finding the volume of rectangular prisms and problem solving

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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.

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.

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:

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 }

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.

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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.

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

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.

2 Find 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.

\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}

3 Write 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.

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

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

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

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.

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

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

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

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

\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}

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.

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}

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}

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 .

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

64=l \times w \times h

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

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 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.

Related volume lessons

• Volume of a cylinder
• Volume of a hemisphere
• Volume of a sphere
• Volume of a cone
• Volume of a triangular prism
• Volume of a pyramid
• Volume of square pyramid
• Volume formula
• Volume of a prism
• Volume of a cube

Practice volume of a rectangular prism questions

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

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}

2. Calculate the volume of the rectangular prism.

\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}

3. Calculate the volume of this rectangular prism.

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.

4. The volume of this rectangular prism is 600 \, cm^{3} . Find the height of the rectangular prism.

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 .

5. Calculate the volume of the rectangular prism.

\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}

6. The base of this prism is a square. The volume of the prism is 450 \, \mathrm{cm}^{3} . Find the height of the prism.

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

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

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 .

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.

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

The next lessons are

• Surface area
• Pythagorean theorem
• Trigonometry

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9.14: Volume of Rectangular Prisms

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Volume equals length times width times height

Ben's mom has grown frustrated with Ben's extensive Lego collection thrown all over his bedroom floor. She gives him a choice of three different boxes in which to store them. Ben wants to choose the box that will hold the most Legos. While the boxes are different in dimensions, Ben cannot figure out which will hold the most. The box measurements are as follows:

Box A: height = 5 inches, length = 18 inches, width = 10

Box B: height = 8 inches, length = 10 inches, width = 12 inches

Box C: height = 6 inches, length = 14 inches, width = 10 inches.

Which box has the greatest volume ?

In this concept, you will learn how to figure out the volume of rectanglar prisms.

Finding the Volume of a Rectangular Prism

Volume is the measure of how much three-dimensional space an object takes up or holds.

Imagine a fish aquarium. Its length, width, and height determine how much water the tank will hold. If you fill it with water, the amount of water is the volume that the tank will hold. You measure volume in cubic units , because you are multiplying three dimensions: length, width, and height.

One way to find the volume of a prism is to consider how many unit cubes it can contain. A unit cube is simply a cube measuring one inch, one centimeter, one foot, or whatever unit of measurement you are using, on all sides. Here are some unit cubes.

To use unit cubes to calculate volume, simply count the number of unit cubes that fit into the prism. Begin by counting the number of cubes that cover the bottom of the prism, and then count each layer. Let’s see how this works.

How many cubes do you see here? If you count all of the cubes, you will see that there are 24 cubes in this prism.

The volume of this prism is \(24 \text{ units}^{3}\) or cubic units.

Find the volume of the following figure using unit cubes.

How many cubes are in this figure? If you count the cubes, you will get a total of 48 cubes. The volume of this prism is 48 cubic units or \(\text{ units}^{3}\).

If you look carefully, you will see that the volume of the rectangular prism is a function of multiplying the length \times the width \times the height. You just discovered the formula for finding the volume of a rectangular prism. Now let’s refine that formula a little further. Here is the formula.

The volume is equal to the B, base area of the prism, times the height of the prism.

Let's look at an example.

Find the volume of the prism below.

Simply put the values for the length, width, and height in for the appropriate variables in the formula, then solve for V, volume.

First find the area of the base. This is the rectangular side on the bottom. Remember, to find the area of a rectangle, multiply the length times the width.

\(\begin{aligned} B&=lw \\ B&=16\times 9 \\ B&=144\text{ cm}^{2}\end{aligned}\)

The base area is 144 square centimeters. Now multiply this by the height.

\(\begin{aligned}V&=Bh \\ V&=144\times 4 \\ V&=576 \text{ cm}^{3}\end{aligned} \)

You can use the following formula for volume of a rectangular prism. This combines the two steps that you completed above:

\(\begin{aligned} V&=lwh \\ V&=(16)(9)(4) \\V&=576\text{ cm}^{3}\end{aligned}\)

The volume of this rectangular prism is 576 cubic centimeters.

