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Volume of Prism

The volume of a prism is the capacity of the prism. There can be different types of prisms like a triangular prism, square prism, rectangular prism, pentagonal prism, hexagonal prism, or octagonal prism. But the method to write the formula of volume of any prism remains the same immaterial of the type of prism. The concept of writing the formula for the volume of the prism stays the same in any case. Like all three-dimensional shapes, you will learn how to calculate the volume of any type of prism. Stay tuned to learn how to use its formula using solved examples.

What is the Volume of Prism?

The volume of a prism is defined as the amount of space a prism occupies. A prism is a solid 3-D shape that has two same faces and other faces that resemble a parallelogram . It is a polyhedron whose naming convention is influenced by the different shapes of the bases. In the case of prisms, every prism has a different base, triangular prism (triangular base), square prism (square base), rectangular prism (rectangular base), pentagonal prism (pentagonal base), hexagonal prism (hexagonal base), or an octagonal prism (octagonal base). Thus, as each prism is a three-dimensional shape, the volume of every prism also lies in a three-dimensional plane. The unit of volume of a prism is given as cubic meters, cubic centimeters, cubic inches or cubic feet, etc.

Formula of Volume of Prism

The formula for the volume of a prism is given by the product of the area of the base and height of the prism.

Thus, as the bases of different types of prisms are different so are the formulas to determine the volume of the prism. Look at the table below to understand this concept better:

Thus, the volume of a prism can be given as V = B × H where V is the volume , B base area, and H height of the prism. The unit of base area is given in square units and the height of the prism is given in units. Thus, the unit of volume of the prism is given as V = (square units) × (units) = cubic units.

How to Calculate the Volume of Prism?

The steps to determine the volume of the prism are:

• Step 1: Write the given dimensions of the prism.
• Step 2: Determine the volume of the prism using the formula V = B × H where V, B, and H are the volume, base area, and height of the prism.
• Step 3: The value of the volume of the prism is once obtained then add the unit of volume of prism in the end (in terms of cubic units).

Example: Find the volume of a prism whose base area is 3 square inches and height is 7 inches.

Solution: As we know, the volume of the prism is V = B × H. Given that: B = 3 square inches, H = 7 inches Thus, the Volume of the prism, V = B × H ⇒ V = 3 × 7 = 21 in 3 Therefore, the volume of the prism is 21 cubic inches.

Important Notes

• The volume of any prism depends on the shape of its base. With the change in the shape of the base, the area of the base changes as well.

Volume of Prism Examples

Example 1: What is the base area of the prism if the volume of the prism is 324 cubic units and the height of the prism is 9 units.

The given dimensions are the volume of the prism = 324 cubic units and the height of the prism = 9 units. Let the base area of the prism be "B".

Substituting the values in the volume of the prism formula, Volume of prism = V = B × H = 324 cubic units ⇒ 9B = 324 ⇒ B = 36 square units

Therefore, the base area of the prism is 36 square units.

Example 2: Find the height of the prism if the volume of the prism is 729 cubic units and the base area is 27 square units.

The given dimensions are the volume of the prism = 729 cubic units and the base area of prism = 27 square units. Let the height of the prism be "H".

Substituting the values in the volume of the prism formula, Volume of prism = V = B × H = 729 cubic units ⇒ 27H = 729 ⇒ H = 27 units

Therefore, the height of the prism is 27 units.

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Practice Questions on Volume of Prism

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

The amount of space occupied by a prism is referred to as the volume of a prism. The volume of the prism depends on the base radius of the prism and the height of the prism. The unit of volume of the prism is expressed in m 3 , cm 3 , in 3 , or ft 3 .

What is the Formula for Volume of Prism?

The formula for the volume of a prism is obtained by taking the product of the base area and height of the prism. The volume of a prism is given as V = B × H where, "V" is the volume of the prism, "B" is the base area of the prism, and "H" is the height of the prism.

How to Find the Volume of Prism?

We can find the volume of the prism using the following steps:

• Step 1: First write the given dimensions of the prism.
• Step 2: Find the volume of the prism using the formula V = B × H where "V", "B", and "H" are the volume, base area, and height of the prism.
• Step 3: Once the value of the volume of the prism is obtained, write the unit of volume of prism in the end (in terms of cubic units).

How Do You Find the Base Area of Prism if the Volume of Prism is Given?

The steps to determine the base area of the prism, if the volume of the prism is given, are:

• Step 2: Substitute the given values in the formula V = B × H where "V", "B", and "H" are the volume, base area, and height of the prism.
• Step 3: Now solve the equation for "B".
• Step 3: Once the value of the base area of the prism is obtained, write the unit of the base area prism in the end (in terms of square units).

What Happens to the Volume of Prism if the Base Area of Prism is Doubled?

The volume of the prism depends on the base radius of the prism. Thus, the volume of the prism doubles if the base area of the prism is doubled as "B" is substituted by "2B" as V = (2B) × H = 2 (B × H) which is double the original volume of the prism.

What Happens to the Volume of Prism When the Base Area and Height are Doubled?

The volume of the prism will quadruple the original volume if the base area and height of the prism are doubled as, radius, "B" is substituted by 2B, and height, "H" is substituted by 2H. Thus, V = (2B) × (2H) = 4 (B × H) which is four times the original volume of the prism.

How Does the Volume of Prism Change if the Type of Prism Changes?

The volume of the prism depends on the base area of the prism. As the type of prism changes, the base of the prism changes thereby changing the base area of the prism. This change in the base area of the prism changes the volume of the prism.

