Mensuration and calculation: including the relationships between lengths, areas and volumes in similar figures (Higher tier only)

The idea

For similar figures, if the length scale factor is k, then the area scale factor is k² and the volume scale factor is k³. This extends the basic similarity idea: corresponding lengths are in ratio k, corresponding areas are in ratio k², and corresponding volumes are in ratio k³. For example, two similar cylinders have heights in the ratio 2 : 3, so their base areas are in the ratio 4 : 9 and their volumes are in the ratio 8 : 27. To solve problems, first find the length scale factor from a pair of corresponding lengths, then square it for areas or cube it for volumes. Conversely, if you know an area ratio, take the square root to find the length ratio; if you know a volume ratio, take the cube root. Always check that the figures are similar and that you are comparing corresponding parts. This relationship is Higher tier only.

What earns the marks

• State that for similar figures the length scale factor is k, the area scale factor is k² and the volume scale factor is k³.
• Calculate an unknown area or volume using the square or cube of the length scale factor.
• Find the length scale factor by taking the square root of an area ratio or the cube root of a volume ratio.
• Apply the relationships to solve problems involving similar 2D shapes and 3D solids.
• Recognise that the relationships hold only when the figures are similar.

Common mistakes and how to put them right

• Using the length scale factor directly for area or volume.

  Put it right by squaring for area and cubing for volume.

• Forgetting to take the square root or cube root when working from an area or volume ratio back to a length ratio.

  Put it right by reversing the power.

• Mixing up which figure is the original and which is the image.

  Put it right: Clearly labelling the scale factor direction.

• Assuming the relationships apply to non-similar figures.

  Put it right by checking that all corresponding angles are equal and all corresponding lengths are in the same ratio.

Exam tips

• Write down the length scale factor first, then show the squaring or cubing step clearly.
• If given an area or volume ratio, state the root you are taking before calculating.
• Use units consistently and check that your final answer has the correct unit (for example cm² for area, cm³ for volume).