You can work with the same rectangular prism, but fill it with unit cubes.

You can count the unit cubes here to find the volume of the rectangular prism. However, you save time by using the formula for volume.

Let's look at another example.

Find the volume of a container with a length of 15 ft, width of 12 ft, and height of 11 ft.

First, plug the values of the dimensions into the formula for volume of a rectangular prism and multiply the values for length and width:

\(\begin{aligned} V&=lwh \\ V&=(15)(12)(11) \\ V&=(180)(11)\end{aligned}\)

Next, multiply the results by the value for the height:

\(\begin{aligned} V&=(180)(11) \\V&=1,980\end{aligned}\)

Then, record the answer including the appropriate unit of measurement:

\(V=1,980\text{ ft}^{3}\)

The answer is the container has a volume of 1,980 cubic feet.

Example \(\PageIndex{1}\)

Earlier, you were given a problem about Ben, who is searching for the box that will hold the most Legos.

Ben needs to figure out which of the following boxes has the greatest volume.

Box A: height = 5 inches, length = 18 inches, width = 10 inches

Box C: height = 6 inches, length = 14 inches, width = 10 inches

Box A: \(V=(5)(18)(10)\)

\(V=(90)(10) \)

Box B: \(V=(8)(10)(12)\)

\(V=(80)(12) \)

Box C: \(V=(6)(14)(10)\)

\(V=(84)(10)\)

Box A: \(V=(90)(10)\)

Box B: \(V=(80)(12)\)

Box C: \(V=(84)(10)\)

Box A: \(V=900\text{ in}^{2}\)

Box B: \(V=960\text{ in}^{2} \)

Box C: \(V=840\text{ in}^{2}\)

The answer is Box B has the greatest volume and therefore can hold the most Legos.

Example \(\PageIndex{2}\)

Carla is cleaning out her fish tank, so she filled the bathtub to the rim with water for her fish to swim in while she empties their tank. If the bathtub is 5.5 feet long, 3.3 feet wide, and 2.2 feet deep, how much water can it hold?

\(\begin{aligned} V&=lwh \\ V&=(5.5)(3.3)(2.2) \\ V&=(18.15)(2.2)\end{aligned}\)

\(\begin{aligned} V&=(18.15)(2.2) \\ V&=39.93\end{aligned}\)

\(V=39.93\text{ ft}^{3}\)

The answer is Carla’s bathtub can hold 39.93 cubic feet of water.

Example \(\PageIndex{3}\)

Find the volume of a container with a length of 10 inches, width of 8 inches, and height of 6 inches.

\(\begin{aligned} V&=lwh \\ V&=(10)(8)(6) \\ V&=(80)(6)\end{aligned}\)

\(\begin{aligned} V&=(80)(6) \\ V&=480\end{aligned}\)

\(V=480\text{ in}^{3}\)

The answer is the container has a volume of 480 cubic inches.

Example \(\PageIndex{4}\)

Find the volume of a container with a length of 8 meters, width of 7 meters, and height of 3 meters.

\(\begin{aligned} V&=lwh \\ V&=(8)(7)(3) \\ V&=(56)(3)\end{aligned}\)

\(\begin{aligned} V&=(56)(3) \\ V&=168\end{aligned}\)

\(V=168\text{ m}^{3}\)

The answer is the container has a volume of 168 cubic meters.

Find the volume of each rectangular prism. Remember to label your answer in cubic units.