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Volume of a Prism Worksheets

This extensive compilation of printable volume of prisms worksheets enables 7th grade, 8th grade, and high school students to find the volume of triangular, rectangular, trapezoidal and polygonal prisms. Featured here are innumerable exercises to practice finding the volume of prisms using dimensions of varying base faces expressed as decimals and integers. The worksheets offer problems presented as 3D shapes and word format with easy and moderate levels classified based on the range of numbers used. Access some of them for free!

Find the Volume of Prisms using Base Area | Integers

The area of the base face of each prism and a dimension are expressed as integers. Multiply the two to compute the volume of the triangular and rectangular prisms featured in these worksheets.

• Download the set

Find the Volume of Prisms using Base Area | Decimals

Intensify the practice in calculating the volume of prisms efficiently and accurately using decimal measures. Determine the volume of six 3D shapes and two-word format problems in each printable worksheet for grade 7.

Volume of Prisms | Bases - Trapezoid and Parallelogram

Incorporate these pdf worksheets on volume of prisms encompassing trapezoidal and parallelogram bases offering an easy level for beginners with dimensions ≤ 20 and a moderate level with 2-digit dimensions.

Volume of Prisms | Bases -Trapezoid and Parallelogram | Decimals

Work out each problem by finding the area of the trapezoidal or parallelogram base faces. To do this, simply plug the known values expressed as decimals into the formula and determine the volume.

Volume of Polygonal Prisms | Level 1

Get a hands-on experience in finding the volume of regular prisms using the side length, apothem and the height given. The polygons featured here range from three-sided to six-sided figures.

Find the Volume of Polygonal Prisms | Level 2

Raise the bar with this set of volume of polygonal prism worksheets. Employ the height and the given perimeter or side length to find the apothem and then determine the volume of each prism.

Volume of Prisms | Revision - Integers | Level 1

Perfect as a review exercise, these printable worksheets present prisms with triangular or quadrilateral bases with two levels of difficulty. Begin with the easy level, then move on to the moderate level.

Volume of Prisms | Mixed Review - Decimals | Level 1

Find the volume of each of the eight prisms that are a mix of triangular and quadrilateral base faces with measures denoted as decimals. Direct students of grade 8 and high school to apply relevant formulas to solve for the volume.

Volume of Prisms | Mixed Review | Level 2

Intensify your practice and level up with this cluster of pdf volume of mixed prisms worksheets involving triangular, quadrilateral and polygonal bases and dimensions depicted as decimals and integers.

Find the Missing Dimensions

Review the concept of finding the volume with prisms whose bases are either triangles, squares, rectangles or parallelograms. Use the volume and the dimension provided to solve for the unknown measure.

Volume of Rectangular Prisms

Get your brains active with this bundle of rectangular prisms pdf worksheets. Compute the volume of rectangular prisms and crosscheck with the answer keys for an instant validation.

(33 Worksheets)

Volume of Triangular Prisms

Learn to apply the area of the cross-section and the known dimensions in the formula to find the volume of each triangular prism. Practice finding the missing dimensions as well.

(24 Worksheets)

Related Worksheets

» Volume of Composite Shapes

» Volume by Counting Cubes

» Volume of Cubes

» Volume of Cylinders

» Volume of Mixed Shapes

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Volume Of A Prism

Here we will learn about the volume of a prism, including how to calculate the volume of a variety of prisms and how to find a missing length given the volume of a prism.

There are also volume and surface area of a prism worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is the volume of a prism?

The volume of a prism is how much space there is inside a prism. 

Imagine filling this L-shaped prism fully with water. The total amount of water inside the prism would represent the volume of the prism in cubic units.

To calculate the volume of a prism, we find the area of the cross section and multiply it by the depth .

Volume of prism = Area of cross section x depth

How to calculate the volume of a prism

In order to calculate the volume of a prism:

Write down the formula.

Calculate the area of the cross section.

Calculate the volume of the prism.

Write the answer, including the units.

Volume of a prism worksheet

Get your free volume of a prism worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on prism shape

Volume of a prism  is part of our series of lessons to support revision on  prism shape . You may find it helpful to start with the main prism shape lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Prism shape
• Surface area of a prism

Volume of a prism examples

Example 1: volume of a triangular prism.

Work out the volume of the triangular prism:

Volume of prism = Area of cross section × depth

2 Calculate the area of the cross section.

The area of the triangle is 12cm^2 .

3 Calculate the volume of the prism.

The depth of the prism is 10cm .

4 Write the answer, including the units.

The measurements on this triangular prism are in centimetres so the volume will be measured in cubic centimetres.

Volume = 120cm^3

Example 2: volume of a rectangular prism (cuboid)

A swimming pool is being built in the shape of a cuboid.

Calculate the volume of water in the pool when it is completely filled, in litres.

The area of the rectangle is 36m^2 .

The depth of the prism is 5m .

The measurements on this prism are in metres so the volume will be measured in cubic metres, however we then need to convert this to litres, as it is stated in the question.

Volume = 180m^3 .

1m^3 = 1000L and so 180m^3 180,000L .

The volume of water in the swimming pool in litres is 180,000L .

Note:  You may also use the formula:

Volume of cuboid = height × width × depth

since the area of a rectangle is equal to height × width.

Example 3: volume of a hexagonal prism

Work out the volume of the prism:

In this example, we are told that the area of the hexagon is 50mm^2 so we can move on to the next step.

