the local aquarium has a volume of 22,712,000,000 cm^{3}. What is the volume in cubic meters?

#### Solution

Figure 9.10 indicates that when converting from a smaller metric volume unit to a larger metric volume unit you divide using the given conversion factor. To convert from cm^{3} to m^{3}, divide the value of the volume by 1,000 to first convert from cm^{3} to dm^{3}, then divide again to convert from dm^{3} to m^{3}.

The holding tank has a volume of 22,712 m^{3}.

### Your Turn 9.24

### 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.

m^{3} to L |
m^{3} to mL |
---|---|

$1{\mathrm{dm}}^{3}=1\mathrm{L}$ | $1{\mathrm{dm}}^{3}=\mathrm{1,000}\mathrm{mL}$ |

$\mathrm{1,000}{\mathrm{cm}}^{3}=1\mathrm{L}$ | $1{\mathrm{cm}}^{3}=1\mathrm{mL}$ |

$\mathrm{1,000,000}{\mathrm{mm}}^{3}=1\mathrm{L}$ | $1{\mathrm{mm}}^{3}=0.001\mathrm{mL}$ |

### Example 9.25

#### Converting Different Units of Measure for Volume

A holding tank at the local aquarium has a volume of 22,712,000,000 cm^{3}? What is the capacity of the holding tank in liters?

#### Solution

Use division to convert from cubic centimeters to liters. To determine the equivalent volume in liters, convert from cm^{3} to L by dividing the value of the volume in cm^{3} by 1,000.

The holding tank holds 22,712,000 L of water.

### Your Turn 9.25

### Example 9.26

#### Converting Different Units of Measure for Volume Using Multiplication

An airplane used 150 m^{3} of fuel to fly from New York to Hawaii. How many liters of fuel did the airplane use?

#### Solution

Because 1 liter is equivalent to 1 cubic decimeter, use multiplication to convert from m^{3} to dm^{3}. Multiply the value of the volume by 1,000 to convert from m^{3} to dm^{3}. Because $1{\mathrm{dm}}^{3}=1\mathrm{L}$, the resulting value is equivalent to the number of liters used.

The airplane used 150,000 liters of fuel.

### Your Turn 9.26

### Example 9.27

#### Converting Different Units of Measure for Volume Using Multi-Step Division

How many liters can a pitcher with a volume of 8,000,000 mm^{3} hold?

#### Solution

Use division to convert from a smaller metric volume unit to a larger metric volume unit. To convert from mm^{3} to dm^{3},

**Step 1:** Divide by 1,000 to convert from mm^{3} to cm^{3}.

**Step 2:** Divide again by 1,000 to convert from cm^{3} to dm^{3}.

**Step 3:** Use the unit value to express the volume in terms of liters.

The pitcher can hold 8 liters of liquid.

### Your Turn 9.27

### 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 9.28

#### Using Volume to Solve Problems

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

#### Solution

**Step 1:** Determine the volume of the shipping carton.

**Step 2:** Use the appropriate conversion factor to convert the volume of the shipping carton from m^{3} to cm^{3}.

**Step 3:** Determine the volume of the smaller boxes.

**Step 4:** Divide the volume of the shipping carton, in cm^{3}, by the volume of the smaller box, in cm^{3}.

The shipping carton will hold 8,000 smaller boxes.

### Your Turn 9.28

### Example 9.29

#### Solving Volume Problems with Different Units

A carton of juice measures 6 cm long, 6 cm wide and 20 cm tall. A factory produces 28,800 liters of orange juice each day. How many cartons of orange juice are produced each day?

#### Solution

**Step 1:** Find the volume of the carton in cubic centimeters.

**Step 2:** Convert the volume in cm^{3} to liters.

**Step 3:** Divide the number of liters of orange juice produced each day by the volume of each carton.

The factory produces 40,000 cartons of orange juice each day.

### Your Turn 9.29

### Example 9.30

#### Solving Complex Volume Problems

A fish tank measures 60 cm long, 15 cm wide and 34 cm tall (Figure 9.11). The tank is 25 percent full. How many liters of water are needed to completely fill the tank?

#### Solution

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

**Step 2:** Convert the volume in cm^{3} to volume in liters.

**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.

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

### Your Turn 9.30

### WORK IT OUT

#### How Does Shape Affect Volume?

Take two large sheets of card stock. Roll one piece to tape the longer edges together to make a cylinder. Tape the cylinder to the other piece of card stock which serves as the base of the cylinder. Fill the cylinder to the top with cereal. Pour the cereal from the cylinder into a plastic storage or shopping bag. Remove the cylinder from the base and the tape from the cylinder. Re-roll the cylinder along the shorter edges a tape together. Attach the new cylinder to the base. Pour the cereal from the plastic bag into the cylinder. What do you observe? How does the shape of a container affect its volume?