Volumes of cuboids Revision Notes for AQA GCSE Maths

Volumes of cuboids

What is volume?

Volume is the amount of space that a 3D shape takes up. When you think about volume, imagine how much water you could pour inside a box - that's its volume.

The most common units for measuring volume are:

cm³ (cubic centimetres) for smaller objects
m³ (cubic metres) for larger objects
Litres for liquids

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Understanding volume as "how much space something takes up" helps you visualise what you're calculating. Think of it as the capacity of a container - how much it can hold inside.

Volume formula for cuboids

A cuboid is a 3D shape with rectangular faces (like a shoebox). To find the volume of any cuboid, you need to know three measurements: length, width, and height.

The formula is straightforward:

$\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}$

This formula works because you're essentially counting how many unit cubes fit inside the cuboid. If you have a length of 5 units, width of 3 units, and height of 2 units, you're fitting $5 \times 3 \times 2 = 30$ unit cubes inside.

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Why does this formula work?

Think of building the cuboid layer by layer. Each layer has length × width squares, and you stack height number of layers. So the total is length × width × height unit cubes.

Calculating volume step by step

When solving volume problems, follow these key steps:

Identify the dimensions - find the length, width, and height
Check the units - make sure all measurements use the same units
Apply the formula - multiply length × width × height
Write the correct units - if your measurements are in cm, your volume will be in cm³

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Worked Example 1: Basic volume calculation

For a cuboid measuring 3.2m × 1.5m × 2.0m:

Step 1: Identify the dimensions

Length = 3.2m
Width = 1.5m
Height = 2.0m

Step 2: Apply the formula Volume = Length × Width × Height Volume = 3.2 × 1.5 × 2.0 = 9.6 m³

Since the length measurements were in metres, the volume is measured in cubic metres (m³).

Converting units

Sometimes you need to convert between different units. A common conversion is from cm³ to litres:

$1 \text{ litre} = 1000 \text{ cm}^3$

So to convert cm³ to litres, divide by 1000.

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Worked Example 2: Converting units

If a volume is 168,000 cm³, convert to litres:

Step 1: Use the conversion factor 1 litre = 1000 cm³

Step 2: Divide by 1000 168,000 ÷ 1000 = 168 litres

Finding missing dimensions

If you know the volume and two dimensions, you can find the missing third dimension by rearranging the formula.

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Worked Example: Finding missing height

If a cuboid has:

Volume = 390 cm³
Length = 15 cm
Width = 4 cm
Height = ? cm

Step 1: Set up the equation $15 \times 4 \times h = 390$

Step 2: Simplify $60 \times h = 390$

Step 3: Solve for h $h = 390 \div 60$ $h = :success[6.5 \text{ cm}]$

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Common Mistake to Avoid

When rearranging the formula, make sure you divide the volume by the product of the two known dimensions, not by each dimension separately.

Incorrect: $h = 390 \div 15 \div 4$ ❌
Correct: $h = 390 \div (15 \times 4) = 390 \div 60$ ✓

Exam tips

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Essential Exam Strategies

Always show your working clearly - marks are awarded for method even if the final answer is incorrect
Check your units match throughout your calculation - mixed units lead to wrong answers
For real-world problems, read carefully to identify what the question is asking
When buying items in whole numbers (like bags of compost), round up to the nearest whole number
Double-check your arithmetic, especially when dealing with decimals

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Key Points to Remember:

Volume measures the space inside a 3D shape
Volume of cuboid = Length × Width × Height
Units must match - if dimensions are in cm, volume is in cm³
1000 cm³ = 1 litre for unit conversions
Show all working clearly in exam questions

Volumes of cuboids (AQA GCSE Maths): Revision Notes