Calculating volumes of regular 3D objects (PDF, 34 KB)
*Measurement progression, 6th step *

In this activity, the learners develop the understanding that the volume of regular 3D objects (for example, shoe boxes and cylinders) can be found by multiplying the area of the face by the height/length.

## The teaching points

The volume of regular 3D objects can be seen as being formed by the area of the face [rectangle for a shoebox, circle for a cylinder] being repeatedly ‘stacked up’.

This means that the volume of a regular 3D object can be found by calculating the area of the face and multiplying it by the height or length of the object. This principle means that it is possible to calculate the **volume** of a variety of regular 3D objects without learning lots of different formulae.

## Resources

- A shoebox and a cardboard cylinder with the face cut out.
- A variety of regular 3D objects, including those with triangle and pentagon faces – fancy chocolate containers are good for this!

## The guided teaching and learning sequence

1. Ask the learners to identify the shape of the face of the shoebox (rectangle) and cylinder (circle).

2. For both the shoebox and the cylinder, push the face through the object and suggest that the object is formed by the face being repeatedly ‘stacked’. Ask the learners how this could help with calculating the volume of the shoebox and the cylinder.

Listen and encourage the idea that if you know the area of the face of an object, you could find the volume by multiplying the area of the face by the height or length.

3. Ask for a volunteer to measure the length of the sides of the shoebox and record the measurements on a diagram on the board.

4. Ask:

“If the volume of the shoebox can be found by multiplying the area of the face by the height or length, what face would you use to find the volume of the shoebox?”

Further questions could include:

“How many faces are there?” “Are they all different?” “Does it matter which one you use to calculate the volume?”

5. To find out whether it matters which face you use, divide the learners into three groups and allocate one face to each group. Ask each group to calculate the volume of the shoebox by finding the area of the face and multiplying it by the height or length.

For example: Shoebox dimensions:

25 centimetres x 15 centimetres x 10 centimetres

Face 1: 25 centimetres x 15 centimetres = 375 cm^{2}

Volume = 375 cm^{2} x 10 centimetres = 3,750cm^{3 }

Face 2: 10 centimetres x 15 centimetres = 150 cm^{2 }

Volume = 150 cm^{2} x 25 centimetres = 3,750 cm^{3 }

Face 3: 25 centimetres x 10 centimetres = 250 cm^{2 }

Volume = 250 cm^{2} x 15 centimetres = 3,750 cm^{3 }

6. Record each group’s results exactly as they are given. For example, if you are told “3,750”, record it as such. Ask whether this is a correct indication of the volume. Listen for and reinforce the idea that it is not because, without the unit of measurement, there is no indication of the size.

7. Ask:

“What is the correct unit of measurement for the volume of the shoebox?”

“Where does it come from?”

Listen for and emphasise the idea that cm^{3} comes from centimetres (the length or height) multiplied by cm^{2} (the face area). You may need to revise the convention cm x cm x cm = cm^{3}.

8. Ask:

“Does the calculated volume seem reasonable – would approximately 3,000 cubic centimetres fit in the box?”

“Does it matter which face you use to find the volume?”

9. Restate the principle that the volume was calculated by multiplying the area of the face by the height or length and ask how this would apply to the cylinder.

## Follow-up activity

Ask the learners to calculate the volumes of the cylinder and other regular 3D containers.

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