Similarity Scale Factors

Statement

When shapes are enlarged or reduced by a scale factor \(k\):

\[ L' = kL, \quad A' = k^2A, \quad V' = k^3V \]

Lengths scale by \(k\), areas by \(k^2\), and volumes by \(k^3\).

Why it’s true

• If lengths multiply by \(k\), areas (products of two lengths) multiply by \(k^2\).
• Volumes (products of three lengths) multiply by \(k^3\).
• This is a direct result of how dimension scaling works in geometry.

Recipe (how to use it)

1. Identify the scale factor \(k\).
2. For lengths: multiply by \(k\).
3. For areas: multiply by \(k^2\).
4. For volumes: multiply by \(k^3\).
5. To work backwards, divide by the same factor.

Spotting it

Look for problems involving enlargements, models, or maps where shapes are similar and scale factors are given.

Common pairings

• Maps and models (linear scale factor).
• Surface area and volume comparisons.
• Similarity proofs in geometry.

Mini examples

1. If a square has side 3 and is enlarged by \(k=2\): new side=6, new area=\(3²×4=36\).
2. A cube with volume 27 enlarged by \(k=3\): new volume=\(27×27=729\).
3. A rectangle with area 20 doubled (\(k=2\)): new area=\(20×4=80\).

Pitfalls

• Using \(k\) for both lengths and areas without squaring.
• Forgetting volume scales with \(k^3\), not \(k^2\).
• Mixing up enlargement factor and ratio of areas/volumes.

Exam strategy

• Always check which dimension (length, area, volume) is asked for.
• Square scale factors for areas, cube them for volumes.
• Check by considering simple cases (like doubling length quadruples area).

Summary

Similarity scale factors: length scales by \(k\), area by \(k^2\), volume by \(k^3\). Essential for enlargement and ratio problems.