*** 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 = pr² × length.
Areas (see also: shapes)
The area of a triangle = half × base × height (there is also
an alternative formula
which uses one of the angles).
The area of a circle = pr² (r is the radius of the circle)
The area of a parallelogram = base × height
Area of a trapezium = half × (sum of the parallel sides) ×
the distance between them [ 1/2(a+b)d ].
Surface area: 4pr²
Curved surface area: 2prh
Volume = 1/3 × area of base × perpendicular height (=1/3pr²h for circular based pyramid).
Curved surface area: prl (l is the "slant height",
i.e. the distance from the edge of the base to the top)
Volume: 1/3pr²h (h is perpendicular height)
USING FORMULAE FOR AREA AND VOLUME IT IS NECESSARY THAT ALL MEASUREMENTS ARE IN
THE SAME UNITS.
1 kilometre (km) = 1000 m
1 metre (m) = 100cm
centimetre (cm) = 10mm
1 litre = 1000 cm³
1 hectare = 10 000 m²
kilogram (kg) = 1000g (grams)
When working with lengths try to use metres
if possible and when working with mass, use kilograms.
1cm² = 100mm²
(10mm × 10mm)
1cm³ = 1000mm³ (10mm × 10mm × 10mm)
Ratios of Lengths, Areas and Volumes
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 1000mm² in a cm²).
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.
The letters r, l, a and b represent lengths. From the
following, tick the three which represent volumes.
3(a² + b²)r
NB: Numbers are dimensionless so
ignore p, 2, 4 and 3.
The first has
three dimensions, since it is r × r × l.
The second has two dimensions (r ×
The third has three dimensions (r × r × r).
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).
Copyright © Matthew Pinkney 2003