Similar solids

Objective. Recover linear factors from area or volume information.

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2. Worked example
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Learn: the key idea

Similar solids have exactly the same shape with all corresponding lengths in the same ratio, called the linear scale factor. If the linear scale factor is k, then areas scale by k² and volumes scale by k³, since area involves two lengths and volume involves three. Given a volume ratio, the linear scale factor is the cube root of that ratio, used to recover missing lengths. For two similar statues with volumes 8 cm³ and 64 cm³, the volume ratio is 8, so the linear scale factor is ∛8 = 2. This lets you scale any linear measurement between the two solids directly.

Worked example

Two similar solids have volumes 27 cm³ and 125 cm³. The smaller solid has height 6 cm. Find the height of the larger solid.

1 $Volume ratio = 125/27$
2 $Linear scale factor = ∛(125/27) = 5/3$
3 $Larger height = 6 \times 5/3$
4 $Height = 10 cm$

Answer: 10 cm

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Common mistake. A frequent mistake is applying the volume ratio directly to lengths, so doubling a volume ratio of 8 onto a length instead of taking the cube root first, which hugely overstates the length change.

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