A cuboid is a three-dimensional shape with [katex] 6 [/katex] rectangular faces, [katex] 8 [/katex] vertices, and [katex] 12 [/katex] edges. It is another name for a rectangular prism.

Volume of a Rectangular Prism - Steps, Examples & Questions

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Volume of a rectangular prism

Here you will learn about the volume of a rectangular prism, including what it is and how to find it.

Students will first learn about the volume of a rectangular prism as part of geometry in $5$ th grade. Students will expand on this knowledge in $6$ th grade to include rectangular prisms with fractional dimensions.

Every week, we teach lessons on finding the volume of a rectangular prism to students in schools and districts across the US as part of our online one-on-one math tutoring programs. On this page we’ve broken down everything we’ve learnt about teaching this topic effectively.

What is volume of a rectangular prism?

The volume of a rectangular prism is the amount of space there is within it. Since rectangular prisms are $3$ dimensional shapes, the space inside them is measured in cubic units.

$\text{Volume of a rectangular prism }=\text { length } \times \text { width } \times \text { height }$

For example,

Volume of a Rectangular Prism image 1 US

This rectangular prism is made from $24$ unit cubes – each side is $1 \, cm$ . That means the space within the rectangular prism, or the volume, is $24 \, \mathrm{cm}^3.$

Even though you can’t see all $24$ unit cubes, you can prove there are $24$ by thinking about the rectangular prism in parts.

Volume of a Rectangular Prism image 2 US

The bottom part of the prism is made up by $2$ rows of $4$ cubes – or $8$ total cubes. The bottom part has a volume of $8 \, \mathrm{cm}^3$ .

The other layers of the rectangular prism are exactly the same. Since the height is $3 \, cm$ , there are $3$ layers of cubes. Each layer has a volume of $8 \, \mathrm{cm}^3$ , so add them to find the total volume.


	$8 \, cm^3 + 8 \, cm^3 + 8 \, cm^3$ $V=24 \, cm^3$

Thinking about this further, the volume of a rectangular prism is the area of the base times the height. Consider the same rectangular prism. If you were to hold it up and look at the bottom, it would look like this:


	$4 \, cm \times 2 \, cm$ Area of the base $= 8 \, cm^2$

And the height of the prism is $3 \, cm$ , so $8 \, \mathrm{cm}^2 \times 3 \mathrm{~cm}=24 \mathrm{~cm}^3$ .

This is why the formula for the volume of a rectangular prism is:

$\text{Volume of a rectangular prism } = \text { length } \times \text { width } \times \text { height }$

What is volume of a rectangular prism?

[FREE] Volume Of A Rectangular Prism Worksheet (Grade 6 to 12)

Use this worksheet to check your grade 6 to 12 students’ understanding of volume of a rectangular prism. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE

[FREE] Volume Of A Rectangular Prism Worksheet (Grade 6 to 12)

Use this worksheet to check your grade 6 to 12 students’ understanding of volume of a rectangular prism. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE

Common Core State Standards

How does this relate to $5$ th grade math and $6$ th grade math?

Grade 5 – Geometry (5.G.C.5b)
Apply the formulas $V=l \times w \times h$ and $V=b \times h$ for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.
Grade 6 – Geometry (6.G.A.2)
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism.

Apply the formulas $V = l \times w \times h$ and $V = b \times h$ to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

How to calculate the volume of a rectangular prism

In order to find the volume of a rectangular prism with cube units shown:

Decide how many cubes make up the first layer.
Find the total of all the layers.
Write the answer and include the units.

In order to calculate the volume of a rectangular prism with the formula:

Write down the formula.
Substitute the values into the formula.
Solve the equation.
Write the answer and include the units.

Volume of a rectangular prism examples

Example 1: volume of a rectangular prism

Each cube has a side length of $1 \, inch$ . Find the volume of the rectangular prism.

Volume of a Rectangular Prism image 5 US

Decide how many cubes make up the first layer.

If you picked up the prism and looked at the bottom, you would see a $9$ by $2$ rectangle:

Volume of a Rectangular Prism image 6 US

The bottom layer of the rectangular prism is $9$ by $2$ cubes, so it is made up of $18$ inch cubes or $18 \text { inches}{ }^3.$

2Find the total of all the layers.

