Measurement

Understanding Volume: Counting Cubes and Cuboids

Year 3 · Year 4 · Year 5 · Year 6

✓By the end of this lesson students will be able to understand what volume is and how it relates to the space a 3D object occupies.
✓By the end of this lesson students will be able to count unit cubes to determine the volume of simple 3D shapes.
✓By the end of this lesson students will be able to recognise and use standard units for volume, such as cubic centimetres (cm³) and cubic metres (m³).
✓By the end of this lesson students will be able to calculate the volume of cuboids by multiplying length, width, and height.
✓By the end of this lesson students will be able to solve practical problems involving the volume of cuboids.

Key concepts

Volume

Volume is the amount of three-dimensional space that an object occupies or contains. Think of it as how much 'stuff' can fit inside a container, or how much space a solid object takes up. It is a measure of space, not just a flat surface.

Unit Cube

A unit cube is a cube where each side has a length of 1 unit. For example, a cube with sides of 1 centimetre (cm) is called a 1 cm³ (one cubic centimetre) cube. We use unit cubes to measure volume by seeing how many of them can fit inside a larger shape.

Cubic Units

Volume is measured in cubic units. This means we are measuring in three dimensions: length, width, and height. Common cubic units include cubic centimetres (cm³) and cubic metres (m³). The small '3' tells us it's a measure of volume.

Cuboid

A cuboid is a three-dimensional shape with six rectangular faces. All its angles are right angles. Examples of cuboids include bricks, cereal boxes, and rooms.

Volume of a Cuboid Formula

To find the volume of a cuboid, you multiply its length by its width, and then multiply that result by its height. This tells you how many unit cubes would fit inside the cuboid.

Volume = length × width × height

Key facts to remember

1Volume measures the amount of space a 3D object occupies.
2Volume is measured in cubic units, such as cubic centimetres (cm³) or cubic metres (m³).
3A unit cube has a volume of 1 cubic unit.
4To find the volume of a simple shape, you can count the number of unit cubes it contains.
5A cuboid is a 3D shape with six rectangular faces.
6The formula for the volume of a cuboid is: Volume = length × width × height.
7The order in which you multiply length, width, and height does not change the final volume.

Worked examples

Example 1

A shape is built from 1 cm³ cubes. It has 4 cubes in the bottom layer, and there are 3 layers stacked on top of each other. What is the total volume of the shape?

IStep 1: Identify the number of cubes in one layer. There are 4 cubes in the bottom layer.

IIStep 2: Identify the number of layers. There are 3 layers.

IIIStep 3: Multiply the number of cubes in one layer by the number of layers to find the total number of cubes. Total cubes = 4 cubes/layer × 3 layers = 12 cubes.

IVStep 4: Since each cube is 1 cm³, the total volume is 12 times 1 cm³.

Answer

The total volume of the shape is 12 cm³.

This method helps visualise how the formula for a cuboid works by thinking about layers.

Example 2

A cuboid is made of 1 cm³ cubes. It is 5 cm long, 2 cm wide, and 3 cm high. Find its volume by first considering layers, and then using the volume formula.

IStep 1: Calculate the number of cubes in the bottom layer. This is length × width = 5 cm × 2 cm = 10 cubes.

IIStep 2: Identify the number of layers (which is the height). The height is 3 cm, so there are 3 layers.

IIIStep 3: Calculate the total volume by multiplying the cubes per layer by the number of layers. Total volume = 10 cubes/layer × 3 layers = 30 cubes.

IVStep 4: State the volume with correct units. Since each cube is 1 cm³, the volume is 30 cm³.

VStep 5: Now, use the formula: Volume = length × width × height.

VIStep 6: Substitute the given values into the formula: Volume = 5 cm × 2 cm × 3 cm.

VIIStep 7: Calculate the product: Volume = 10 cm² × 3 cm = 30 cm³.

Answer

The volume of the cuboid is 30 cm³.

Both methods give the same answer, showing how counting layers leads to the formula.

Example 3

Calculate the volume of a cuboid with a length of 6 metres, a width of 4 metres, and a height of 2 metres.

IStep 1: Write down the formula for the volume of a cuboid: Volume = length × width × height.

IIStep 2: Identify the given dimensions: length = 6 m, width = 4 m, height = 2 m.

IIIStep 3: Substitute these values into the formula: Volume = 6 m × 4 m × 2 m.

IVStep 4: Perform the multiplication: Volume = 24 m² × 2 m = 48 m³.

Answer

The volume of the cuboid is 48 m³.

Always remember to include the correct cubic units in your final answer.

Common mistakes

✗Confusing volume with area: Area is for 2D shapes (length × width) and is measured in square units (e.g., cm²), while volume is for 3D shapes and is measured in cubic units (e.g., cm³).
✗Forgetting to include the correct cubic units (e.g., writing 30 instead of 30 cm³).
✗Only multiplying two of the dimensions (e.g., just length × width) instead of all three for a cuboid.
✗Miscounting cubes in a diagram, especially if some cubes are hidden from view.
✗Using different units for length, width, and height without converting them first (e.g., cm and m in the same calculation).

Exam tips

★Always read the question carefully to identify if it's asking for area or volume.
★Draw a simple diagram if it helps you visualise the shape and its dimensions.
★Write down the formula you are using before substituting the values.
★Double-check your calculations, especially when multiplying three numbers.
★Ensure your final answer includes the correct cubic units (e.g., cm³, m³).

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