Volume Scaling

Statement

When a solid is enlarged or reduced by a scale factor \(k\), its volume scales by the cube of that factor:

\[ V' = k^3 V \]

where \(V\) is the original volume and \(V'\) is the new volume.

Why it’s true

• Scaling multiplies every linear dimension by \(k\).
• Area scales by \(k^2\).
• Volume scales by \(k^3\), because it involves three dimensions (length × width × height).

Recipe (how to use it)

1. Identify the scale factor \(k\).
2. Cube it to get \(k^3\).
3. Multiply the original volume by \(k^3\).
4. This gives the new volume.

Spotting it

Look for enlargement or reduction problems where two similar 3D shapes are compared, often cones, spheres, pyramids, or cuboids.

Common pairings

• Similar shapes problems (ratios of lengths, areas, and volumes).
• Density and mass problems linked with enlargement.

Mini examples

1. Cube enlarged by scale factor 2: original volume 27 → new volume \(2^3×27=216\).
2. Sphere reduced by scale factor 0.5: original volume 288π → new volume \(0.5^3×288π=36π\).

Pitfalls

• Using \(k\) instead of \(k^3\).
• Mixing up area and volume scaling (area uses \(k^2\)).
• Forgetting that reduction uses a scale factor less than 1.

Exam strategy

• Always write the relation \(V' = k^3V\).
• Check whether the question gives linear scale or volume ratio.
• If given volume ratio, cube root it to get scale factor.

Summary

Volume scales by the cube of the scale factor. Double the scale factor, volume increases eightfold; halve it, volume reduces to one-eighth.