The height is $2 \, inches$ , so there are $2$ layers of cubes.

Since each layer has a volume of $18 \text { inches}^3$ , you can add the volume of each layer to find the total volume.

Volume of a Rectangular Prism image 6.1 US

$\begin{aligned} \text { Volume } & = 18 \, \text {inches}^3 + 18 \, \text {inches}^3 \\\\ & = 36 \, \text {inches}^3\end{aligned}$

You can also multiply the area of the base ( $\text {length } \times \text { width}$ ) times the $\text { height}$ .

$\begin{aligned} \text { Volume } & =18 \text { inches}^2 \times 2 \text { inches } \\\\ & =36 \text { inches}^3 \end{aligned}$

3Write the answer and include the units.

The dimensions of the rectangular prism are in inches, therefore the volume is in cubic inches ( $\text { inches}^3.$ )

$\text { Volume }=36 \text { inches}^3$

Example 2: volume of a cube

Calculate the volume of this cube.

Volume of a Rectangular Prism image 7 US

Write down the formula.

$\text {Volume }=\text { length } \times \text { width } \times \text { height }$

Substitute the values into the formula.

Since this is a cube, the length of the rectangular prism, width of the rectangular prism, and height of the rectangular prism are all $6 \,ft$ :

$\text {Volume }=6 \times 6 \times 6$

Solve the equation.

$\begin{aligned} & \text {Volume }=6 \times 6 \times 6 \\\\ & \text {Volume }=216 \end{aligned}$

Write the answer and include the units.

The dimensions of the rectangular prism are in feet, therefore the volume is in cubic feet ( $ft^3$ ).

$\text {Volume }=216 \, \mathrm{ft}^3$

Example 3: volume of a rectangular prism – different units

Calculate the volume of this rectangular prism.

Volume of a Rectangular Prism image 8 US

Write down the formula.

$\text {Volume }=\text { length } \times \text { width } \times \text { height }$

Substitute the values into the formula.

Notice here that one of the units is in $cm$ and the others are in $m$ . All the units should be the same to calculate the volume.

Change $cm$ to $m$ : $50 \,cm = 0.5 \,m$ .

Now that all of the measurements are in m, calculate the volume:

$\text {Volume }=4 \times 2 \times 0.5$

Solve the equation.

$\begin{aligned} & \text {Volume }=4 \times 2 \times 0.5 \\\\ & \text {Volume }=4 \end{aligned}$

Write the answer and include the units.

Since the dimensions of the rectangular prism were calculated in meters, the volume is in cubic meters.

$\text {Volume }=4 \, m^3$

Example 4: volume of a rectangular prism – fractions

Calculate the volume of this rectangular prism.

Volume of a Rectangular Prism image 9 US

Write down the formula.

$\text {Volume }=\text { length } \times \text { width } \times \text { height }$

Substitute the values into the formula.

$\text {Volume }=9 \times 13 \, \cfrac{2}{3} \, \times 5 \, \cfrac{3}{4}$

Solve the equation.

$\begin{aligned} \text {Volume } & =9 \times 13 \, \cfrac{2}{3} \, \times 5 \, \cfrac{3}{4} \\\\ & =\cfrac{9}{1} \, \times \, \cfrac{41}{3} \, \times \, \cfrac{23}{4} \\\\ & =\cfrac{8,487}{12} \\\\ & =707 \, \cfrac{3}{12} \text { or } 707 \, \cfrac{1}{4} \end{aligned}$

Write the answer and include the units.

$V=707 \, \cfrac{1}{4} \, f t^3$

Example 5: find the length of a rectangular prism given the volume

The rectangular prism below has a square base.

The height of the rectangular prism is $8 \, \cfrac{2}{9} \, m$ and the volume of the rectangular prism is $33 \, \mathrm{m}^3$ .

Find the area of the base.

Volume of a Rectangular Prism image 10 US

Write down the formula.

$\text {Volume }=\text { length } \times \text { width } \times \text { height }$

Substitute the values into the formula.

The volume of the rectangular prism is $33 \, \mathrm{m}^3$ and the height is $8 \, \cfrac{2}{9} \, m$ – fill those into the formula.

