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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.
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,
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.
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:
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.
[FREE] Volume Check for Understanding Quiz (Grade 6 to 8)
Use this quiz to check your grade 6 to 8 students’ understanding of volume. 10+ questions with answers covering a range of 6th, 7th and 8th grade volume topics to identify areas of strength and support!
[FREE] Volume Check for Understanding Quiz (Grade 6 to 8)
Use this quiz to check your grade 6 to 8 students’ understanding of volume. 10+ questions with answers covering a range of 6th, 7th and 8th grade volume topics to identify areas of strength and support!
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}.
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,
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.
1. Each cube has a side length of 1 \, ft . Find the volume of the rectangular prism.
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:
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.
\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}
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.
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