Area and Volume Scale Factors

Grade 7

Why 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.
Nova
Try this one

\( ABCD \) is a parallelogram. The diagonal \( AC \) is drawn, splitting it into triangle \( ABC \) and triangle \( CDA \).

Prove that triangle \( ABC \) is congruent to triangle \( CDA \). State the congruence condition you use. [3]

Nova's hint:

Use the properties of a parallelogram, together with the diagonal, to match up parts of the two triangles.

Ask yourself: 'which pairs of sides can I show are equal, and is there a side that both triangles share?' Then decide which congruence condition those facts satisfy.

Try to Solve it with Nova?