• Length = 5 in, width = 3 in, height = 4 in
• Length = 7 m, width = 6 m, height = 5 m
• Length = 8 cm, width = 4 cm, height = 9 cm
• Length = 8 cm, width = 4 cm, height = 12 cm
• Length = 10 ft, width = 5 ft, height = 6 ft
• Length = 9 m, width = 8 m, height = 11 m
• Length = 5.5 in, width = 3 in, height = 5 in
• Length = 6.6 cm, width = 5 cm, height = 7 cm
• Length = 7 ft, width = 4 ft, height = 6 ft
• Length = 15 m, width = 8 m, height = 10 m
• Length = 10.5 m, width = 11 m, height = 4 m
• Length = 12 ft, width = 12 ft, height = 8 ft
• Length = 16 in, width = 8 in, height = 8 in
• Length = 12 m, width = 12 m, height = 12 m
• Length = 24 in, width = 6 in, height = 6 in

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.10.

Video: Solid Geometry Volume

Practice: Volume of Rectangular Prisms

Volume of Rectangular Prism

The volume of a rectangular prism is the measurement of the total space inside it. Imagine a rectangular container filled with water. In this case, the total quantity of water that the container can hold is its volume. A prism is a polyhedron having identical bases, flat rectangular side faces, and the same cross-section all along its length. Prisms are classified on the basis of the shape of their base. A rectangular prism is categorized as a three-dimensional shape. It has six faces and all the faces of the prism are rectangles. Let us learn the formula to find the volume of a rectangular prism in this article.

What is the Volume of Rectangular Prism?

The volume of a rectangular prism is defined as the space occupied within a rectangular prism . A rectangular prism is a polyhedron that has two pairs of congruent and parallel bases. It has 6 faces (all are rectangular),12 sides, and 8 vertices. As the rectangular prism is a three-dimensional shape (3D shape) , the unit that is used to express the volume of the rectangular prism is cm 3 , m 3 and so on. In mathematics, any polyhedron , having all such characteristics can be referred to as a Cuboid.

Volume of Rectangular Prism Formula

The formula for the volume of a rectangular prism = base area × height of the prism. Since the base of a rectangular prism is a rectangle, its area will be l × w. This area is then multiplied by the height of the prism to get the volume of the prism. Therefore, another way to express this formula is by multiplying the length, width, and height of the prism and write the value in cubic units (cm 3 , m 3 , in 3 , etc).

Therefore, the formula for the volume of a rectangular prism is, volume of a rectangular prism (V) = l × w × h, where

• “l” is the base length
• “w” is the base width
• “h” is the height of the prism

To find the volume of the rectangular prism, we multiply the length, width, and height, that is the area of the base is multiplied by the height. It should be noted that the volume is measured in cubic units.

There are two types of rectangular prisms - right rectangular prisms and oblique prisms.

• In the case of a right rectangular prism, the bases are perpendicular to the other faces.
• In the case of an oblique rectangular prism, the bases are not perpendicular to the other faces. Thus, the perpendicular drawn from the vertex of one base to the other base of the prism will be taken as its height.

It should be noted that we can apply the same formula to calculate the volume of the prism, that is the rectangular prism volume formula, v = lwh, irrespective of the type of rectangular prism.

How to Find the Volume of a Rectangular Prism?

Before calculating the volume of a rectangular prism using the formula, we need to make sure that all the dimensions are of the same units. The following steps are used to calculate the volume of a rectangular prism.

• Step 1: Identify the type of the base and find its area using a suitable formula (as explained in the previous section).
• Step 2: Identify the height of the prism, which is perpendicular from the top vertex to the base of the prism.
• Step 3: Multiply the base area and the height of the prism to get the volume of the rectangular prism in cubic units. Volume = base area × height of the prism

Example: Calculate the volume of a rectangular prism whose height is 8 in and whose base area is 90 square inches.

Solution: We can calculate the volume of the rectangular prism using the following steps:

• Step 1: The base area is already given as 90 square inches.
• Step 2: The height of the prism is 8 in.
• Step 3: The volume of the given rectangular prism = base area × height of the prism = 90 × 8 = 720 cubic inches.

Volume of Rectangular Prism Examples

Example 1: If the volume of a rectangular prism is 40 cubic units and its base area is 10 square units, what is its height?

Solution: Given, the volume of the rectangular prism = 40 cubic units; and the base area of the rectangular prism= l × b = 10 square units.