The measurements on this prism are in millimetres so the volume will be measured in cubic millimetres.

Volume = 750mm^3

Example 4: volume of a compound prism

Work out the volume of the L-shaped prism:

To calculate the area of the cross section we need to split it into two rectangles and work out the missing side lengths. We can then work out the area of each rectangle:

Rectangle A:

Rectangle B:

Total area:

The depth of the prism is 12cm .

The measurements on this prism are in cm so the volume will be measured in cm^3 .

Volume = 696cm^3

Example 5: volume of a trapezoidal prism

Volume = 45cm^3 .

How to work out a missing length given the volume

Sometimes we might know the volume and some of the measurements of a prism and we might want to work out the other measurements. We can do this by substituting the values that we know into the formula for the volume of a prism and solving the equation that is formed.

• Write down the formula. Volume of a prism = Area of cross section × depth

Substitute known values into the formula, and solve the equation.

Missing length examples

Example 6: missing length in a pentagonal prism.

The volume of this prism is 225cm^2 . Work out the depth, L, of the prism:

In this example, we are told the area of the cross section is 25cm^2

Since the units in this question are in cm and cm^3 , the depth of the prism is 9cm .

Example 7: missing height in a trapezoidal prism

The volume of this prism is 336cm^3 . Work out the height of the prism.

Since the units in this question are in cm and cm^3 , the height of the prism is 6cm .

Common misconceptions

• Missing/incorrect units

You should always include units in your answer. Remember, volume is measured in units cubed (e.g. mm^3, cm^3, m^3 etc)

• Calculating with different units

You need to make sure all measurements are in the same units before calculating volume. (E.g. you can’t have some in cm and some in m )

• Using the incorrect formula

Be careful to apply the correct prism related formula to the correct question type.

Practice volume of a prism questions

1. Work out the volume of the prism:

2. Calculate the volume of the prism:

3. Work out the volume of the prism:

Area of cross section = 24cm^2

4. Work out the volume of the prism:

Area of triangle A :

Area of rectangle B :

5. The volume of this prism is 156cm^3 . Work out the depth, x , of the prism.

6. The volume of this prism is 270mm^3 . Work out the height, h , of the prism.

Volume of a prism GCSE questions

1. Work out the volume of the prism. State the units in your solution.

\text{Area of cross section }=2 \times 4 + 4 \times 1=12\text{cm}^2 or \text{Area of cross section }=2 \times 3 + 6 \times 1=12\text{cm}^2 For calculating the cross-sectional area of the prism

\text{Volume of prism: }12 \times 5=60 For calculating the volume of the prism

60cm^3 For correct units

2. The volume of the cuboid is twice the volume of the triangular prism. Work out the height, y , of the cuboid.

\frac{1}{2} \times 9 \times 4=18\mathrm{cm}^{2} For area of the cross section (triangle)

18 \times 5=90 \mathrm{cm}^{3} For volume of the triangular prism

3 \times y \times 6=18y For volume of the cuboid

90 \times 2=180\text{ and } 18y=180 Forming an equation to calculate the height of the cuboid

y=10cm For the correct answer

3. (a) Calculate the volume of the trapezoidal prism.

(b) The prism is made from aluminum, which has a density of 2.7g/cm^3 . Work out the mass of the prism. State the units in your answer.

(a) \frac{1}{2}(2+6) \times 4 = 16\mathrm{cm}^{2} For area of the cross section (trapezium)

16 \times 8=128 \mathrm{cm}^{3} For volume of the prism

(b) 128 \times 2.7 For using Mass = Density \times Volume

=345.6g For correct solution including units

Learning checklist

You have now learned how to:

• Know and apply formulae to calculate the volume of prisms
• Use the properties of faces, surfaces, edges and vertices to solve problems in 3-D
• Calculate the volume of composite solids

The next lessons are

• How to calculate volume
• Triangular prism

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Course: 5th grade   >   Unit 11

• Measuring volume as area times length
• Volume as area of base times height

Volume of a rectangular prism

• Volume of rectangular prisms

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9.10: Surface Area and Volume of Prisms

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3-D figures with 2 congruent bases in parallel planes and rectangles for their other faces.

A prism is a 3-dimensional figure with 2 congruent bases, in parallel planes, in which the other faces are rectangles.

The non-base faces are l ateral faces . The edges between the lateral faces are lateral edges .

This particular example is a pentagonal prism because its base is a pentagon. Prisms are named by the shape of their base. Prisms are classified as either right prisms (prisms where all the lateral faces are perpendicular to the bases) or oblique prisms (prisms that lean to one side, whose base is a parallelogram rather than a rectangle, and whose height is perpendicular to the base's plane), as shown below.

Surface Area

To find the surface area of a prism, find the sum of the areas of its faces. The lateral area is the sum of the areas of the lateral faces. The basic unit of area is the square unit.

\(Surface Area=B_{1}+B_{2}+L_{1}+L_{2}+L_{3}\)

\(Lateral Area=L_{1}+L_{2}+L_{3}\)

To find the volume of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit.

For prisms in particular, to find the volume you must find the area of the base and multiply it by the height.

Volume of a Prism: \(V=B\cdot h\), where \(B= area\: of\: base\).

If an oblique prism and a right prism have the same base area and height, then they will have the same volume. This is due to Cavalieri’s Principle , which states that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

What if you were given a solid three-dimensional figure with two congruent bases in which the other faces were rectangles? How could you determine how much two-dimensional and three-dimensional space that figure occupies?

