Contemporary Mathematics

# 9.3Measuring Volume

Contemporary Mathematics9.3 Measuring Volume

Figure 9.8 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:

1. Identify reasonable values for volume applications.
2. Convert between like units of measures of volume.
3. Convert between different unit values.
4. 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:

$1⁢cm×1⁢cm×1⁢cm=1⁢cm31⁢cm×1⁢cm×1⁢cm=1⁢cm3$

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:

$Volume= length (l)×width (w)×height(h)Volume= length (l)×width (w)×height(h)$
$orV=lwhorV=lwh$
Figure 9.9 Rectangular Prism with Height $(h)(h)$, Length $(l)(l)$, and Width $(w)(w)$ Labeled.

## 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 $100×100×100,⁢or⁢1,000,000100×100×100,⁢or⁢1,000,000$ 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 $km×km×km=km3km×km×km=km3$
$1km=1,000m1km=1,000m$
$1,000⁢m×1,000m×1,000m=1,000,000,000m31,000⁢m×1,000m×1,000m=1,000,000,000m3$
1 km3 = 1,000,000,000 m3
m3 to dm3 $m×m×m=m3m×m×m=m3$
$1⁢m=10⁢dm1⁢m=10⁢dm$
$10⁢dm×10⁢dm×10⁢dm=1,000⁢dm310⁢dm×10⁢dm×10⁢dm=1,000⁢dm3$
1 m3 = 1,000 dm3
dm3 to cm3 $dm×dm×dm=dm3dm×dm×dm=dm3$
$1⁢dm=10⁢cm1⁢dm=10⁢cm$
$10⁢cm×10⁢cm×10⁢cm=1,000,000⁢cm310⁢cm×10⁢cm×10⁢cm=1,000,000⁢cm3$
1 cm3 = 1,000 dm3
cm3 to mm3 $cm×cm×cm=cm3cm×cm×cm=cm3$
$1⁢cm=10⁢mm1⁢cm=10⁢mm$
$10⁢mm×10⁢mm×10⁢mm=1,000⁢mm310⁢mm×10⁢mm×10⁢mm=1,000⁢mm3$
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.

## 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 km3,
• 3,375 m3, or
• 3,375 cm3?

1.
Which represents a reasonable value for the volume of a storage area:
8 m3, 8 cm3, or 8 mm3?

## 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 dm3,
• 2 cm3, or
• 2 mm3?

1.
Which represents a reasonable value for the volume of a fish tank:
40,000 mm3, 40,000 cm3, or 40,000 m3?

## Example 9.21

### Explaining Reasonable Values for Volume

A farmer has a hay loft. They calculate the volume of the hayloft as 64 cm3. Does the calculation make sense? Explain your answer.

1.
An artist creates a glass paperweight. They decide they want to box and ship the paperweight, so they measure and determine that the volume of the cubic box is 125,000 mm3. Does their calculation make sense? Explain your answer.

## 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.

Figure 9.10 Common Conversion Factors for Metric Volume Units

## Example 9.22

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

A pencil case has a volume of 1,700 cm3. What is the volume in cubic millimeters?

1.
A jewelry box has a volume of 8 cm3. What is the volume of the jewelry box in cubic millimeters?

## Example 9.23

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

A shipping container has a volume of 33.2 m3. What is the volume in cubic centimeters?

1.
A gasoline storage tank has a volume of 37.854 m3. What is the volume of the storage tank in cubic centimeters?

## 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 cm3. What is the volume in cubic meters?

1.
A warehouse has a volume of 465,000,000 cm3. What is the volume of the warehouse in cubic meters?

## 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.

m3 to L m3 to mL
$1⁢dm3=1⁢L1⁢dm3=1⁢L$ $1⁢dm3=1,000⁢mL1⁢dm3=1,000⁢mL$
$1,000⁢cm3=1⁢L1,000⁢cm3=1⁢L$ $1⁢cm3=1⁢mL1⁢cm3=1⁢mL$
$1,000,000⁢mm3=1⁢L1,000,000⁢mm3=1⁢L$ $1⁢mm3=0.001⁢mL1⁢mm3=0.001⁢mL$
Table 9.2 Relationships Between Metric Volume and Metric Capacity Units

## 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 cm3? What is the capacity of the holding tank in liters?

1.
A gas can has a volume of 19,000 cm3. How much gas, in liters, does the gas can hold?

## Example 9.26

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

An airplane used 150 m3 of fuel to fly from New York to Hawaii. How many liters of fuel did the airplane use?