Using the volume of rectangular prism formula, base area × height of the prism = 40 The height of the given rectangular prism = 10 × height = 40.

Thus, the height of the rectangular prism = 40/10 = 4 units

Example 2: If the base length of a rectangular prism is 8 inches, the base width is 5 inches and the height of the prism is 16 inches, find the volume of the rectangular prism.

Solution: Given: The base length of the rectangular prism (l) = 8 in, base width (w) = 5 in.

So the base area = l × w = 8 × 5 = 40 sq inches. The height of the prism is h = 16 in.

Using the volume of the rectangular prism formula, the volume of rectangular prism = base area × height of the prism = (40 × 16) = 640 cubic inches.

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Practice Questions on Volume of Rectangular Prism

Faqs on volume of rectangular prism, what is the volume of a rectangular prism.

The volume of a rectangular prism is the capacity that it can hold or the space occupied by it. Thus, the volume of a rectangular prism can be calculated by multiplying its base area by its height. The formula that is used to find the volume of a rectangular prism is, Volume (V) = height of the prism × base area. It is expressed in cubic units such as cm 3 , m 3 , in 3 , etc.

What is the Formula for the Volume of a Rectangular Prism?

The formula for the volume of a rectangular prism is, Volume (V) = base area × height of the prism. Another way to express this formula is, Volume = l × w × h; where 'l' is the length, 'w' is the width, and 'h' is the height of the prism.

In order to find the volume of a rectangular prism, follow the steps given below:

• Step 1: Identify the length (l) and width (w) of the base of the prism.
• Step 2: Find the height (h) of the prism.
• Step 3: Substitute the respective values in the formula, V = lwh
• Step 4: Find the product of these values to get the volume of the prism in cubic units.

How to Find the Height of a Rectangular Prism when the Volume of a Rectangular Prism is given?

The height of a rectangular prism can be calculated if the volume of the prism is given using the same formula, Volume (V) = base area × height of the prism. For example, the volume of a prism is 600 cubic units and its base area is 60 square units. Then, after substituting the values in the equation, we get 600 = 60 × height of the prism. This means, height = 600/60 = 10. Therefore, the height is 10 units.

What Happens to the Volume of Rectangular Prism if its Height is Doubled?

We know that the volume of a rectangular prism is the product of its three dimensions, that is, volume = length × width × height. If its height is doubled, its volume will be l × w × (2h) = 2lwh = 2 × v. Thus, we can say that the volume of the rectangular prism also gets doubled when its height is doubled.

What Happens to the Volume of Rectangular Prism if the Length is Doubled and the Height is Reduced to One Half?

The formula for calculating the volume of a rectangular prism is volume = length × width × height. If its length is doubled, and the height is reduced to half, then its volume can be written as, V = (2l) × (w) × (1/2h). After simplifying this, we get l × w × h which is the regular formula for volume. Thus, we can say that the volume of the rectangular prism will remain the same if its length gets doubled and the height is reduced to half.

What Happens to the Volume of Rectangular Prism if the Length, Width, and Height of Prism are Doubled?

The volume of a rectangular prism is the product of its three dimensions, that is volume = length × width × height. If its length, width, and height is doubled, then its volume will be (2l) × (2w) × (2h) = 8lwh = 8 × v. Thus, we can conclude that if its length, width, and height is doubled, the volume of the rectangular prism will be 8 times the original value.

How to Find the Volume of a Rectangular Prism with Fractions?

The volume of a rectangular prism can be calculated even if the values are given in fractions . For example, if the length of a rectangular prism is \(1\dfrac{3}{5}\) units, the width is 3/4, and its height is 2/3 units, the volume of the prism = l × w × h = \(1\dfrac{3}{5}\) × 3/4 × 2/3. Now, let us convert the mixed fraction into an improper fraction and then multiply all the fractions. We will first multiply the numerators and then the denominators and then reduce the resultant fraction, if needed. This means Volume = 8/5 × 3/4 × 2/3 = 48/60 = 4/5. Therefore the volume of the prism is 4/5 cubic units.