Example \(\PageIndex{1}\)

The total surface area of the triangular prism is \(540\text{ units}^{2}\). What is \(x\)?

The total surface area is equal to:

\(A_{2\: triangles}+A_{3\: rectangles}=540\)

The hypotenuse of the triangle bases is 13, \(5^{2}+12^{2}\). Let’s fill in what we know.

\(\begin{aligned} A_{2\: triangles}=2(\dfrac{1}{2}\cdot 5\cdot 12)=60 \\ A_{3\: rectangles}&=5x+12x+13x=30x \\ 60+30x &=540 \\ 30x&=480 \\ x&=16\text{ units }\qquad \text{ The height is 16 units.}\end{aligned}\)

Example \(\PageIndex{2}\)

Find the volume of the right rectangular prism below.

The area of the base is \((5)(4)=20\) and the height is 3. So the total volume is \((20)(3)=60\text{ unit}^{3}\)

Example \(\PageIndex{3}\)

Find the surface area of the prism below.

To solve, draw the net of the prism so that we can make sure we find the area of ALL faces.

Using the net, we have:

\(\begin{aligned} SA_{prism}&=2(4)(10)+2(10)(17)+2(17)(4)\\ &=80+340+136 \\ &=556 \text{ cm}^{2}\end{aligned}\)

Example \(\PageIndex{4}\)

This is a right triangular prism. To find the surface area, we need to find the length of the hypotenuse of the base because it is the width of one of the lateral faces. We can use the Pythagorean Theorem to find this length.

\(\begin{aligned} 7^{2}+24^{2}&=c^{2} \\ 49+576&=c^{2} \\ 625&=c^{2} \qquad c=25\end{aligned}\)

Looking at the net, the surface area is:

\(\begin{aligned} SA&=28(7)+28(24)+28(25)+2(\dfrac{1}{2}\cdot 7\cdot 24) \\ SA&=196+672+700+168=1736 units^{2}\end{aligned}\)

Example \(\PageIndex{5}\)

You have a small, triangular prism-shaped tent. How much volume does it have once it is set up?

First, we need to find the area of the base.

\(\begin{aligned} B&=\dfrac{1}{2}(3)(4)=6 \text{ ft}^{2} \\ V&=Bh=6(7)=42 \text{ ft}^{3}\end{aligned}\)

Even though the height in this problem does not look like a “height,” it is because it is the perpendicular segment connecting the two bases.

• Draw the net of this prism.
• Find the area of the bases.
• Find the area of lateral faces, or the lateral surface area.
• Find the total surface area of the prism.
• How many one-inch cubes can fit into a box that is 8 inches wide, 10 inches long, and 12 inches tall? Is this the same as the volume of the box?
• A cereal box in 2 inches wide, 10 inches long and 14 inches tall. How much cereal does the box hold?
• A can of soda is 4 inches tall and has a diameter of 2 inches. How much soda does the can hold? Round your answer to the nearest hundredth.
• A cube holds \(216\text{ in}^{3}\). What is the length of each edge?
• A cube has sides that are 8 inches. What is the volume?

Use the right triangular prism to answer questions 11-15.

• Find the volume of the prism.
• What shape are the bases of this prism? What are their areas?
• What are the dimensions of each of the lateral faces? What are their areas?
• Find the lateral surface area of the prism.
• Describe the difference between lateral surface area and total surface area.

• What is the volume and surface area of one die?
• What is the volume and surface area of both dice?

Find the volume of the following solids. Round your answers to the nearest hundredth.

Find the value of \(x\), given the surface area.

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.3.

Additional Resources

Interactive Element

Video: Prisms Principles - Basic

Activities: Prisms Discussion Questions

Study Aids: Prisms and Cylinders Study Guide

Practice: Surface Area and Volume of Prism

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High School Math : How to find the volume of a prism

Study concepts, example questions & explanations for high school math, all high school math resources, example questions, example question #91 : solid geometry.

Find the volume of the following triangular prism.

The formula for the volume of a triangular prism is:

Plugging in our values, we get:

Example Question #92 : Solid Geometry

Example Question #93 : Solid Geometry

Find the volume of the following triangular prism:

The formula for the volume of an equilateral, triangular prism is:

Example Question #94 : Solid Geometry

What is the volume?

The volume is calculated using the equation:

Example Question #1 : How To Find The Volume Of A Prism

A rectangular box has two sides with the following lengths: 

The volume of a rectangular prism is found using the following formula:

If we substitute our known values, then we can solve for the missing side.

Divide both sides of the equation by 12.

We now know that the missing length equals 7 centimeters.

This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm. 

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How to Solve Word Problems of Volume of Cubes and Rectangular Prisms

Today, we're exploring the world of volume, specifically focusing on cubes and rectangular prisms. We're taking it a step further by diving into word problems, a vital skill for applying math to real-world situations!

1. Understanding Volume

Volume is the measure of space occupied by a three-dimensional object. For a cube or rectangular prism, it’s calculated by multiplying the length, width, and height of the object.

2. Solving Volume Word Problems

Word problems involve taking a real-world situation and translating it into mathematical terms. For volume problems, we’re usually given or need to find the dimensions of a shape and calculate the volume.

Step-By-Step Guide to Solving Volume Word Problems

Let’s break down the process:

Step 1: Understand the Problem

First, read the problem carefully to identify the known values (usually dimensions of the shape) and what you need to find (typically the volume).

Step 2: Visualize or Draw the Shape

It’s often helpful to draw the shape and label its dimensions.

