If the length scale factor between two similar shapes is k, then the area scale factor is k² and the volume scale factor is k³. Doubling every length makes the area 4 times as big and the volume 8 times as big.
To go backwards, take roots. If the areas are 28 cm² and 63 cm², the area factor is 63 ÷ 28 = 2.25, so the length factor is √2.25 = 1.5 and the volume factor is 1.5³ = 3.375.
Worked example
Two similar solids have lengths 4 cm and 12 cm. The smaller has volume 10 cm³. Find the larger volume.
- Length factor k = 3, so volume factor = 27.
- 10 × 27 = 270.
Answer:270 cm³
Key words
- length factor
- k
- area factor
- k²
- volume factor
- k³
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