Rectangular Prism Calculator

What is a rectangular prism, how do i find the volume of a rectangular prism, how do i find the area of a rectangular prism, how do i calculate the diagonal of a rectangular prism, how to calculate volumes of the other solids.

Thanks to our rectangular prism calculator, you can easily find the cuboid volume, surface area, and rectangular prism diagonal. Whether you are wondering how much water your fish tank holds or trying to find out how much paper you need to wrap a gift, give this rectangular prism calculator a go! If you are still unsure how it works, keep scrolling to learn about rectangular prism formulas.

A right rectangular prism is a box-shaped object, that is, a 3-dimensional solid with six rectangular faces.

Rectangular prisms can also be oblique - leaning to one side - but in this instance, the side faces are parallelograms, not rectangles. When this happens, they are called oblique rectangular prism.

A right rectangular prism is also called a cuboid, box, or rectangular hexahedron. Moreover, the names "rectangular prism" and " right rectangular prisms" are often used interchangeably.

The most common math problems related to this solid are of the type right rectangular prism calc find V or find A , where the letters stand for the V olume and A rea , respectively. Let's see the necessary rectangular prism formula and learn how to solve t these problems quickly and easily.

We calculate the volume of a rectangular prism with the following formula:

volume = h × w × l ,

where h is the height of the prism, w is its width, and l is its length. To calculate the volume of a cardboard box:

• Find the length of the box , say 18 in.
• Determine its width - 12 in.
• Find out the height of the rectangular prism - 15 in.
• Calculate the volume of the cuboid . Using the rectangular prism volume formula, we get: volume = (18 × 12 × 15) in³ volume = 3240 in³ .

The surface area of the cuboid consists of 6 faces - three pairs of parallel rectangles. To find the rectangular prism surface area, add the areas of all faces:

surface_area = 2 × (h × w) + 2 × (h × l) + 2 × (l × w) = 2 × (h × w + h × l + l × w) ,

where h is prism height, w is its width, and l is its length.

Let's see an example of how to solve the right rectangular prism calc - find A problem. We'll come back to our example with the box and calculate its surface area:

• Calculate the rectangular prism surface area . First rectangle area is 15in × 12in = 180in² , second 15in × 18in = 270in² and third one 18in × 12in = 216in² . Add all three rectangles' areas - it's equal to 666 in² ( what a number! ) - and finally multiply by 2. The surface area of our cardboard box is 1332in².
• Or save yourself some time and use our rectangular prism calculator .

Finally, let's attack the right rectangular prism calc find d (that is, the diagonal) type of problem.

To determine the diagonal of a rectangular prism, follow these steps:

Apply the formula:

diagonal = √(l² + h² + w²)

where h is the height of the prism, w is its width, and l is its length.

Substitute the values.

Do you have the feeling that you saw the formula before? Yes, that's possible because this equation resembles the famous one from the Pythagorean theorem.

That rectangular prism was a piece of cake! If you are amazed at how easily you can calculate the volume with our tool, try out the other volume calculators:

• triangular prism calculator
• cylinder volume calculator
• sphere volume calculator
• cone volume calculator
• pyramid volume calculator

Be sure to check out the volume calculator - the volume of basic 3D solids, all in one place!

How many edges does a rectangular prism have?

The answer is twelve edges . A rectangular prism has:

• Eight vertices (or corners); and
• Twelve edges.

If you are not sure about the result, you can try to draw a rectangular prism and count its faces, vertices, and edges.

How do I calculate the volume of a rectangular prism with only its length?

You cannot calculate the volume of a rectangular prism, knowing only its length . You need to know its length, width, and height to calculate it. Once you have these parameters, you can use the equation:

volume = h × w × l .

What is the volume of a box with all sides equal?

Assuming that all sides are equal to 20 in, the volume is 8,000 in³ . We arrive at this answer by following these steps:

• Get the length, width, and height of the box.
• Using the rectangular prism volume formula, we get: volume = (20 × 20 × 20) in³ volume = 8,000 in³ .