Step 3: Use the Volume Formula

For a cube or rectangular prism, the volume is calculated as length x width x height.

Step 4: Solve and Check

Plug in the known values into the volume formula and solve. Then, check your answer to make sure it makes sense in the context of the problem.

For example, let’s consider a problem: “A rectangular box has a length of \(5\ cm\), a width of \(3\ cm\), and a height of \(2\ cm\). What is the volume of the box?”

• Understand the problem: We know the dimensions of the box and need to find the volume.
• Visualize the shape: Draw a rectangular box and label the dimensions.
• Use the volume formula: \(Volume = length\times width\times height = 5\ cm\times 3\ cm\times 2\ cm\).
• Solve and check: The volume is \(30\) cubic cm. This makes sense given the dimensions of the box.

As always, practice is key. Try solving a variety of volume word problems to solidify your understanding.

Keep practicing, keep exploring, and enjoy your mathematical journey!

In this blog post, we’ve explained how to solve word problems involving the volume of cubes and rectangular prisms. We also provided a step-by-step guide to practice this essential skill. Keep practicing, and you’ll master this useful real-world mathematical skill in no time. Happy solving!

by: Effortless Math Team about 8 months ago (category: Articles )

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Volume of a Prism – Definition With Examples

Volume of a prism – introduction, what is the volume of a prism, how to find the volume of a prism, solved examples, practice problems, frequently asked questions.

The definition of the volume of a prism can be simply given as the space occupied by the prism. A prism is a three-dimensional figure that has flat sides and two identical bases.

The bases of a prism are polygons, like a triangle, a square, a rectangle, or a hexagram. Prisms are often named after the polygons that form the base.

To find the volume, we need to know the prism’s base area and height. 

The volume of a prism can be defined as the space it occupies in cubic units. Cubic units measure the volume of a three-dimensional figure. 

To find the volume, we multiply the base area with the height of the prism.

The base area is the surface area of the base. The height is the column connecting the two bases of the prism.

When we multiply the base with the height, the resulting volume will be measured in cubic units such as cubic inches, cubic yards, cubic feet, cubic centimeters, or cubic meters. 

To find the volume , the base area and height are the only values we need to know. 

What Is the Formula for the Volume of a Prism?

The volume of a prism formula is:

Volume $=$ base area $\times$ height 

We can find the volume of any prism using this formula. 

Formulas for different prisms are discussed in the following table.

Table of Formulas for Different Types of Prisms

The formula to calculate the prism volume can be written as $\text{V} = b \times h$, where V is the volume, b is the base area, and h is the height of the prism. 

Finding volume of a prism example: 

Let us find the volume of a prism whose base area is 5 square inches and height is 10 inches. 

Applying the formula:

$\text{V} = b \times h$

Volume $= 5 \times 10 = 50$ cubic inches. 

• We can apply the volume of a prism equation to find the volume of different types of prisms.
• The volume depends on the shape of its base since that will determine the base area. 
• The base area of a prism is measured in square units, square inches, or square meters since it represents a two-dimensional area, i.e., the surface area of the prism base.
• The height of a prism is measured in regular units like inches or meters. This is because it represents the length of the column that connects the bases. 

Finding the volume of a prism is easy once you get the hang of the formula. Remember, it doesn’t matter what type of prism it is. As long as you know the height and the base area, you can find its volume!

1. What is the base area of a prism if the volume of the prism is 150 cubic feet and the height of the prism is 10 feet?  

Solution : Applying the formula:

$150 = b \times 10$

$b = 150 / 10 = 15$ square feet 

2. The base of a prism has a surface area of 277 square centimeters. The volume is 2770 cubic centimeters. Calculate the height. 

Solution: Applying the formula:

$2770 = 277 \times h$

$h = 2770 / 277 = 10$ centimeters 

3. The height of a square prism is 11 yards. The volume is 99 cubic yards. What is the length of the side of the square at the base of the prism? 

Solution: Volume = base area x height 

$99 = b \times 11$

$b = \frac{99}{11} = 9$ square yards 

Since the base area is 9 square yards, this is the area of the square base of the prism. So, the side of the square will be the square root of $9 = 3$ yards. 

4. The base of a prism has a surface area of 475 square centimeters. The volume is 3800 cubic centimeters. Calculate the height. 

$3800 = 475 \times h$

$h = \frac{3800}{475} = 8$ centimeters

5. The column of a prism has a length of 20 meters. The base of the prism has an area of 25 square meters. What is the volume of the prism? 

$\text{V} = 25 \times 20 = 500$ cubic meters

Volume of a Prism - Definition With Examples

Attend this quiz & Test your knowledge.

What is the base area of a prism if the volume of the prism is 300 cubic feet and the height of the prism is 6 feet?

The base of a prism has a surface area of 25 square centimeters. its volume is 125 cubic centimeters. calculate the height., the base area of a prism is 123 square yards. the height is 9 yards. find the volume., the base of a prism has a surface area of 12 square centimeters. the volume of the prism is 144 cubic centimeters. calculate the height., the column of a prism has a length of 100 feet. the base of the prism has an area of 25 square feet. what is its volume.

Irregular prisms have irregular polygons as their bases. This means the base shapes have different angles, and the sides are not equal in length. Though the base area will differ, the formula for finding volume will be the same, i.e., base area $\times$ height.

Is the volume of a prism affected by the number of faces it has?

No, the volume of a prism is not affected by the number of faces because its base area and height are not changed by the number of faces.

How does the volume differ between different types of prisms?

