20 min read

Scaling Solids

When every dimension of a solid scales by factor k, surface area scales by k² and volume scales by k³.

Area and Volume Scale Differently

Example 1 — scaling a cube

A cube with side 2 cm is enlarged to a cube with side 6 cm (scale factor 3). Find how many times bigger the volume becomes. 1 Volume scale factor = k^{3} = 3^{3} = 27 Answer: 27 times bigger . Check: original volume = 2^{3} = 8 . New volume = 6^{3} = 216 = 8 \times 27 ✓

Example 2 — scaling a surface area

A model statue is scaled up by factor 5 to make a full-size version. The model's surface area is 40 cm². Find the full-size statue's surface area. 1 Surface area scale factor = k^{2} = 5^{2} = 25 2 Multiply: 40 \times 25 Answer: 1000 cm²

Practice

A solid is enlarged by scale factor 4 . Its original volume was 10 cm³. Find the new volume.

Your answer

Area and Volume Scale Differently

Example 1 — scaling a cube

A cube with side 2 cm is enlarged to a cube with side 6 cm (scale factor 3). Find how many times bigger the volume becomes. 1 Volume scale factor = k^{3} = 3^{3} = 27 Answer: 27 times bigger . Check: original volume = 2^{3} = 8 . New volume = 6^{3} = 216 = 8 \times 27 ✓

Example 2 — scaling a surface area

A model statue is scaled up by factor 5 to make a full-size version. The model's surface area is 40 cm². Find the full-size statue's surface area. 1 Surface area scale factor = k^{2} = 5^{2} = 25 2 Multiply: 40 \times 25 Answer: 1000 cm²

Practice

A solid is enlarged by scale factor 4 . Its original volume was 10 cm³. Find the new volume.

Your answer