Lengths, Area and Volumes
Imagine two cubes, one with sides of length 4cm and one with sides of length 8cm. The ratio of these lengths is 4 : 8 (= 1 : 2).
Since these are cubes, each face has base 4 and 8 respectively, and height 4 and 8 respectively. The area of a face of the first is 16cm 2 and the area of a face of the second is 64cm 2 . The ratio of these areas is 16 : 64 (= 1 : 4) .
The volume of the first is 64 cm 3 and the volume of the second is 512 cm 3 . 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
The ratio of the length of a mm to a cm is 1:10 (there are 10 mm in a cm). The ratio of their areas expressed in these units (i.e. mm² to cm²) is 10²:1 (there are 100mm² in a cm²) and the ratio of their volumes (mm³ to cm³) is 10³:1 (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, m 2 and m 3 respectively. We obtain areas by multiplying two lengths together and volumes by multiplying three 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 letters
and
represent lengths. 
is an volume since
is a number with no units and
has the units m 3 . 
is an area. Ignore
and r² has units m². 
is a volume. Ignore
and 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.