Similar shapes: area and volume

Higher only

The idea

If the lengths of similar shapes are in the ratio $k$ , their areas are in the ratio $k^2$ and their volumes in the ratio $k^3$ .

Find the length scale factor $k$ .
Square it for area; cube it for volume.
To go from area to length, square root; from volume to length, cube root.

Watch out Do not use the length scale factor on an area or a volume.

See it worked

Worked exampleNon-calculator3 marks

Two cylinders are mathematically similar. The smaller has height 6 cm and volume $160\text{ cm}^3$ . The larger has height 9 cm.

Work out the volume of the larger cylinder.

Length scale factor $k = \frac{9}{6} = 1.5$ .
Volumes scale by $k^3$ . $1.5^3 = 3.375$
$160 \times 3.375 = 540$

Answer $540\text{ cm}^3$

Your turn

Question 1Non-calculator1 mark

Circle B has four times the radius of circle A. Which statement about their areas is correct?

Question 2Non-calculator4 marks

Two solid shapes are mathematically similar. Their surface areas are $72\text{ cm}^2$ and $162\text{ cm}^2$ . The smaller shape has volume $96\text{ cm}^3$ .

Work out the volume of the larger shape.

Question 3Calculator allowed3 marks

Two similar closed solid models are made from the same material. Their surface areas differ by 84 cm² and are in the ratio 9 : 16. The smaller model has volume 81 cm³. A third similar model has surface area equal to the sum of the first two surface areas. Find the volume of the third model. You must show your working.

In past papers

39 official questions in 38 papers since 2017. The latest:

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