Apply the correct scale factor to length, area and volume

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Apply the correct scale factor to length, area and volume

Similar figures have the same shape, with corresponding lengths multiplied by one scale factor k. Areas scale by k² because two dimensions change; volumes of similar solids scale by k³ because three dimensions change.

In the illustration, square sides change from 2 cm to 6 cm, giving k = 3. Their areas change from 4 cm² to 36 cm², a factor of 9. A corresponding similar solid would have volume factor 27, not 3 or 9.

Same-angle right triangles illustrate why trigonometric side ratios are constant: corresponding sides scale together, so their quotient is unchanged. Identifying corresponding sides matters; similarity does not mean that any two lengths within different figures have the same ratio.

Worked example: Similar solids have length ratio 2: 3. Find their volume ratio.

Cube each length-ratio entry.

Use corresponding volumes.

Answer: 8: 27.

Check your understanding: If the area factor is 16, what is the positive length factor?

Answer: 4.

Why: Take the square root, not the cube root.