1.
A gasoline storage tank has a volume of 37.854 m3. What is the volume of the storage tank in liters?

## 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 mm3 hold?

1.
A glass jar has a volume of 800,000 mm3. How many mL of liquid can the glass jar hold?

## 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 $2m×2m×2m2m×2m×2m$. A company wants to fill the carton with smaller cubic boxes that measure $10⁢cm×10⁢cm×10⁢cm10⁢cm×10⁢cm×10⁢cm$. How many of the smaller boxes will fit in each shipping carton?

1.
A factory can mill 300 cubic meters of flour each day. They package the flour in boxes that measure $20\,{\text{cm}} \times 5\,{\text{cm}} \times 30\,{\text{cm}}$. How many boxes of flour does the factory produce each day?

## 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?

1.
An ice cream maker boxes frozen yogurt mix in boxes that measure 25 cm long, 8 cm wide and 35 cm tall. They produce 42,000 liters of frozen yogurt mix each day. How many boxes of frozen yogurt mix are produced each day?

## 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?

Figure 9.11

1.
A fish tank measures 75 cm long, 20 cm wide and 25 cm tall. The tank is 50 percent full. How many liters of water are needed to completely fill the tank?

## 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?

For the following exercises, determine the most reasonable value for each volume.
16.
Terrarium: 50,000 km3, 50,000 m3, 50,000 cm3, or 50,000 mm3
17.
Milk carton: 236,000 L, 236 L, 236,000 mL, or 236 mL
18.
Box of crackers: 1,500 km3, 1,500 m3, 1,500 cm3, or 1,500 mm3
For the following exercises, Convert the given volume to the units shown.
19.
42,500 mm3 = __________ cm3
20.
1.5 dm3 = __________ mL
21.
6.75 cm3 = __________ mm3
For the following exercises, determine the volume of objects with the dimensions shown.
22.
$20\,{\text{cm}} \times 20\,{\text{m}} \times 20\,\text{cm}$
$V =$ ________ L
23.
$12\,{\text{mm}} \times 5\,{\text{cm}} \times 1.7\,{\text{cm}}$
$V =$ ________ mL
24.
$7.3\,{\text{m}} \times 3.2\,{\text{m}} \times 7\,{\text{m}}$
$V =$ ________ m3

