Ratios of Lengths, Areas and Volumes

Imagine two squares, one with sides of length 4cm and one with sides of length 8cm. The ratio of these lengths is 4 : 8 (= 1 : 2). The area of the first is 16cm and the area of the second is 64cm. The ratio of these areas is 16 : 64 (= 1 : 4) .
In general, if the ratio of two lengths (of similar shapes) is a : b, the ratio of their areas is The ratio of their volumes is
This is why the ratio of the length of a mm to a cm is 1:10 (there are 10mm in a cm). The ratio of their areas expressed (i.e. mm² to cm²) is 1:10² (there are 100mm² in a cm²) and the ratio of their volumes (mm³ to cm³) is 1:10³ (there are 1000mm² in a cm²).

Dimensions

Lines have one dimension, areas have two dimensions and volumes have three. We can see this from their respective units: m, m2 and m3 respectively. We obtain areas by multiplying two length together and volumes by multiplying 3 lengths together. If you use also the fact that you can only add lengths to lengths, areas to areas and volumes to volumes, is is quite easy to pick out those expressions which identify lengths, areas or volumes , or represent nothing at all.

Examples

The lettersandrepresent lengths. .
is an volume sinceis a number with no units andhas the units m3.
is an area. Ignoreand r² has units m²
is a volume. Ignoreand r3 has units m3.
is not a length, area or volume since the units are m4
is an area since the units are m3/m=m2
is a volume
is an area