Different types of prisms have different bases of different shapes and sizes. Since the base area differs, the volume will differ accordingly.

How is volume related to the side of the base polygon in a regular prism?

The longer the side of the base polygon, the greater the base area and, therefore, the more the volume.

How do you calculate the volume of a right prism?

A right prism is a prism in which the angles between the base and the sides (lateral faces) of the prism are right angles, for example, a rectangular prism. The formula to find the volume is the same: V $= b \times h$. 

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Volume of a Prism

A prism is a polyhedron in which all the faces are flat, and the bases are parallel to each other. It is a solid object with flat faces, identical ends, and the same cross-section along with its length. In Geometry, we will learn the different types of prisms, such as triangular prism, pentagonal prism, and hexagonal prism. Since it is a three-dimensional shape , a prism has a surface area and volume. In this article, we are going to discuss the volume of a prism, its formulas and solved problems.

What is the Volume of a Prism?

The volume of a prism is defined as the total space occupied by the three-dimensional object. Mathematically, it is defined as the product of the area of the base and the length.

The volume of a Prism = Base Area × Length

The measurement unit used to represent the volume of a three-dimensional object is cubic units. 

Volume of a Prism Formula

Now, let us discuss the volume of the different prism formulas, such as the volume of the triangular prism, rectangular prism, pentagonal prism, and so on.

Volume of a Triangular Prism

A triangular prism is a prism that has three rectangular faces and two triangular bases. Since the cross-section of the triangular prism is a triangle, the formula for the volume of a triangular prism is given as:

The volume of a Triangular Prism = (½) abh cubic units.

a = Apothem length of a triangular prism

b = Base length of a triangular prism

h = height of a triangular prism

Volume of a Rectangular prism

A rectangular prism has four rectangular faces and two parallel rectangular bases. We know that the cross-section of a rectangular prism is a rectangle. The rectangular prism is also known as a “Cuboid”.

Hence, the formula to find the volume of a rectangular prism is given:

The volume of a Rectangular Prism = l.b.h cubic units.

l = Base width of a rectangular prism

b = Base length of a rectangular prism

h = height of a rectangular prism

Volume of a Pentagonal Prism

A pentagonal prism has five rectangular faces and two parallel pentagonal bases. Since the base area of the pentagonal prism is (5/2) ab, the volume of the pentagonal prism is given as:

The Volume of a Pentagonal Prism = (5/2) a.b.h cubic units

a – Apothem length of the pentagonal prism.

b – Base length of the pentagonal prism.

h – Height of the pentagonal prism

Volume of a Hexagonal Prism

A hexagonal prism is a prism with six rectangular faces and two parallel hexagonal bases. The base area of the hexagonal prism is 3ab, the formula to find the volume of a hexagonal prism is given as:

The volume of a Hexagonal Prism = 3abh cubic units

a – Apothem length of the hexagonal prism.

b – Base length of the hexagonal prism.

h – Height of the hexagonal prism.

Volume of a Prism Example

Question 1:

What is the volume of a triangular prism with dimensions of 12 m, 16 m and 20 m as given in figure.

The volume of a triangular prism can be found by V = Area of base × Height of Prism

As the base is triangular, so,

Area of triangle = ½ × base × height = ½ × base × height =½ × 12 × 16 = 96

Frequently Asked Questions – FAQs

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Volume of a Rectangular Prism Calculator

What is a rectangular prism, what is the formula for the volume of a rectangular prism, how to find the volume of a rectangular prism.

With this volume of a rectangular prism calculator – a.k.a., a box volume calculator – you'll find the volume of any box-shaped container in a blink of an eye. No fuss is required. You need to enter only three values, and we'll calculate the volume for you (though it's not so tricky, you could figure it out yourself 😊).

If you're searching for the definition, check the section What is a rectangular prism . However, if you're still wondering how to find the volume of a rectangular prism, we'll show you step-by-step how to use our calculator on an (almost) real-life example – the volume of a cat 🐈.

If you're looking for the area of a rectangular prism, try our surface area of a rectangular prism calculator , or go to the rectangular prism calculator for an all-in-one rectangular prism tool.

A rectangular prism is a 3D shape with 6 faces, all of which are rectangles . Other names for a rectangular prism are a cuboid , or simply a box .

Every rectangular prism has:

• 8 vertices;
• 12 edges; and

Finding the volume of a rectangular prism is a straightforward task – all you need to do is to multiply the length, width, and height together:

Rectangular prism volume = length × width × height

Where can you use this formula in real life? Let's imagine three possible scenarios:

You bought a fish tank for your golden fish 🐠. It's in a regular box shape, nothing fancy, like a corner bow-front aquarium. If you're wondering how much water you need to fill it, simply use the volume of a rectangular prism formula. It is a similar story for other pets kept in tanks and cages, like turtles or rats – if you want a happy pet, then you should guarantee them enough living space.

The time has come – you've decided that this year you'd like to grow your own carrots 🥕 and salad 🥗. For that, you need to construct a raised bed and fill it with potting soil. But how much dirt should you buy? Well, that's the same question as how to find the volume of a rectangular prism: measure your raised bed, use the formula, and run to the gardening center.

You are going on the vacation of your dreams 🌴. You have to pack your stuff for the three weeks, and you're wondering which suitcase 🧳 will fit more in :

Your good old large suitcase, 30 × 19 × 11 inches; or

The new fancy one, 28 × 21 × 12 inches

That's again the problem solved by the volume of a rectangular prism formula. Solve it manually, or find it using our calculator.

