Volumes of 3D shapes Revision Notes for AQA GCSE Maths

Volumes of 3D shapes

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In your GCSE exam, these formulas will be provided on the formula sheet, but you need to know how to apply them correctly.

Key volume formulas

Understanding how to work with volume formulas is essential for success in GCSE mathematics. Each 3D shape has its own specific formula that you must learn to apply correctly.

Cone volume

The volume of a cone uses the formula: $V = \frac{1}{3}\pi r^2 h$

This means you take one third of the base area ( $\pi r^2$ ) multiplied by the vertical height (h).

Sphere volume

The volume of a sphere uses the formula: $V = \frac{4}{3}\pi r^3$

This formula only needs the radius - you multiply four thirds by pi by the radius cubed.

Pyramid volume

The volume of a pyramid uses the formula: $V = \frac{1}{3}Ah$

This means one third of the base area (A) multiplied by the vertical height (h).

Comparing volumes - worked example

When comparing volumes of different shapes, you calculate each volume separately then work out the relationship between them. This type of question frequently appears in GCSE exams and requires a systematic approach.

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Worked Example: Comparing Cone and Cylinder Volumes

Compare the volumes of cone A (radius 3cm, height 8cm) and cylinder B (radius 4cm, height 12cm).

Step 1: Calculate volume of cone A $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \times 3^2 \times 8 = \frac{1}{3}\pi \times 9 \times 8 = 24\pi$

Step 2: Calculate volume of cylinder B
$V = \pi r^2 h = \pi \times 4^2 \times 12 = \pi \times 16 \times 12 = 192\pi$

Step 3: Find the ratio $192\pi ÷ 24\pi = 8$

So cylinder B has 8 times the volume of cone A

Exam techniques

Mastering volume calculations requires understanding the common question types and developing effective strategies for tackling them.

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Common volume questions in your exam might involve:

Working out ratios between two different shapes or volumes
Finding unknown quantities represented by letters in the formulas
Finding expressions for length, area or volume in terms of unknowns

Top tip: When comparing volumes, calculate both volumes completely, then write a clear conclusion showing the ratio. You can often leave your working in terms of π to make calculations easier.

Practice method

When approaching volume problems involving equal volumes or unknown quantities, following a structured method will help you avoid common mistakes and reach the correct answer efficiently.

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When you have questions involving equal volumes, follow this systematic approach:

Step 1: Write the volume formula for each shape Step 2: Set the expressions equal to each other
Step 3: Rearrange to make the unknown quantity the subject Step 4: Substitute known values and solve

For example, if a cone and sphere both have radius 3cm, and their volumes are equal, you would set $\frac{1}{3}\pi(3^2)h = \frac{4}{3}\pi(3^3)$ and solve for h.

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Key Points to Remember:

Cone and pyramid formulas both include the $\frac{1}{3}$ factor - don't forget this
Sphere volume needs $\frac{4}{3}\pi r^3$ - remember it's "four thirds"
Always show your working clearly in exam questions, especially when comparing volumes
Leave answers in terms of π when possible to avoid rounding errors
Check your ratio calculations by substituting back into the original volumes

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