Revision:Areas and Volumes - The Student Room

Register

About Us | Help | Sign in

Create New Article | Editing Help

Advanced Search

Edit \| Hide
Welcome \| Study Help \| A Levels \| Applications, UCAS etc. \| University Discussion \| Careers \| General Discussion \| Have Your Say \| Health & Relationships \| Debate & Discussion \| Technology \| Sport \| Entertainment \| Chat \| Ask A Moderator

Revision:Areas and Volumes

From The Student Room

Jump to:navigation, search

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Areas and Volumes

1 Formulae
2 Units
3 Ratios of lengths, areas and volumes
4 Dimensions
5 Comments

Formulae

Bleow are the area and volume forumlae for a number of shapes.

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.

	The area of a triangle is $\frac{1}{2}bh.$
	The area of a circle is $\pi r^2.$
	The area of a parallelogram is $\ell h.$
	The area of a trapezium is $\frac{1}{2}(a+b)h.$
	The volume of a sphere is $\frac{4}{3}\pi r^3.$ The curved surface area of a sphere is $4\pi r^2.$
	The volume of a cylinder is $\pi r^2 h.$ (Note that this follows from the volume of a prism.) The curved surface area is $2\pi rh.$
	The volume of a pyramid (with any base) is $\frac{1}{3} Ah,$ where A is the area of the base.
	The volume of a cone is $\frac{1}{3} \pi r^2h.$ (See pyramid.)
	The volume of a prism (with any base) is where A is the area of the base.

When using these formulae, make sure all of the lengths are in the same units!

Units

1km = 1000m
1m = 100cm
1cm = 10mm
1 litre = 1000 cm³
1 hectare = 10 000 m²
1 kilogram = 1000g
1cm² = 100mm² (= 10mm × 10mm)
1cm³ = 1000mm³ (= 10mm × 10mm × 10mm)

Ratios of lengths, areas and volumes

Two squares A and B have side lengths 3cm and 6cm respectively. 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 (1² : 2²) . In general, if the ratio of the lengths of two corresponding sides of two similar shapes is a : b, the ratio of their areas is a² : b² . The ratio of their volumes is a³ : b³ .

Dimensions

Lengths 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 , , $\ell$ and represent lengths. From the following, tick the three which represent volumes.

$\pi r^2\ell$
$2\pi r^2$
$4\pi r^3$
$abr\ell$
$\frac{ab\ell}{r}$
$\pi r\ell$

NB: Numbers are dimensionless, so $\pi,$ 2, 3 and 4 do not have any effect on the dimension of a formula.

The first has three dimensions, since it is the product of three lengths. The second has two dimensions; the third has three dimensions; 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.