5.6 Comparing lengths, areas and volumes — Learn

Prerequisites

You should be able to multiply and divide whole numbers and fractions. Check units before comparing quantities.

Key terms

Similar shapes have equal angles and proportional corresponding lengths. A scale factor compares their matching lengths.

Objectives

• Convert a length scale factor into area and volume scale factors, in both directions.
• Check that the result matches the units and direction of change.

Learn the method

For similar shapes, matching lengths share a scale factor $s$. Areas scale by $s^2$. Volumes scale by $s^3$.

Worked example. Similar solids have length scale factor 3. The smaller volume is 20 cm$^3$. Find the larger volume.

• Cube the length factor: $3^3=27$.
• Multiply: $20\times27=540$ cm$^3$.
• Check the direction: the larger solid has greater volume.

To work backwards, divide by the area or volume factor.

Watch out. Do not use the length factor directly for area or volume. Square it for area and cube it for volume, then check which shape is larger.

You should now be able to

• I can square a length scale factor to find an area factor.
• I can cube a length scale factor to find a volume factor.
• I can work backwards from area or volume factors.