18.6 Similarity: areas and volumes — Practise

Work through these on your own. Questions marked H are for Higher only.

H Similar shapes have length scale factor $3$ . Find the area scale factor.
H Similar solids have length scale factor $2$ . Find the volume scale factor.
H A small similar shape has area $12$ cm $^2$ . Length scale factor to the large shape is $4$ . Find the large area.
H A solid has volume $5$ cm $^3$ . It is enlarged by length factor $3$ . Find the new volume.
H Similar shapes have area ratio small:large $=1:25$ . Find the length scale factor small to large.
H Similar solids have volume ratio small:large $=8:125$ . Find the length scale factor.
H A shape has area $20$ cm $^2$ . Its enlargement has area $180$ cm $^2$ . Find the length scale factor.
H Similar solids have volumes $27$ cm $^3$ and $216$ cm $^3$ . Find the length scale factor from smaller to larger.
H A similar solid is enlarged by length factor $5$ . Its volume is $40$ cm $^3$ . Find the new volume.
H Why is the area scale factor the square of the length scale factor?
H A model of a statue is enlarged by length factor $4$ . Which scale factor do you need for the paint (surface area), and which for the material (volume)? Give a reason, then find both.

Answers and concise solutions

1. Similar shapes have length scale factor $3$ . Find the area scale factor.

Answer: 9

2. Similar solids have length scale factor $2$ . Find the volume scale factor.

Answer: 8

3. A small similar shape has area $12$ cm $^2$ . Length scale factor to the large shape is $4$ . Find the large area.

Answer: 192 cm^2

4. A solid has volume $5$ cm $^3$ . It is enlarged by length factor $3$ . Find the new volume.

Answer: 135 cm^3

5. Similar shapes have area ratio small:large $=1:25$ . Find the length scale factor small to large.

Answer: 5

6. Similar solids have volume ratio small:large $=8:125$ . Find the length scale factor.

Answer: 2/5

7. A shape has area $20$ cm $^2$ . Its enlargement has area $180$ cm $^2$ . Find the length scale factor.

Answer: 3

8. Similar solids have volumes $27$ cm $^3$ and $216$ cm $^3$ . Find the length scale factor from smaller to larger.

Answer: 2

9. A similar solid is enlarged by length factor $5$ . Its volume is $40$ cm $^3$ . Find the new volume.

Answer: 5000 cm^3

10. Why is the area scale factor the square of the length scale factor?

Answer: Area multiplies two lengths, each by the length factor, so the multiplier is its square.

11. Method choice: area or volume factor

Answer: Paint covers a surface, so use the area factor: $4^2=16$ . Material fills a volume, so use the volume factor: $4^3=64$ .