Comparing dimensions; similarity ratios
When shapes are similar, their corresponding lengths are in a fixed ratio. WJEC Higher extends this to areas and volumes.
Linear, area, volume scale factors
If linear scale factor (LSF) is k, then:
- Area scale factor (ASF) = k².
- Volume scale factor (VSF) = k³.
This works in both directions:
- LSF = √ASF.
- LSF = ³√VSF.
Why?
Length is one-dimensional, area is length × length, volume is length × length × length. Each multiplicative factor is applied that many times.
Linking back from area to length
Two similar tins have areas 50 cm² and 200 cm². ASF = 200/50 = 4 → LSF = √4 = 2. Larger tin's lengths are double the smaller.
Linking back from volume to length
Two similar bottles have volumes 27 cm³ and 216 cm³. VSF = 216/27 = 8 → LSF = ³√8 = 2.
If the smaller bottle has surface area 60 cm², the larger has surface area 60 × 2² = 240 cm².
⚠Common mistakes— Pitfalls
- Students square or cube the wrong number. ALWAYS identify the LSF first; never apply the area/volume ratio directly to a length.
- Mixed units. Convert to the same unit before forming a ratio (e.g. 0.5 m vs 25 cm → 50 cm vs 25 cm → 2 : 1).
WJEC exam tip
When a question gives areas and asks about a length, state explicitly: "ASF = ..., so LSF = √(ASF) = ...". The B1/M1 split rewards stating the link before any calculation.
AI-generated · claude-opus-4-7 · v3-wjec-maths-leaves