I have two mathematically similar shapes, one of which has a volume of 1 litre and one which has a volume of half a litre. If the largest shape has a height of 40 cm, what (in centimetres) is the height of the smaller shape?

Answer 1

The volumes of similar shapes are proportional to the cubes of their corresponding sides, so

1 L / .5 L = 40^3 cm^3 / s^3 cm^3
cancelling the units
1 / .5 = 40^3/s^3
s^3 = .5(64,000) = 32,000
s = 32000^(1/3) = appr 31.748 cm

check"
40^3 : 31.748^3
= 64000 : 32000
= 2 : 1

Answer 2

the ratio of volumes is the cube of the ratio of heights.
So the ratio of heights is 1:∛2
40/∛2 = 31.748