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Measuring Volume

Identify reasonable values for volume applications.

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.

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 \(100\times 100\times 100,\text{or}1,000,000\) 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	\(\text{km}\times \text{km}\times \text{km}={\text{km}}^{3}\) \(1\text{km}=1,000m\) \(1,000m\times 1,000m\times 1,000m=1,000,000,000{m}^{3}\)	1 km³ = 1,000,000,000 m³
m³ to dm³	\(m\times m\times m={m}^{3}\) \(1m=10\text{dm}\) \(10\text{dm}\times 10\text{dm}\times 10\text{dm}=1,000{\text{dm}}^{3}\)	1 m³ = 1,000 dm³
dm³ to cm³	\(\text{dm}\times \text{dm}\times \text{dm}={\text{dm}}^{3}\) \(1\text{dm}=10\text{cm}\) \(10\text{cm}\times 10\text{cm}\times 10\text{cm}=1,000,000{\text{cm}}^{3}\)	1 cm³ = 1,000 dm³
cm³ to mm³	\(\text{cm}\times \text{cm}\times \text{cm}={\text{cm}}^{3}\) \(1\text{cm}=10\text{mm}\) \(10\text{mm}\times 10\text{mm}\times 10\text{mm}=1,000{\text{mm}}^{3}\)	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.

Determining Reasonable Values for Volume

Try it.

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?

Solution

A volume of 3,375 km³ is equivalent to a rectangular prism with dimensions of \(15\text{km}\times 15\text{km}\times 15\text{km},\) which is far too large for a shipping box. An area of 3,375 m³ is equivalent to a surface of \(15m\times 15m\times 15m,\) which is also too large. A reasonable value for the volume of the box is 3,375 cm³.

Identifying Reasonable Values for Volume

Try it.

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?

Solution

A decimeter is equal to 10 centimeters. A box with a volume of 2 dm³ might have the dimensions \(1\text{dm}\times 1\text{dm}\times 2\text{dm}\), or \(10\text{cm}\times 10\text{cm}\times 20\text{cm}\), which is reasonable. A box with a volume of 2 cm³ or 2 mm³ would be too small.

Condensed — the full section is in OpenStax Contemporary Mathematics.

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 .

Converting Like Units of Measure for Volume Using Multiplication

Try it.

A pencil case has a volume of 1,700 cm³. 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 cm³ to mm³, multiply the value of the volume by 1,000.

\[1,700\times 1,000=1,700,000\]

The pencil case has a volume of 1,700,000 mm³.

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

Try it.

A shipping container has a volume of 33.2 m³. 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 m³ to cm³, first multiply the value of the volume by 1,000 to convert from m³ to dm³, and then multiply again by 1,000 to convert from dm³ to cm³.

\[33.2\times 1,000\times 1,000=33,200,000\]

The shipping container has a volume of 32,200,000 cm³.

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

Try it.

A holding tank at the local aquarium has a volume of 22,712,000,000 cm³. What is the volume in cubic meters?

Solution

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 cm³ to m³, divide the value of the volume by 1,000 to first convert from cm³ to dm³, then divide again to convert from dm³ to m³.

\[\begin{array}{l}{\text{cm}}^{3}{to dm}^{3}:\frac{22,712,000,000}{1,000}=22,712,000 \\ {\text{dm}}^{3}{to m}^{3}:\frac{22,712,000}{1,000}=22,712\end{array}\]

The holding tank has a volume of 22,712 m³.

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 .

m³ to L	m³ to mL
\(1{\text{dm}}^{3}=1L\)	\(1{\text{dm}}^{3}=1,000\text{mL}\)
\(1,000{\text{cm}}^{3}=1L\)	\(1{\text{cm}}^{3}=1\text{mL}\)
\(1,000,000{\text{mm}}^{3}=1L\)	\(1{\text{mm}}^{3}=0.001\text{mL}\)

Converting Different Units of Measure for Volume

Try it.

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?

Solution

Use division to convert from cubic centimeters to liters. To determine the equivalent volume in liters, convert from cm³ to L by dividing the value of the volume in cm³ by 1,000.

\[\frac{22,712,000,000{\text{cm}}^{3}}{1,000}=22,712,000L\]

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

Converting Different Units of Measure for Volume Using Multiplication

Try it.

An airplane used 150 m³ 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 m³ to dm³. Multiply the value of the volume by 1,000 to convert from m³ to dm³. Because \(1{\text{dm}}^{3}=1L\), the resulting value is equivalent to the number of liters used.

\[150\times 1,000=150,000{\text{dm}}^{3}=150,000L\]

The airplane used 150,000 liters of fuel.

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

Try it.

How many liters can a pitcher with a volume of 8,000,000 mm³ hold?

