Areas and Volumes

*** 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.

Areas (see also: shapes)

• The area of a triangle = half ×2; base × height (there is also an alternative formula which uses one of the angles).

• The area of a circle = π r²   (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 ].

Spheres

Volume: 4/3πr³
Surface area: 4πr²

Cylinder

Curved surface area: 2πrh
Volume: πr²h

Pyramid

Volume = 1/3 × area of base × perpendicular height (=1/3πr²h for circular based pyramid).

Cone

Curved surface area: πrl (l is the "slant height", i.e. the distance from the edge of the base to the top)
Volume: 1/3πr²h (h is perpendicular height)

WHEN USING FORMULAE FOR AREA AND VOLUME IT IS NECESSARY THAT ALL MEASUREMENTS ARE IN THE SAME UNITS.

Units

1 kilometre (km) = 1000 m
1 metre (m) = 100cm
1 centimetre (cm) = 10mm
1 litre = 1000 cm³
1 hectare = 10 000 m²
1 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 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).

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© Matthew Pinkney 2007