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MathidiaMathematics Academy
Similarity, congruence and scale
Lesson 48 of 75
Standard practice

Area and volume scale factors

Objective. Move fluently between linear, area and volume factors.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

When a shape is enlarged by a linear scale factor k, its area scales by k² and its volume scales by k³, since area involves two dimensions and volume involves three. For a shape enlarged with linear scale factor 3, its area increases by a factor of 3²=9, and any corresponding solid's volume increases by 3³=27. To move between factors, take the square root of an area ratio to find the linear scale factor, or the cube root of a volume ratio. If two similar cylinders have volumes 8cm³ and 64cm³, the volume ratio is 8, so the linear scale factor is the cube root of 8, which is 2. Always identify whether you are given a linear, area or volume ratio before choosing which root or power to apply.

Worked example

Two similar solids have a linear scale factor of 4. The smaller solid has volume 15 cm³. Find the volume of the larger solid.

4 … 15
Illustrated card showing 4 … 15
  1. 1Volume scale factor = (linear scale factor)³ = 4³
  2. 2Calculate: 4³ = 64
  3. 3Multiply the smaller volume by this factor: 15 × 64
  4. 4Calculate: 15 × 64 = 960

Answer: 960 cm³

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Common mistake. Pupils apply the linear scale factor directly to area or volume without squaring or cubing it. For a linear scale factor of 2, they might say the area doubles instead of correctly quadrupling (2²=4). Always square the linear factor for area and cube it for volume.

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