Solution

Use division to convert from a smaller metric volume unit to a larger metric volume unit. To convert from mm³ to dm³,

Step 1: Divide by 1,000 to convert from mm³ to cm³.

Step 2: Divide again by 1,000 to convert from cm³ to dm³.

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

\[\begin{array}{lll}\frac{8,000,000\ {\text{mm}}^{3}\ }{1,000} & = & 8,000\ {\text{cm}}^{3} \\ \frac{8,000\ {\text{cm}}^{3}}{1,000} & = & 8{\text{dm}}^{3}=8\ L\end{array}\]

The pitcher can hold 8 liters of liquid.

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.

Using Volume to Solve Problems

Try it.

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

Solution

Step 1: Determine the volume of the shipping carton.

\[2m\times 2m\times 2m=8{m}^{3}\]

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

\[\begin{array}{l}8{m}^{3}\times 1,000=8,000{\text{dm}}^{3} \\ 8,000{\text{dm}}^{3}\times 1,000=8,000,000{\text{cm}}^{3}\end{array}\]

Step 3: Determine the volume of the smaller boxes.

\[10\text{cm}\times 10\text{cm}\times 10\text{cm}=1,000{\text{cm}}^{3}\]

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

\[\frac{8,000,000{\text{cm}}^{3}}{1,000{\text{cm}}^{3}}=8,000\]

The shipping carton will hold 8,000 smaller boxes.

Solving Volume Problems with Different Units

Try it.

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.

\[6\text{cm}\times 6\text{cm}\times 20\text{cm}=720{\text{cm}}^{3}\]

Step 2: Convert the volume in cm³ to liters.

\[\begin{array}{lll}1\ {\text{cm}}^{3} & = & 1\ \text{mL} \\ 720\ {\text{cm}}^{3} & = & 720\ \text{mL} \\ \frac{720\ \text{mL}}{1,000} & = & 0.72\ L\end{array}\]

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

\[\frac{28,800L}{0.72L}=40,000\]

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

Condensed — the full section is in OpenStax Contemporary Mathematics.

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 \(10\text{cm}\times 10\text{cm}\times 10\text{cm}\).

Formulas

To determine the volume of a rectangular prism:

\[\text{Volume}=\text{ length }(l)\times \text{width }(w)\times \text{height}\ (h)\]\[\begin{array}{l}\text{or} \\ V=lwh\end{array}\]

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

Practice (12)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

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?
Onthul het antwoord

A volume of 3,375 km³ is equivalent to a rectangular prism with dimensions of \(15\text{km}\times 15\text{km}\times 15\text{km},\) which is far too large for a shipping box. An area of 3,375 m³ is equivalent to a surface of \(15m\times 15m\times 15m,\) which is also too large. A reasonable value for the volume of the box is 3,375 cm³.
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?
Onthul het antwoord

A decimeter is equal to 10 centimeters. A box with a volume of 2 dm³ might have the dimensions \(1\text{dm}\times 1\text{dm}\times 2\text{dm}\), or \(10\text{cm}\times 10\text{cm}\times 20\text{cm}\), which is reasonable. A box with a volume of 2 cm³ or 2 mm³ would be too small.
A farmer has a hay loft. They calculate the volume of the hayloft as 64 cm³. Does the calculation make sense? Explain your answer.

Onthul het antwoord

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 m². A volume of 64 m³ 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 m³.
A pencil case has a volume of 1,700 cm³. What is the volume in cubic millimeters?

Onthul het antwoord

Use multiplication to convert from a larger metric volume unit to a smaller metric volume unit. To convert from cm³ to mm³, multiply the value of the volume by 1,000.
\[1,700\times 1,000=1,700,000\]
The pencil case has a volume of 1,700,000 mm³.
A shipping container has a volume of 33.2 m³. What is the volume in cubic centimeters?

Onthul het antwoord

Use multiplication to convert a larger metric volume unit to a smaller metric volume unit. To convert from m³ to cm³, first multiply the value of the volume by 1,000 to convert from m³ to dm³, and then multiply again by 1,000 to convert from dm³ to cm³.
\[33.2\times 1,000\times 1,000=33,200,000\]
The shipping container has a volume of 32,200,000 cm³.
A holding tank at the local aquarium has a volume of 22,712,000,000 cm³. What is the volume in cubic meters?

Onthul het antwoord

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 cm³ to m³, divide the value of the volume by 1,000 to first convert from cm³ to dm³, then divide again to convert from dm³ to m³.
\[\begin{array}{l}{\text{cm}}^{3}{to dm}^{3}:\frac{22,712,000,000}{1,000}=22,712,000 \\ {\text{dm}}^{3}{to m}^{3}:\frac{22,712,000}{1,000}=22,712\end{array}\]
The holding tank has a volume of 22,712 m³.
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?

