Volume (AQA GCSE Maths)

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Amber

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16 March 2023

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Volume

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

Cuboid volume, IGCSE & GCSE Maths revision notes

- 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

Prism volume, IGCSE & GCSE Maths revision notes

- 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 = πr²h

Cylinder volume, IGCSE & GCSE Maths revision notes

- Note that a cylinder is in fact a circular-based prism: its cross-section is a circle with area πr², 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

Pyramid volume, IGCSE & GCSE Maths revision notes

- 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 πr²h

Cone volume, IGCSE & GCSE Maths revision notes

- 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 πr³

Sphere Radius r, IGCSE & GCSE Maths revision notes

Examiner Tip

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 m³, 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 m³ 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.

$\begin{array}{rcl} Volume of cuboid & = & length \times width \times height \\ = & 0.35 \times 0.35 \times 2 \\ = & 0.245 m^{3} \end{array}$

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

$\begin{array}{rcl} Radius of base of cone & = & \frac{1}{2} (diameter) \\ = & \frac{1}{2} (0.35) \\ = & 0.175 m \end{array}$

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

$\begin{array}{rcl} Volume of cone & = & \frac{1}{3} \times area base \times height \\ = & \frac{1}{3} \times π r^{2} \times 2 \\ = & \frac{2}{3} \times π \times {(0.175)}^{2} \\ = & 0.0641409 . . . \end{array}$

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 m³ (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
It will likely either be:
- A prism (3D shape with the same cross-section running through it)
- A portion or fraction of a standard shape (a hemisphere for example)
If the shape is a prism, recall that the volume of a prism is the cross sectional area × its length
- 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
If the shape is a fraction of a standard shape, consider the "full" version of the shape and then find the appropriate fraction of it
- A hemisphere is half a sphere
- A frustum is a truncated (chopped-off) cone or pyramid
  - 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

Examiner Tip

Before you start calculating, make a quick note of your plan to tackle the question
- e.g. "find the area of the triangle and the rectangle, add together, times by the length"

Worked example

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.

3-7-1-problem-solving-with-volume-1

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

As it is an isosceles triangle, length $a = 20 cm$

We can then use Pythagoras to find length $b$

$b = \sqrt{20^{2} + 20^{2}} = \sqrt{800} = 20 \sqrt{2}$

Length $c$ will also be $20 \sqrt{2}$

Finding the area of the triangle using $Area = \frac{1}{2} \times base \times height$

$Triangle Area = \frac{1}{2} \times 20 \times 20 = 200 {cm}^{2}$

Finding the area of the rectangle

$40 \times 20 \sqrt{2} = 800 \sqrt{2} {cm}^{2}$

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

$200 + 800 \sqrt{2} {cm}^{2} = 1 331.37085 . . . {cm}^{2}$

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

$(200 + 800 \sqrt{2}) \times 60 = 12000 + 48000 \sqrt{2} {cm}^{3} = 79 882.25099 . . . {cm}^{3}$

Rounding to 3 significant figures

79 900 cm³

Worked example

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

3-7-1-problem-solving-with-volume-2

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)

Formula for the volume of a cone: $\frac{1}{3} π r^{2} h$

Calculate the volume of the larger cone

$V_{L} = \frac{1}{3} \times π \times 20^{2} \times 30 = 4000 π = 12 566.37061 . . . {cm}^{3}$

Calculate the volume of the smaller cone

$V_{S} = \frac{1}{3} \times π \times 10^{2} \times 15 = 500 π = 1 570.796327 . . . {cm}^{3}$

Find the difference

$V_{L} - V_{S} = 4000 π - 500 π = 3500 π = 10 995.57429 . . . {cm}^{3}$

Round to 3 significant figures

11 000 cm³

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Volume (AQA GCSE Maths)

Revision Note

What is volume?

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

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

How can I solve problems when its not a "standard" 3D shape?

You've read 0 of your 10 free revision notes

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Amber