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20th May 2013

## GCSE Maths

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# 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.

• The area of a triangle = half x base x 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).