Onthul het antwoord

Use division to convert from cubic centimeters to liters. To determine the equivalent volume in liters, convert from cm³ to L by dividing the value of the volume in cm³ by 1,000.
\[\frac{22,712,000,000{\text{cm}}^{3}}{1,000}=22,712,000L\]
The holding tank holds 22,712,000 L of water.
An airplane used 150 m³ of fuel to fly from New York to Hawaii. How many liters of fuel did the airplane use?

Onthul het antwoord

Because 1 liter is equivalent to 1 cubic decimeter, use multiplication to convert from m³ to dm³. Multiply the value of the volume by 1,000 to convert from m³ to dm³. Because \(1{\text{dm}}^{3}=1L\), the resulting value is equivalent to the number of liters used.
\[150\times 1,000=150,000{\text{dm}}^{3}=150,000L\]
The airplane used 150,000 liters of fuel.
How many liters can a pitcher with a volume of 8,000,000 mm³ hold?

Onthul het antwoord

Use division to convert from a smaller metric volume unit to a larger metric volume unit. To convert from mm³ to dm³,
Step 1: Divide by 1,000 to convert from mm³ to cm³.
Step 2: Divide again by 1,000 to convert from cm³ to dm³.
Step 3: Use the unit value to express the volume in terms of liters.
\[\begin{array}{lll}\frac{8,000,000\ {\text{mm}}^{3}\ }{1,000} & = & 8,000\ {\text{cm}}^{3} \\ \frac{8,000\ {\text{cm}}^{3}}{1,000} & = & 8{\text{dm}}^{3}=8\ L\end{array}\]
The pitcher can hold 8 liters of liquid.
A cubic shipping carton’s dimensions measure \(2m\times 2m\times 2m\). A company wants to fill the carton with smaller cubic boxes that measure \(10\text{cm}\times 10\text{cm}\times 10\text{cm}\). How many of the smaller boxes will fit in each shipping carton?

Onthul het antwoord

Step 1: Determine the volume of the shipping carton.
\[2m\times 2m\times 2m=8{m}^{3}\]
Step 2: Use the appropriate conversion factor to convert the volume of the shipping carton from m³ to cm³.
\[\begin{array}{l}8{m}^{3}\times 1,000=8,000{\text{dm}}^{3} \\ 8,000{\text{dm}}^{3}\times 1,000=8,000,000{\text{cm}}^{3}\end{array}\]
Step 3: Determine the volume of the smaller boxes.
\[10\text{cm}\times 10\text{cm}\times 10\text{cm}=1,000{\text{cm}}^{3}\]
Step 4: Divide the volume of the shipping carton, in cm³, by the volume of the smaller box, in cm³.
\[\frac{8,000,000{\text{cm}}^{3}}{1,000{\text{cm}}^{3}}=8,000\]
The shipping carton will hold 8,000 smaller boxes.
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?

Onthul het antwoord

Step 1: Find the volume of the carton in cubic centimeters.
\[6\text{cm}\times 6\text{cm}\times 20\text{cm}=720{\text{cm}}^{3}\]
Step 2: Convert the volume in cm³ to liters.
\[\begin{array}{lll}1\ {\text{cm}}^{3} & = & 1\ \text{mL} \\ 720\ {\text{cm}}^{3} & = & 720\ \text{mL} \\ \frac{720\ \text{mL}}{1,000} & = & 0.72\ L\end{array}\]
Step 3: Divide the number of liters of orange juice produced each day by the volume of each carton.
\[\frac{28,800L}{0.72L}=40,000\]
The factory produces 40,000 cartons of orange juice each day.
A fish tank measures 60 cm long, 15 cm wide and 34 cm tall (). The tank is 25 percent full. How many liters of water are needed to completely fill the tank?

Onthul het antwoord

Step 1: Determine the volume of the fish tank in cubic centimeters.
\[60\text{cm}\times 15\text{cm}\times 34\text{cm}=30,600{\text{cm}}^{3}\]
Step 2: Convert the volume in cm³ to volume in liters.
\[\begin{array}{lll}1{\text{cm}}^{3} & = & 1\text{mL} \\ 30,600{\text{cm}}^{3} & = & 30,600\text{mL} \\ \frac{30,600\text{mL}}{1,000} & = & 30.6L\end{array}\]
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.
\[\begin{array}{lll}75\% & = & 0.75 \\ 30.6\times 0.75 & = & 22.95\end{array}\]
So, 22.95 liters of water are needed to fill the tank.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Measuring Volume

Identify reasonable values for volume applications.
Convert between like units of measures of volume.
Convert between different unit values.
Solve application problems involving volume.
3,375 km
3,375 m
3,375 cm
2 dm

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Probeer je eigen

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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