$\begin{aligned} & 33=l \times w \times 8 \, \cfrac{2}{9} \\\\ & 33=(\text {area of the base}) \times 8 \, \cfrac{2}{9} \end{aligned}$

Solve the equation.

To find the missing area, you can divide $33$ by $8 \, \cfrac{2}{9}$ :

$\begin{aligned} & 33 \, \div 8 \, \cfrac{2}{9} \\\\ & =33 \, \div \, \cfrac{74}{9} \\\\ & =\cfrac{33}{1} \, \times \, \cfrac{9}{74} \\\\ & =\cfrac{297}{74} \\\\ & =4 \, \cfrac{1}{74} \end{aligned}$

Write the answer and include the units.

Since the area of the base is calculated by $\text {length } \times \text { width}$ , the measurement is $2D$ and the units are squared.

The area of the base is $4 \, \cfrac{1}{74} \, m^2.$

Example 6: dimensions of a cube given the volume

Find the dimensions of a cube that has a volume of $64 \, \mathrm{mm}^3$ .

Write down the formula.

$\text {Volume }=\text { length } \times \text { width } \times \text { height }$

Substitute the values into the formula.

The only value known is the volume which is $64 \, \mathrm{mm}^3$ – fill it into the formula.

$64=l \times w \times h$

Solve the equation.

Since the shape is a cube, the length, width and height are all the same. The missing number, when multiplied by itself three times, makes $64$ .

Since $64$ is even, the number multiplied will also be even. It will also be much smaller than $64$ , since it was multiplied $3$ times by itself to get to $64$ .

With this in mind, start guessing and checking with smaller, even numbers.

Let’s try $2$ …

$2 \times 2 \times 2=8$

Let’s try $4$ …

$4 \times 4 \times 4=64$

Write the answer and include the units.

The dimensions of the cube are $4 \, \mathrm{mm} \times 4 \, \mathrm{mm} \times 4 \, \mathrm{mm}.$

[FREE] Volume Of A Rectangular Prism Worksheet (Grade 6 to 12)

Use this worksheet to check your grade 6 to 12 students’ understanding of volume of a rectangular prism. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE

[FREE] Volume Of A Rectangular Prism Worksheet (Grade 6 to 12)

Use this worksheet to check your grade 6 to 12 students’ understanding of volume of a rectangular prism. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE

Teaching tips for volume of rectangular prism

In the beginning, focus on activities and discussions that show the units (cubes) within rectangular prisms and connect such representations to the volume formula.

Worksheets are easy ways to provide practice problems but be sure to include ones that include word problems or real-life applications of volume.

Easy mistakes to make

Writing the incorrect units or forgetting to include the units
Always include units when recording a measurement. Volume is measured in cubic units.
For example, $\mathrm{mm}^3, \mathrm{~cm}^3, \mathrm{~m}^3$ etc.

Calculating volume with different units
All measurements must be in the same units before calculating volume.
For example,

The units need to all be the same before calculating the volume. Convert the $cm$ to $mm$ or the $mm$ to $cm$ and then multiply the three.

Thinking that converting the units changes the volume (actual space taken up)
The number representing the volume will change based on the units it is being calculated in, but the volume has the same value.
For example,

$\text { Volume: } 4 \times 1 \times 2=8 \, m^3 \quad \text { Volume: } 400 \times 100 \times 200=32,000,000 \, \mathrm{~cm}^3$

The rectangular prisms above are the same size – but there is one measured in $\text{meters}$ and one measured in $\text{centimeters.}$ Notice how the number value for the volume is very different, but this is only because it was calculated in different units – not because their actual volumes are different.

Calculating surface area instead of volume
Surface area and volume are different types of measurement – surface area is the total area of the faces and is measured in square units, and volume is the space within the shape and is measured in cubic units.

Practice volume of a rectangular prism questions

128 \, \mathrm{ft}^3

64 \, \mathrm{ft}^3

88 \, \mathrm{ft}^3

96 \, \mathrm{ft}^3

If you picked up the prism and looked at the bottom, you would see a $6$ by $4$ rectangle:

Volume of a Rectangular Prism image 14 US

The bottom layer of the rectangular prism is $6$ by $4$ cubes, so it is made up of $24$ feet cubes or $24 \, \text {ft}^3.$

The height is $4 \, feet$ , so there are $4$ layers of cubes. Since each layer has a volume of $24 \, \text {ft}^3$ , you can add the volume of each layer to find the total volume.

