P5 Volume with Unit Cubes

Area covers a surface. Volume describes the space inside a solid or the amount of liquid in a tank. Imagine building the solid from identical 1-unit cubes: one complete layer is a rectangle of cubes, and more layers repeat that same rectangle.

Parents: ask your child to predict the cube count before changing a dimension, then ask which layer stayed the same.

By the end you will be able to build and count unit-cube solids, draw cubes and cuboids on an isometric grid, use cubic units, find volume, and connect millilitres and litres to cubic centimetres.

Build a solid and count its layers

A unit cube has side length 1 unit. A solid that is 3 cubes long, 2 cubes broad and 2 cubes high has two equal layers. Each layer has 3 × 2 = 6 cubes, so the total is 6 + 6 = 12 unit cubes. Cubes behind the visible faces still belong to the solid and must be counted.

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Unit-cube volume lab

🤔 Predict first: A solid is 3 × 2 × 2 unit cubes. How many unit cubes does it contain?

6 unit cubes12 unit cubes18 unit cubes

Cubic units name the space

When the side of each unit cube is 1 cm, one cube has a volume of 1 cm³. When the side is 1 m, one cube has a volume of 1 m³. These are different units for different-sized solids. At P5, keep cm³ and m³ separate; do not convert between them.

Name the cubic unit

$$1\text{ cm}\times1\text{ cm}\times1\text{ cm}=1\text{ cm}^3$$

$$1\text{ m}\times1\text{ m}\times1\text{ m}=1\text{ m}^3$$

Show the worked solution

Cuboid volume

$$4\times2\times3=24\text{ cubic units}$$

Draw a cube on an isometric grid

An isometric grid uses three directions for equal unit edges: one vertical direction and two sloping directions. In the worked cube below, start at the near lower corner A, draw equal edges AB, AD and AE, then join matching endpoints. The labels A, B, D and E identify the edges used in the construction; projected points that sit on top of each other do not become the same 3-D point.

Worked construction — cube with edge 3 units:

1. Mark A and draw AB three grid steps along one sloping direction.
2. From A draw AD three grid steps along the other sloping direction.
3. Draw AE three grid steps vertically upwards.
4. Join B to the matching top point, D to its matching top point, and the top endpoints in parallel directions. Keep hidden back edges dashed.
5. Count three equal steps on every axis. Since all three edge lengths are 3, this solid is a cube.

Draw a cuboid on an isometric grid

The same three directions work for a cuboid; only the counts differ. The worked drawing below uses length 4, breadth 2 and height 3. The three dimension labels sit beside their own exterior edges, and hidden edges remain part of the solid even when they are dashed.

Worked construction — cuboid 4 × 2 × 3 units:

1. From A, count 4 grid steps for AB, 2 steps for AD and 3 vertical steps for AE.
2. Join B to C and D to C along the matching sloping directions to close the base parallelogram.
3. Draw verticals from the four base corners for height 3.
4. Join the top endpoints in pairs parallel to AB and AD. Use dashed lines for BC, CD and CG because they are behind the chosen view.

Learner construction task

Draw a cuboid with length 3 units, breadth 2 units and height 2 units. Draw all three directions on an isometric grid, label the near lower corner A, and show the three hidden edges with dashes. Before opening the solution, predict the number of unit cubes in the solid.

Show the worked drawing and count

Count the learner task

$$3\times2=6\text{ unit cubes in each layer}$$

$$6\times2=12\text{ unit cubes}$$

$$V=3\times2\times2=12\text{ cubic units}$$

The two fully hidden cubes are included in the count. The diagram shows the same three visible faces, while the dashed edges show where the hidden part continues.

Find cube and cuboid volume

The repeated-layer idea gives the formula volume = length × breadth × height. For a cube, all three lengths are the same, so multiply the edge three times.

Show the worked solution

Cuboid volume

$$\text{Volume}=\text{length}\times\text{breadth}\times\text{height}$$

$$V=8\times5\times3=120\text{ cm}^3$$

For a cube with edge 4 cm, the volume is 4 × 4 × 4 = 64 cm³. Do not write square units: three edge lengths are multiplied.

Volume of liquid in a rectangular tank

Water in a rectangular tank forms a cuboid. Use the water depth, not the empty tank height. A tank with base 8 cm by 5 cm and full height 10 cm has water only 6 cm deep in the drawing below.

Show the worked solution

Use the water depth

$$\text{Water volume}=8\times5\times6=240\text{ cm}^3$$

$$240\text{ cm}^3=240\text{ ml}$$

Litres, millilitres and cubic centimetres

One millilitre occupies the same volume as a 1 cm³ cube: 1 ml = 1 cm³. One litre is 1000 millilitres, so 1 l = 1000 cm³. These are relationships between litres or millilitres and cm³; they do not create a cm³-to-m³ conversion.

Liquid volume relationships

$$1\text{ ml}=1\text{ cm}^3$$

$$1\text{ l}=1{,}000\text{ ml}=1{,}000\text{ cm}^3$$

Graduated practice

Try each complete question before opening its solution.

Practice 1 — count by layers. A solid has length 5 unit cubes, breadth 2 unit cubes and height 3 unit cubes. How many unit cubes does it contain?

Show the solution

Practice 1

$$5\times2=10\text{ unit cubes in each layer}$$

$$10\times3=30\text{ unit cubes}$$

$$V=5\times2\times3=30\text{ cubic units}$$

Practice 2 — use measured edges. Find the volume of a cube with edge 6 cm.

Show the solution

Practice 2

$$V=6\times6\times6$$

$$V=216\text{ cm}^3$$

Practice 3 — tank water. A rectangular tank has a base of 12 cm by 4 cm. Its empty height is 9 cm, but water is 5 cm deep. Find the water volume in cm³ and ml.

Show the solution

Practice 3

$$\text{Water volume}=12\times4\times5=240\text{ cm}^3$$

$$240\text{ cm}^3=240\text{ ml}$$

The empty height 9 cm is not used because the water depth is 5 cm.

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 7 — your progress is saved on this device.

1. A solid is 3 unit cubes long, 2 broad and 2 high. How many unit cubes does it contain?

   61218

2. A cuboid is 4 cm by 2 cm by 3 cm. What is its volume?

   9 cm³24 cm³12 cm³

3. What unit describes a 1 cm by 1 cm by 1 cm cube?

   1 cm1 cm²1 cm³

4. A cube has edge 5 m. What is its volume?

   15 m³25 m³125 m³

5. A tank base is 8 cm by 5 cm and the water is 6 cm deep. What is the water volume?

   240 cm³400 cm³80 cm³

6. Which relationship is correct?

   1 ml = 1 cm³1 ml = 100 cm³1 l = 1 cm³

7. Which statement about drawing a cuboid on an isometric grid is correct?

   All three edge directions can be random.Edges in each sloping direction are parallel, the two directions cross at each corner, and hidden back edges are dashed.A vertical edge must be sloping.

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