If you're searching for a calculator for other 3D shapes – like e.g. a cube , which is a special case of a rectangular prism – you may want to check out our comprehensive volume calculator . It has a gazillion different shapes! (Fourteen, to be exact.)

Well, now that you know what a rectangular prism is and its volume formula, all the calculations should be a piece of cake! Just measure the three dimensions of your rectangular prism, and use the method from the previous paragraph. Alternatively, you can simply use our box volume calculator.

So, let's have a look at the example – let's calculate the volume of a cat🐈. Yes, a cat, as cats are almost like liquids (they take on the shape of whatever container they are in). Assuming that the cat completely fills a plastic container with dimensions 12 inches × 10 inches × 8 inches:

Input the container's length into the first field of our volume of a rectangular prism calculator. It's 12 inches in our case.

Enter the width of the box . Put 10 inches into the proper field.

Finally, input the height of your container, 8 inches.

And there it is: the volume of a rectangular prism calculator did the job. Now we know that our cat's volume is 960 cubic inches . Isn't that paw esome? 🐾

(Of course, it's a really rough estimate. Take it with a pinch of salt 🧂 or even two pinches.)

So, now that you know your cat's volume, you can go and search for a perfect cardboard box📦 for your pet!

How do I calculate the volume of a rectangular prism?

To find the volume of a rectangular prism, you need to:

• Determine the lengths of the sides : width, length, and height.
• Multiply together the three values from Step 1.
• The result you've got is the volume of your solid.

Don't forget to include the units if it is given! Since it's volume, you need cubic units.

What is the volume of a rectangular prism with sides 2, 5, and 7?

The answer is 70 . To see how to get this result, recall the formula for the volume of a rectangular prism:

volume = length × height × width

Hence, we compute the volume as 2 × 5 × 7 = 70 .

Remember to include the units: for instance, if your measurements are in inches (in), the volume will be in cubic inches (in³).

How do I calculate the volume of a rectangular prism given diagonals?

To determine the volume of a rectangular prism when you know the diagonals of its three faces, you need to apply the formula:

volume = 1/8 × √(a² - b² + c²)(a² + b² - c²)(-a² + b² + c²) ,

where a, b, and c are the diagonals you're given. This formula can be easily derived by using the Pythagorean theorem.

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Volume of rectangular prisms

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Volume of Prisms: How to Engage & Excite 5th Graders

There is something about geometry standards that I just love to teach! I feel like I love every math topic and I probably do, but the volume of prisms is something special. This is the unit when I see my students gain excitement and have all the lightbulb moments. Typically I have a few students who claim they hate math and that it’s really hard and boring. But in the volume of prisms unit, those students change their tune. They start to like math!

Volume for 5th Grade is one of 2 geometry units that I teach. The other unit is more about quadrilaterals and types of triangles. Today, I’m just focusing on the volume of prisms. 

While the volume of prisms is already a hands-on topic, I’m sharing my favorite ways to make it even more interactive for your students!

Volume of Prisms Manipulatives

With any math concept, I like to start with the foundation. Making math as hands-on as possible is key to keeping students excited and engaged, especially in the upper elementary and middle school standards. I’ve linked some of my favorite math manipulatives for the volume of prisms below.

Plastic Fillable 3D Shapes

Foam Blocks

Math Link Cubes

General Volume for 5th Grade Teaching Tips

In general, keep your unit short and sweet. The volume for 5th grade standards relies on concrete understanding, vocabulary, and the formula. Composite volume is also a key common core standard. Make sure you build to all the standards in a logical order. 

Let students explore volume. To gain a rich understanding of the volume of prisms, students need to explore what volume is. Use manipulatives and let kids build. This is a very hands-on unit, so let the messy block building happen.

Explain the difference between area, perimeter, and volume. While area and perimeter are covered in the lower grades, still include both in your volume for 5th grade unit. Talking about word problems and the application of all 3 is important to gain that deeper understanding. If you want to learn more about how to solve math word problems, be sure to check out this blog post .

Build to the formula. Hands-on is the best place to start but don’t stop there. Make sure that students walk away with a deeper understanding of volume AND be able to use the formula. The middle school math teachers will thank you!

Make your unit fun! There is nothing worse than a boring math unit. Let students play games, use math stations, and utilize projects. If you are looking to start math learning centers, check out this blog post . I lay out all the information on starting stations and give you lots of math station rotation ideas to get started. I also have a 5 part video series all about transforming your math classroom from frustrating to fun! Just drop your email below and the video series will be sent to your inbox!

Volume of Prisms Curriculum

Once you have the manipulatives, we need a good foundation for teaching. I don’t have a set curriculum so I made my own! This one has all of the main pieces that I need while still being student-focused. 

This unit has a bunch of different worksheets for notes, homework, tests, quizzes, study guide, exit tickets, and data tracking sheets! Everything you need to teach for 8 days!

• Fill in the blank notes with short videos (5-10 minutes) to match the notes sheet. These are perfect for a flipped classroom or for absent students.
• Worksheets on Volume of Rectangular Prisms for homework or guided practice. These also make easy, independent work for math station ideas.
• Math Exit Tickets for each day of notes are included. They are available in 3 different forms: paper slips, Google Slides to present to the class, and Google Forms for self-grading.
• A mid-unit quiz
• Study Guide to review your entire volume of prisms unit
• 2 Versions of the Test for retake options or they can be used for a pretest and post-test data point
• Data Tracking Spreadsheets for simple data tracking on exit tickets, assessments, or Common Core Standards

If you want to know more about the unit, click here to be taken to my TPT shop.

