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Lesson overview

Geometry - volume of 3D solids

Volume measures how much space is inside a 3D shape. For a cuboid, the volume is length x width x height because each layer has the same number of unit cubes. For any prism, the volume is area of cross-section x length because the same cross-section repeats all the way through the solid.

Cuboid volume is built from equal layers

This cuboid has length $5$ cm, depth $3$ cm and height $4$ cm. Its volume $V = l \times w \times h$ .

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Drag the shape to rotate it. $V = l \times w \times h$ — count layers: area of one layer × number of layers.

A prism repeats the same cross-section

If the triangular cross-section has area $12$ ${cm}^{2}$ and the prism length is $7$ cm, then $V = 12 \times 7$ .

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For any prism: $V = A_{\text{cross-section}} \times l$ . Find the cross-section area first, then multiply by length.

Volume uses cubic units

Area is measured in square units, but volume is measured in cubic units because the shape is three-dimensional.

Lengthcm

one dimension

Area ${cm}^{2}$

two dimensions

Volume ${cm}^{3}$

three dimensions

Check the units at the end. Volume should finish in cubic units.

Worked example

Find the volume of a prism with cross-section area 15 ${cm}^{2}$ and length 8 cm.

Use the prism rule: volume = area of cross-section x length.
The cross-section area is already given as 15 ${cm}^{2}$ .
Multiply 15 by 8 to get 120.
Write the final answer as 120 ${cm}^{3}$ because volume uses cubic units.

Try it

Decide first whether you are using cuboid volume directly or finding a prism cross-section area before multiplying.

Question 1

What is the volume of a cuboid with dimensions 4 cm, 3 cm and 5 cm?

Question 2

A prism has cross-section area 9 ${cm}^{2}$ and length 6 cm. What is its volume?

Question 3

Which formula matches the volume of any prism?

Cross-sectionsame shape all the way through

Lengthhow far it extends

Question 4

A cuboid has a base area of 18 ${cm}^{2}$ and height 4 cm. What is its volume?

Question 5

Which unit is correct for volume?

Lengthcm

Area ${cm}^{2}$

Volume?

Common mistakes

Watch for these when working through the lesson.

Adding dimensions instead of multiplying them.
Forgetting to find the prism cross-section area before multiplying by the length.
Writing square units instead of cubic units for volume.

Volume of cuboids and prisms

Lesson overview

Worked example

Try it

Common mistakes

Related topics

Independent practice will plug in here