Volumes of 3D shapes Revision Notes for Edexcel GCSE Maths

Volumes of 3D shapes

Key volume formulae

Understanding the volume formulae for different 3D shapes is essential for GCSE maths. These formulae will be provided in your exam, but you need to know how to apply them correctly.

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While the formulae are provided in your exam, understanding when and how to use each one is crucial for success. Focus on recognising which formula applies to each shape type.

Cone

The volume of a cone uses one-third of the base area multiplied by the vertical height:

Volume of cone = $\frac{1}{3} \times \text{base area} \times \text{vertical height}$
For a circular cone: Volume = $\frac{1}{3}\pi r^2h$
Where $r =$ radius of the circular base, $h =$ vertical height

Sphere

A sphere's volume depends entirely on its radius:

Volume of sphere = $\frac{4}{3}\pi r^3$
Where $r =$ radius of the sphere
This is the only volume formula that uses $\frac{4}{3}$

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The sphere formula is unique because it uses $\frac{4}{3}$ rather than $\frac{1}{3}$ . This makes it easy to distinguish from cone and pyramid formulae in exam questions.

Pyramid

Like cones, pyramids also use one-third in their volume calculation:

Volume of pyramid = $\frac{1}{3} \times \text{base area} \times \text{vertical height}$
This can be written as Volume = $\frac{1}{3}Ah$
Where $A =$ area of the base, $h =$ vertical height

Worked example: comparing volumes

When comparing the volumes of different shapes, follow these clear steps:

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Worked Example: Comparing Cone and Cylinder Volumes

Problem: A cone has radius 3cm and height 8cm. A cylinder has radius 4cm and height 12cm. How do their volumes compare?

Step 1: Calculate the cone's volume

Volume of cone = $\frac{1}{3}\pi r^2h = \frac{1}{3}\pi \times 3^2 \times 8 = \frac{1}{3}\pi \times 9 \times 8 = 24\pi \text{ cm}^3$

Step 2: Calculate the cylinder's volume

Volume of cylinder = $\pi r^2h = \pi \times 4^2 \times 12 = \pi \times 16 \times 12 = 192\pi \text{ cm}^3$

Step 3: Compare the volumes

Cylinder volume ÷ cone volume = $192\pi \div 24\pi = 8$
Therefore, the cylinder has 8 times the volume of the cone

Exam guidance

Common question types

Exam questions about volumes often involve:

Comparing two volumes by finding ratios between them
Finding unknown quantities when volumes are equal or in a given ratio
Working with expressions rather than just numbers
Leaving answers in terms of π to make calculations simpler

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Key Exam Tips:

Always show your working clearly when calculating volumes
When comparing volumes, calculate both first, then write your conclusion
You can often leave your working in terms of π - don't always convert to decimals
Look out for questions asking you to "show that" one volume is a multiple of another

Practice question approach

When tackling volume problems where shapes have equal volumes, use this strategic approach:

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Strategy: Set up equations by making the volume expressions equal to each other, then rearrange to find the unknown.

For example, if a cone and sphere have equal volumes, write:

Volume of cone = Volume of sphere
$\frac{1}{3}\pi r_1^2h = \frac{4}{3}\pi r_2^3$
Then rearrange to solve for the unknown measurement

This approach works for any combination of 3D shapes with equal volumes.

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Key Points to Remember:

Volume formulae for cones and pyramids always start with $\frac{1}{3}$
Sphere volume uses $\frac{4}{3}\pi r^3$ - the only formula with $\frac{4}{3}$
Show all working clearly in exam questions, especially when comparing volumes
Leave answers in terms of π when possible to avoid rounding errors
Set volume expressions equal when shapes have the same volume to find unknown measurements

Volumes of 3D shapes (Edexcel GCSE Maths): Revision Notes