Volume for 5th Grade Actvities

Now that you have a volume of prisms unit to teach from, let’s add some fun! I do a few different lessons during the unit to shake things up and keep students excited. 

Learning Labs

I put those manipulatives that we talked about above to good use with these learning labs. Like a science lab, students follow directions that lead them to learn and explore the volume of prisms. 

One of the labs I do is about discovering what volume is so I use a boatload of cubes and have students have at it. Letting students build 3-D models and determining how many cubes they used to build their creation shows them what volume means.

The second lab I do in my volume for 5th grade unit is to bridge the gap between counting cubes and the geometry formula. This lab is always a hit and students see the reason behind using the formula. This also makes an awesome lesson to be observed on 😉 

If you want to see more about learning labs, check out this bundle of activities on my TPT shop.

Choice Board Templates

This is a quick math station rotation idea or uses in place of an assessment. I like to give students choice in their learning so choice board templates are an easy way to do that. These choice board prompts are ready to assign to your students. However, they are only available in this bundle! I have that product linked here if you are looking for blank digital choice board templates.

If you want to see more about choice board templates in action, be sure to check out this blog post !

Error Analysis for Volume of Prisms

Helping students learn to find mistakes is such a huge blessing in the math classroom. Whether it is when they solve math word problems or just answering a 5th grade math problem, students need to know how to check their work. These worksheets on volume of rectangular prisms help students identify mistakes in others’ work. Check out the resource here.

Want to know how I teach how to solve math word problems? This blog post has all the answers.

Games for Geometry

It’s no secret that I love using math learning centers in my classroom. I’m always looking for new math games and this set is perfect for leaving out year-round. Dice games for kids but make it the volume of prisms. Students roll a dice or 2 and create their own volume problems to solve. Students can choose to use the formula (once we learn it) or to use their hands and build the prism. The differentiation is key for a student centered learning classroom. 

Solve Math Word Problems in Geometry

This math station rotation idea is so much fun and the best part is that students are practicing volume of prisms AND word problems. It’s a win-win situation! Check out this blog post where I go more in-depth about how to solve math word problems. 

If you want to purchase the word problems in geometry game, grab them here !

Digital Task Cards & Worksheets on Volume of Rectangular Prisms

Looking for an independent practice task or gearing up for a sub day? I still use digital task cards in my classroom for these reasons! They are easy to assign to students digitally, but if you want a more hands-on approach, paper-pencil task cards are included too. Check out the digital task cards for the volume of prisms here .

Want to see ways I use task cards in my classroom? Check out this blog post where I dive into how to use task cards for math .

Looking for more ideas on how to teach volume of prisms?

Make your volume of prisms unit engaging and exciting with these ideas! I’ve rounded up my favorite blog posts that are full of ideas and activities for your volume for 5th grade unit.

• Teaching Volume of Rectangular Prisms – I love the visuals in this post
• Volume of Rectangular Prisms – This is an entire lesson plan all about volume of prisms
• 9 Volume of Prisms Activities Your Students Will Love – escape rooms are always a fun go to activity
• Teaching Volume with Hands-On Activities – use sugar cubes to show volume!
• Can’t Miss Easy STEM Challenges – the popcorn volume challenge is so fun and delicious!
• Understanding the Concept of Volume of a Rectangular Prism – letting kids build their own models out of paper
• 5 Tips for Teaching Volume – number 3 is so key!

I’d love to hear about your volume of prisms unit! What’s your favorite activity? How do you make it fun while still rigorous? Drop your responses in the comments!

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[…] minds-on is my strategy with volume for 5th grade but also for algebra. Using index cards, physical manipulatives and even simplifying moving around […]

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Problem solving with volumes of prisms

A problem-solving activity using volumes of prisms and cylinders, plus constructing and solving simple equations.

Students are shown prisms and a cylinder with measurements given either numerically or as unknowns x, y and z.

They must use the information to form and solve simple equations to find all the unknowns as well as the radius of the cylinder.

Students can then create their own prism with the same volume to go in the empty box.

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Volume of prisms

Subject: Mathematics

Age range: 11-14

Resource type: Lesson (complete)

Last updated

15 May 2019

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TWO FULL LESSONS on finding the volume of prisms.

Contents of download:

• Higher ability full lesson (5.2.1h) on volume of prisms (could be taught over one or two lessons depending on previous knowledge).
• Lower ability full lesson (5.2.1f) on volume of prisms (to be taught over one or two lessons depending on previous knowledge). Ex4 and Ex5 are a suggested starting point for lesson two .
• Examples (to go with PowerPoint)
• Worksheets with solutions (printed and displayed on PowerPoint).

We are learning about: The volume of a prism We are learning to: Calculate the volume of a prism.

Differentiated objectives:

• Developing learners will be able to calculate the volume of a prism.
• Secure learners will be able to find the missing length of a prism given its volume and cross-section.
• Excelling learners will be able to solve unfamiliar problems using their knowledge of calculating the volume of a prism.

Main: Walkthrough examples followed by practice questions on worksheets. All solutions given on PPT and in worksheet format.

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Great resource!

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I hope it helps!

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Very nicely presented powerpoint!

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Well thought out and very organised. Thanks

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latoyaabatan

OMG! you have just saved me a lot of time planning. these resources has everything we has dicussed to plan our lessons collaboratively. Thank you!

Glad to help!

Amazing resources!!!!! Thank you for all your work!

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