Volume of a Rectangular Prism image 15 US

$\begin{aligned} \text {Volume } & =24 \, f t^3+24 \, f t^3+24 \, f t^3+24 \, f t^3 \\\\ & =96 \, f t^3 \end{aligned}$

You can also multiply the area of the base ( $\text {length } \times \text { width}$ ) times the $\text {height}$ .

$\begin{aligned} \text {Volume } =24 \, \mathrm{ft}^2 \times 4 \, f t \\\\ & =96 \, \mathrm{ft}^3 \end{aligned}$

144 \, \mathrm{cm}^{2}

12 \, \mathrm{cm}^{3}

144 \, \mathrm{cm}^{3}

36 \, \mathrm{cm}^{3}

$\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height } \\\\ \text{Volume }&= 12 \times 3 \times 4 \\\\ \text{Volume }&= 144 \, \mathrm{cm}^{3} \end{aligned}$

1,748 \, \mathrm{cm}^{3}

174.8 \, \mathrm{m}^{3}

17.48 \, \mathrm{m}^{3}

17.48 \, \mathrm{cm}^{3}

There are measurements in $cm$ and $m$ , so convert the units before calculating the volume: $380 \, cm$ to $3.8 \, m$ .

$\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\ \text{Volume }&= 2.3 \times 2 \times 3.8 \\\\ \text{Volume }&=17.48 \,\mathrm{m}^{3} \end{aligned}$

Since the measurements used were in meters, the volume will be in cubic meters.

3 \, cm

24 \, cm

200 \, cm

75 \, cm

$\text {Volume }=\text { length } \times \text { width } \times \text { height }$

Fill in the known values:

$\begin{aligned} & 600=8 \times 25 \times h \\\\ & 600=200 \times h \end{aligned}$

The missing height times $200$ is equal to $600$ , so $h=3 \, \mathrm{cm}$ because $200 \times 3=600$ .

186 \, \cfrac{25}{36} \, \mathrm{ft}^3

132 \, \cfrac{1}{6} \, \mathrm{ft}^3

219 \, \cfrac{7}{9} \, \mathrm{ft}^3

360 \, \cfrac{1}{12} \, \mathrm{ft}^3

$\begin{aligned} & \text {Volume }=\text { length } \times \text { width } \times \text { height } \\\\ & \text {Volume }=11 \, \cfrac{3}{4} \, \times 4 \, \cfrac{1}{3} \, \times 3 \, \cfrac{2}{3} \\\\ & \text {Volume }=\cfrac{47}{4} \, \times \, \cfrac{13}{3} \, \times \, \cfrac{11}{3} \\\\ & \text {Volume }=\cfrac{6,721}{36} \\\\ & \text {Volume }=186 \, \cfrac{25}{36} \, \mathrm{ft}^3 \end{aligned}$

45 \, cm

4.5 \, cm

2.1 \, cm

6.7 \, cm

Since the base of the prism is a square, the length and the width are both $10 \, cm$ .

$\begin{aligned} & \text {Volume }=\text { length } \times \text { width } \times \text { height } \\\\ & 450=10 \times 10 \times h \\\\ & 450=100 \, h \end{aligned}$

The missing height times $100$ is equal to $450$ , so $h=4.5 \, \mathrm{cm}$ because $100 \times 4.5=450$ .

Volume of a rectangular prism FAQs

What is a cuboid?

A cuboid is a three-dimensional shape with $6$ rectangular faces, $8$ vertices, and $12$ edges. It is another name for a rectangular prism.

What is the volume of a rectangular prism formula?

Volume of a rectangular prism is the area of the base (length times width) times the height of the rectangular prism or $l \times w \times h$ .

Is finding the volume of a triangular prism the same as a rectangular prism?

No, but the formula for both can be stated as the $\text {area of the base } \times \text {height of the prism}$ . However, since there are different formulas for finding the area of a rectangle and finding the area of a triangle, they are not found with the exact same formula.

How do you calculate the surface area of a rectangular prism?

Find the area of the six faces and then add them together.

The next lessons are

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