*** Remember, with many exam boards, formulae will be given
to you in the exam. However, you need to know how to apply the formulae and
learning them (especially the simpler ones) will help you in the exam. ***
A prism is a shape with a constant cross section, in other words the
cross-section looks the same anywhere along the length of the solid (examples:
cylinder, cuboid).
The volume of a prism = the area of the cross-section × the length. So, for example, the volume of a cylinder = π r2 × length.
Imagine two squares, one with sides of length 3cm and one with sides of length
6cm. The ratio of these lengths is 3 : 6 (= 1 : 2). The area of the first is
9cm and the area of the second is 36cm. The ratio of these areas is 9 : 36 (= 1
: 4) .
In general, if the ratio of two lengths (of similar shapes) is a : b, the ratio
of their areas is a2 : b2 . The ratio of their volumes is a3 : b3 .
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 (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 1000mm3
in a cm3).
Dimensions
Lines have one dimension, areas have two dimensions and volumes have three.
Therefore if you are asked to choose a formula for the volume of an object from
a list, you will know that it is the one with three dimensions.
Example
The letters r, l, a and b represent lengths. From the following, tick the three
which represent volumes.
πr2l
2πr2
4πr3
abrl
abl/r
3(a2 + b2)r
πrl
NB: Numbers are dimensionless so ignore π,
2, 4 and 3.
The first has three dimensions, since it is r × r × l.
The second has two dimensions (r × r).
The third has three dimensions (r × r × r).
etc.
3(a2 + b2)r is the third formula with three dimensions. The expanded version of
this formula is 3a2r + 3b2r and 3 dimensions + 3 dimensions = 3 dimensions (the
dimension can only be increased or reduced by multiplication or division).