Area and Volume Scale Factors
Grade 7Why lengths, areas, and volumes scale differently
When two shapes are similar with a linear scale factor of k, their areas and volumes do not scale by the same factor.
Area scale factor = k²
Volume scale factor = k³
If the lengths are in ratio 2:3, areas are in ratio 4:9, and volumes are in ratio 8:27.
The logic behind it
Area involves two dimensions (length × width) — each scales by k, so area scales by k × k = k². Volume involves three dimensions, so it scales by k³. This is not a rule to memorise blindly — it falls out naturally once you see it.
Never apply the linear scale factor to an area or volume directly. Double-check what you are scaling: a length? Use k. An area? Use k². A volume? Use k³.
Worked example
EXAMPLE
Two similar cones have radii of 3 cm and 6 cm. The smaller cone has a volume of 24 cm³. Find the volume of the larger cone.
Linear scale factor k = 6 ÷ 3 = 2. Volume scale factor = 2³ = 8. Volume of larger cone = 24 × 8 = 192 cm³.