How do I find the perimeter of a rectangular prism?

In a 3D solid, you find the surface area of the solid instead of the perimeter. For the rectangular prism, you can find its surface area using the formula:

ASA triangle

Christmas tree, exponential notation.

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How to Calculate the Volume of a Rectangular Prism

Last Updated: April 25, 2023 Fact Checked

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,271,537 times.

Volume is the amount of three-dimensional space taken up by an object. The computer or phone you're using right now has volume, and even you have volume. Finding the volume of a rectangular prism is actually really easy. Just multiply the length, the width, and the height of the rectangular prism. That's it! This article will walk you through the process step-by-step and show you an example.

• Ex: Length = 5 in.

• Ex: Width = 4 in.

• Ex: Height = 3 in.

• Ex: V = 5 in. * 4 in. * 3 in. = 60 in.

• 60 will become 60 in 3 .

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• ↑ http://www.softschools.com/math/geometry/topics/volume_of_a_rectangular_prism/
• ↑ https://www.khanacademy.org/math/cc-fifth-grade-math/5th-volume/volume-word-problems/a/volume-of-rectangular-prisms-review
• ↑ https://www.omnicalculator.com/math/rectangular-prism-volume
• ↑ http://lrd.kangan.edu.au/numbers/content/03_volume/03_page.htm
• ↑ https://sciencing.com/calculate-volume-rectangular-prism-2040920.html

About This Article

1. Find the length, width, and height of the rectangular prism. 2. Multiply the length, width, and height to get the volume. 3. Write the answer in cubic units. Did this summary help you? Yes No

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5th Grade Volume Worksheets

Welcome to our 5th Grade Volume Worksheets page.

Here you will find our collection of worksheets to introduce and help you learn about volume.

These worksheets will help you to understand and practice how to find the volume of rectangular prisms and other simple shapes.

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Volume of Rectangular Prisms

On this webpage you will find our range of worksheets to help you work out the volume of simple 3d shapes such as rectangular prisms.

They are also very useful for introducing the concept of volume being the number of cubes that fill up a space.

These sheets are graded from easiest to hardest, and each sheet comes complete with answers.

Using these sheets will help your child to:

• learn how to find the volume of simple 3d shapes by counting cubes;
• learn how to find the volume of rectangular prisms by multiplying length x width x height
• practice using their knowledge to solve basic volume problems.

What is Volume?

• Volume is the amount of space that is inside a 3 dimensional shape.
• Because it is an amount of space, it has to be measured in cubes.
• If the shape is measured in cm, then the volume would be measured in cubic cm or cm 3
• If the shape is measured in inches, then the volume would be measured in cubic inches or in 3

Volume of a Rectangular Prism

• The volume of a rectangular prism is the number of cubes it is made from.
• To find the number of cubes, we need to multiply the length by the width by the height.
• So Volume = length x width x height or l x w x h.
• We could also multiply the area of the base (which is the length x width) by the height.
• So Volume = l x w x h or b x h (where b is the area of the base)

In the example above, the length is 3, the width is 6 and the height is 2.

So the volume is 3 x 6 x 2 = 36cm 3 or 36 cubic cm.

This tells us that there are 36 cm cubes that make up the shape.

We have split our worksheets up into different sections, to make it easier for you to select the right sheets for your needs.

• Section 1 - Find the Volume by Counting Cubes
• Section 2 - Finding the Volume by multiplication
• Section 3 - Match the Volume (multiplication)
• Section 3 - Volume Problem Solving Riddles

5th Grade Volume Worksheets - Counting Cubes

• Volume - Count the Cubes Sheet 1
• PDF version
• Volume - Count the Cubes Sheet 2

5th Grade Volume Worksheets - Find the Volume by Multiplication

The first sheet is supported, the other two sheets are more independent.

