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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

Still stuck?

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The resources on this page will hopefully help you teach AO2 and AO3 of the new GCSE specification - problem solving and reasoning.

This brief lesson is designed to lead students into thinking about how to solve mathematical problems. It features ideas of strategies to use, clear steps to follow and plenty of opportunities for discussion.

The PixiMaths problem solving booklets are aimed at "crossover" marks (questions that will be on both higher and foundation) so will be accessed by most students. The booklets are collated Edexcel exam questions; you may well recognise them from elsewhere. Each booklet has 70 marks worth of questions and will probably last two lessons, including time to go through answers with your students. There is one for each area of the new GCSE specification and they are designed to complement the PixiMaths year 11 SOL.

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Some of my favourite websites have plenty of other excellent resources to support you and your students in these assessment objectives.

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Volume ( Edexcel GCSE Maths )

Revision note.

What is volume?

• The volume of a 3D shape is a measure of how much space it takes up
• You need to be able to calculate the volumes of a number of common 3D shapes

How do I find the volume of cuboids, prisms, and cylinders?

• To find the volume of a cuboid use the formula Volume of a cuboid = length × width × height

• You will sometimes see the terms  'depth' or 'breadth' instead of 'height' or 'width'
• A cuboid is in fact another name for a rectangular-based prism
• To find the volume of a prism use the formula Volume of a prism = area of cross-section × length

• Note that the cross-section can be any shape, so as long as you know its area and length , you can calculate the volume of the prism
• Or if you know the volume and length of the prism, you can calculate the cross-section area
• To calculate the volume of a cylinder with radius, r and height, h, use the formula Volume of a cylinder = πr 2 h

• Note that a cylinder is in fact a circular-based prism : its cross-section is a circle with area πr 2 , and its length is h

How do I find the volume of pyramids, cones, & spheres?

• To calculate the volume of a pyramid with height h, use the formula Volume of a pyramid = 1/3 × area of base × h

• Note that to use this formula the height must be a line from the top of the pyramid that is perpendicular to the base
• To calculate the volume of a cone with base radius r and height h, use the formula Volume of a cone = 1/3 πr 2 h

• Note that a cone is in fact a circular-based pyramid
• As with a pyramid, to use the cone volume formula the height must be a line from the top of the cone that is perpendicular to the base
• To calculate the volume of a sphere with radius r, use the formula Volume of a sphere = 4/3 πr 3

• The formula for volume of a sphere or volume of a cone will be given to you in an exam question if you need it
• You need to memorise the other volume formulae

Worked example

A sculptor has a block of marble in the shape of a cuboid, with a square base of side 35 cm and a height of 2 m.

He carves the block into a cone, with the same height as the original block and with a base diameter equal to the side length of the original square base.

What is the volume of the marble he removes from the block whilst carving the cone.

Give your answer in m 3 , rounded to 3 significant figures.

The volume of the removed material will be equal to the volume of the original marble minus the volume of the cone.

Find the volume of the original marble.

Convert the units of the length, width and height of the cuboid into the same units, either metres or cm. The question asks for the answer in m 3 so it makes sense to use this throughout your calculations.

Length = width = 0.35 m

Height = 2 m 

Substitute the values into the formula for the volume of a cuboid.

Find the radius of the base of the cone, this will be half of the diameter.

Find the volume of the cone by substituting the radius and the height into the formula for the volume of a cone.

Find the volume of the marble that was removed by subtracting the volume of the cone from the volume of the cuboid.

Volume removed = 0.245 - 0.0641409 = 0.180859...

Round the answer to 3 significant figures.

Volume of removed marble = 0.181 m 3 (3 s.f.)

Problem Solving with Volumes

How can i solve problems when its not a "standard" 3d shape.

• Often the shape in a question will not be a standard cuboid, cone, sphere, etc
• A prism (3D shape with the same cross-section running through it)
• A portion or fraction of a standard shape (a hemisphere for example)
• The cross-sectional area may be a compound shape, such an an L-shape, or a combination of a rectangle and a triangle for example
• A hemisphere is half a sphere
• The volume of a frustum will be the volume of the smaller cone or pyramid subtracted from the volume of the larger cone or pyramid
• e.g. "find the area of the triangle and the rectangle, add together, times by the length"

A doll's house is in the shape of a prism pictured below. The prism consists of a cuboid with a triangular prism on top of it. The cross section of the triangular prism is an isosceles right-angled triangle. Find the volume of the doll's house.

Our strategy is to find the area of the triangle and the rectangle and add them together to find the cross-sectional area, and then multiply this by the length to find the volume

Finding the area of the rectangle

The total cross-sectional area is therefore the triangle plus the rectangle

Finding the volume of the prism by multiplying the cross-sectional area by the length

Rounding to 3 significant figures

79 900 cm 3

The diagram shows a truncated cone (a frustum). Using the given dimensions, find the volume of the frustum.

To find the volume of the frustum, find the volume of the larger cone (30 cm tall, with a radius of 20 cm), and subtract the volume of the smaller cone (15 cm tall, with a radius of 10 cm)

Calculate the volume of the larger cone

Calculate the volume of the smaller cone

Find the difference

Round to 3 significant figures 

11 000 cm 3  

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Author: Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

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Volume question with algebra – Level 4 GCSE maths

November 30, 2017 by Simon Deacon Leave a Comment

Volume question with algebra

These types of questions are all about working out a volume using algebra. They tend to be aimed at around level 4 GCSE maths and it’s really about reading, and re- reading, the question until you are sure what they are asking.

Volume is better calculated as area x depth and, with a triangular prism you need to remember that:

area of a triangle is 1/2 x base x height.

Once you’ve got the area, then multiply by the depth to get the volume.

This is always the same for any type of volume question and here’s some more examples of cones , hemispheres , cylinders , and prisms . These questions are fairly popular, and are usually calculator papers for anything involving a cylinder (because of pi), or non calc if they are testing arithmetic skills.

• Keep track of how you are answering the question by working down the page
• Remember that they are not asking you to solve, or work out a value
• Be careful with cube roots

If you’d like to ask for any more detail, or you’re not sure about anything, please do ask a question in the comments section.

All best with your studies.

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Place value and Estimating - Full lesson including MCQs, Problem Solving, Exam Questions

Subject: Mathematics

Age range: 11-14

Resource type: Lesson (complete)

Last updated

27 August 2024

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A comprehensive lesson on the topic of Place Value and Estimating, ideal for students studying the GCSE Mathematics curriculum. The lesson is designed to build a solid understanding of place value, which is crucial for accurate estimation and other mathematical operations.

The resource include:

*** Exam Questions: Practice questions that align with the GCSE syllabus, helping students prepare effectively for their exams.

*** Multiple Choice Questions (MCQs): Quick, focused questions that test students’ grasp of key concepts and reinforce learning.

*** Literacy Task: A task designed to enhance students’ mathematical vocabulary and understanding of key terms related to place value and estimation.

*** Problem-Solving Activities: Engaging challenges that require students to apply their knowledge in real-world contexts, fostering critical thinking and practical application.

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