## Section 9.3 Exercises

For the following exercises, determine the most reasonable value for each volume.
1 .
Fish tank:
71,120 km3, 71,120 m3, 71,120 cm3, or 71,120 mm3
2 .
Juice box:
125,000 L, 125 L, 125,000 mL, or 125 mL
3 .
Box of cereal:
2,700 km3, 2,700 m3, 2,700 cm3, or 2,700 mm3
4 .
Water bottle:
5 L, 0.5 L, 5 mL, or 0.5 mL
5 .
Shoe box:
3,600 km3, 3.6 m3, 3,600 cm3, or 3,600 mm3
6 .
Swimming pool:
45 L, 45,000 L, 45 mL, or 45,000 mL
For the following exercises, convert the given volume to the units shown.
7 .
38,861 mm3 = __________ cm3
8 .
13 dm3 = __________ mL
9 .
874 cm3 = __________ mm3
10 .
4 m3 = __________ cm3
11 .
0.00003 m3 = _________ mm3
12 .
57,500 mm3 = _______ L
13 .
0.007 m3 = __________ L
14 .
8,600 cm3 = _________ m3
15 .
45.65 m3 = _______ cm3
16 .
0.06 m3 = __________ dm3
17 .
0.081 m3 = _________ mL
18 .
3,884,000 mm3 = _______ m3
For the following exercises, determine the volume of objects with the dimensions shown.
19 .
$30\,{\text{cm}} \times 20\,{\text{m}} \times 10\,{\text{cm}}$
$V =$ ________ L
20 .
$17\,{\text{mm}} \times 3\,{\text{cm}} \times 2.5\,{\text{cm}}$
$V =$ ________ mL
21 .
$3.4\,{\text{m}} \times 2.5\,{\text{m}} \times 10\,{\text{m}}$
$V =$ ________ m3
22 .
$325\,{\text{mm}} \times 20\,{\text{cm}} \times 0.05\,{\text{m}}$
$V =$ ________ cm3
23 .
$3.7\,{\text{m}} \times 4\,{\text{m}} \times 5.5\,{\text{m}}$
$V =$ ________ m3
24 .
$18\,{\text{dm}} \times 0.8\,{\text{m}} \times 150\,{\text{cm}}$
$V =$ ________ L
25 .
$15\,{\text{cm}} \times 400\,{\text{mm}} \times 3\,{\text{dm}}$
$V =$ ________ mL
26 .
$3.5\,{\text{cm}} \times 200\,{\text{mm}} \times 0.7\,{\text{dm}}$
$V =$ ________ cm3
27 .
$35\,{\text{m}} \times 1.2\,{\text{m}} \times 0.007\,{\text{km}}$
$V =$ ________ m3
28 .
A box has dimensions of $20\,{\text{cm}} \times 15\,{\text{cm}} \times 30\,{\text{cm}}$. The box currently holds 1,250 cm3 of rice. How many cubic centimeters of rice are needed to completely fill the box?
29 .
The dimensions of a medium storage unit are $4\,{\text{m}} \times 4\,{\text{m}} \times 8\,{\text{m}}$. What is the volume of a small storage area with dimensions half the size of the medium unit?
30 .
How much liquid, in liters, can a container with dimensions of $40\,{\text{cm}} \times 20\,{\text{cm}} \times 120\,{\text{cm}}$ hold?
31 .
What is the volume of the rectangular prism that is shown?
32 .
A box is 15 centimeters long and 5 centimeters wide. The volume of the box is 225 cm3. What is the height of the box?
33 .
Kareem mixed two cartons of orange juice, three 2-liter bottles of soda water and six cans of cocktail fruits to make a fruit punch for a party. The cartons of orange juice and cans of cocktail fruits each have a volume of 500 cm3. How much punch, in liters, did Kareem make?
34 .
A holding tank has dimensions of $16\,{\text{m}} \times 8\,{\text{m}} \times 8\,{\text{m}}$. If the tank is half-full, how more liters of liquid can the tank hold?
35 .
A large plastic storage bin has dimensions of $16\,{\text{cm}} \times 16\,{\text{cm}} \times 16\,{\text{cm}}$. A medium bin’s dimensions are half the size of the large bin. A small bin’s dimensions are the size of the medium bin. If the storage bins come in a set of 3—small, medium, and large—what is the total volume of the storage bin set in cubic centimeters?
36 .
A soft serve ice cream machine holds a 19.2 liter bag of ice cream mix. If the average serving size of an ice cream cone is 120 mL, how many cones can be made from each bag of mix?
37 .
A shipping carton has dimensions of $0.5\,{\text{m}} \times 0.5\,{\text{m}} \times 0.5\,{\text{m}}$. How many boxes with dimensions of $50\,{\text{mm}} \times 50\,{\text{mm}} \times 50\,{\text{mm}}$ will fit in the shipping carton?
38 .
A recipe for chili makes 3.5 liters of chili. If a restaurant serves chili in 250 mL bowls, how many bowls of chili can they serve?
39 .
A contractor is building an in-ground pool. They excavate a pit that measures $12\,{\text{m}} \times 9\,{\text{m}} \times 2.5\,{\text{m}}$. The dirt is being taken away in a truck that holds 30 m3. How many trips will the truck have to make to cart away all of the dirt?
40 .
A juice dispenser measures $30\,{\text{cm}} \times 30\,{\text{cm}} \times 30\,{\text{cm}}$. How many 375 mL servings will a full dispenser serve?
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