You can choose between the standard or metric versions of sheets 2 and 3 (the measurements are the same)

• Find the Volume Sheet 1 (supported)
• Find the Volume Sheet 2 (standard)
• Find the Volume Sheet 2 (metric)
• Find the Volume Sheet 3 (standard)
• Find the Volume Sheet 3 (metric)

5th Grade Volume Worksheets - Match the Volume

• Match the Volume Sheet 1
• Match the Volume Sheet 2

5th Grade Volume Worksheets - Volume Riddles

• Volume Riddles Sheet 5A
• Volume Riddles Sheet 5B

Volume of Rectangular Prisms Walkthrough Video

This short video walkthrough shows several problems from our Find the Volume Sheet 2 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, check out the video below!

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Volume of a Cube/Cuboid/Box Calculators

Each of the pages below includes a handy calculator to help you find the volume of cubes, cuboids and boxes.

• Volume of a Cube Calculator

• Volume of a Box Calculator

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Word Problems Involving Volume of Cubes and Rectangular Prisms

Here, you will learn how to solve word problems involving the volume of cubes and rectangular prisms.

A step-by-step guide to using a table to write down a two-variable equation

here’s a step-by-step guide to solving word problems involving volume of cubes and rectangular prisms:

• Read the problem carefully: Make sure you understand the information provided in the problem and what is being asked of you.
• Identify the dimensions of the cube or rectangular prism: Look for the measurements of the length, width, and height of the object. Make sure you identify which dimension corresponds to which measurement.
• Calculate the volume: Once you have identified the dimensions of the object, use the formula for finding the volume of a cube or rectangular prism, depending on the shape of the object. The formula for the volume of a cube is \(V = s^3\), where s is the length of one side of the cube. The formula for the volume of a rectangular prism is \(V = lwh\), where l is the length, w is the width, and h is the height of the rectangular prism. Substitute the appropriate values into the formula to calculate the volume.
• Check your units: Make sure that the units of measurement used in the problem are consistent with the units used in the formula for finding the volume. If not, convert the units as necessary.
• Round your answer: Round your answer to the appropriate number of significant figures, as specified in the problem.
• Interpret your answer: Make sure you answer the question that is being asked in the problem. For example, if the problem asks for the volume of a rectangular prism in cubic inches, make sure your answer is expressed in cubic inches.
• Check your work: Double-check your calculations to make sure you have not made any errors.

It’s always a good idea to practice with a variety of problems to become comfortable with these steps.

Word Problems Involving Volume of Cubes and Rectangular Prisms – Example 1

Emma has a rectangular prism box for packing birthday presents that has a volume of 16600 cubic centimeters. The prism has a width of 28 centimeters and a height of 17 centimeters. What is the length of the box? Solution: To find out the length of the present box: Step 1: Multiply the width by height. \(28×17=476\) Step 2: Divide 16600 by 476 to find the length of the box. \(16600÷476=35\) So, the length is 35 cm.

Word Problems Involving Volume of Cubes and Rectangular Prisms – Example 2

Luci bought a handmade rug for her mother’s birthday party. She returned home with a box that was 32 inches long, 24 inches wide, and 9 inches tall. What is the volume of the box? Solution: To find out the volume of the box (V), use the formula for the volume of a rectangular prism. \(V=lwh\). \(32×24×9=6912 in^3\) So, the area of the box is \(6912 in^3\).

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Finding the Volume of Rectangular Prisms

5th - 7th grade, mathematics.

15 questions

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• 7. Multiple Choice Edit 5 minutes 1 pt What is volume? The amount of space of a 2-D object The amount of space in a 3-D object Water Area
• 9. Multiple Choice Edit 5 minutes 1 pt Find the volume of a rectangular prism with side lengths of 6m, 12 m, and 11 m. 58 m 3 270 m 3 540 m 3 792 m 3
• 11. Multiple Choice Edit 5 minutes 1 pt The laundry basket is 10 feet tall. It is 2 feet wide. The volume is 160 cubic feet. How long is the laundry basket?  20 ft.  8 ft. 172 ft.  160 ft

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