Scaling solids: areas and volumes

Warm-up

Three quick questions from earlier lessons.

Discover

A 1 cm cube is enlarged to a 2 cm cube. How many small cubes fit inside the big one?

Not to scale

Key idea

Lengths, areas and volumes scale differently.

Under a scale factor

k

: lengths ×

k

, areas (including surface area) ×

k^2

, volumes ×

k^3

. Conversely, a volume ratio

V

gives scale factor

\sqrt[3]{V}

Worked example

A statue is copied at scale 3.

Paint needed: 9 times as much

Areas × 9.

Material needed: 27 times as much

Volumes × 27.

Your turn, guided

Your turn: volumes in ratio 64 : 1. The scale factor.

4

∛64.

Practice ★

A solid has volume 25 cm³. It is enlarged with scale factor 2. Find the new volume.
A solid has volume 5 cm³. It is enlarged with scale factor 3. Find the new volume.

A solid has surface area 49 cm². It is enlarged with scale factor 3. Find the new surface area.

Watch out!

Practice ★★

Two similar solids have volumes in the ratio $27 : 1$ . Find the ratio of their heights (scale factor).
Two similar solids have volumes in the ratio $125 : 1$ . Find the ratio of their heights (scale factor).

Exit ticket

Three questions to check you have got it.

A solid has volume 25 cm³. It is enlarged with scale factor 2. Find the new volume.
A solid has volume 5 cm³. It is enlarged with scale factor 3. Find the new volume.
A solid has volume 9 cm³. It is enlarged with scale factor 2. Find the new volume.

Summary card

Scaling solids

Length $k$ , area $k^2$ , volume $k^3$ .

Watch out: cube roots undo volume ratios.

End of the lesson

Next lesson: Volume of spheres; density Back to the chapter

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