## Formulas

### Introduction

A formula is just a statement about the relationship between various quantities.

Example

P = mv

is a formula encountered in Physics - it says that the momentum (P) of a body is equal to the mass (m) of the body multiplied by its velocity (v).

Example

Another physics formula is

$E = \frac{1}{2} m v^2$

which says that the kinetic energy (E) of a body is equal to half multiplied by its mass multiplied by its velocity squared ('velocity squared' means v × v - note that it is only v that being squared in this formula).

Example

The area of a rectangle is given by multiplying its length by its breadth. Stated as a formula, if A is area, l is the length and w the breadth (or width), then

A = l × w,     or     A = lw

If the length and breadth were identical, then we would have a square, and the above formula would reduce to

A = d2

where d is the length of each side (compare this with the equation

$E = \frac{1}{2} mv^2$

where we descibed the last term as 'v squared'. Hopefully you can now see the derivation of this phrase)

### Lines, Areas, Volumes

The area of a rectangle (or square) was mentioned in an example above. This concept of an area being given by two lengths multiplied together generalizes to other expressions. So

p r2

could be an expression for area because it has two lengths multiplied together, and in fact it is the area of a circle of radius r.

However an expression like

2 p r

could not be a formula for an area because it does not contain two lengths multiplied together.

Quick Quiz
 Which of the following could be a formula for an area (stressing the word could - we are not guaranteeing that it is a valid formula for a definite shape. At this stage, we are just eliminating those formulas that are definitely not area formulas) abc a2 ab2 3ab 6ab 7a2

Following the same logic

• A formula consisting of three lengths multiplied together could represent a volume. So if we are told that

$\frac{4}{3} \pi r^3$

is a valid geometrical formula, then we know straightaway - without being given any more information - that it will represent a volume.

• Going the other way, a formula for the length of a line will tend to contain just one length (which is not squared or raised to any power). For example

2 p r

which has previously been mentioned as the length of the circumference of a circle.

### Perimeters

 The formula for the perimeter of a shape is straightforward - it is just the addition of all the separate sides of the shape. So for a rectangle, the perimeter (P) would be P = l + l + w + w     = 2l + 2w which also equals P = 2 ( l + w ) if you know about factorization.

### Standard Formulas

Just a few more formulas that are quite common

Area of a Triangle

 $\frac{1}{2} \times \ \mbox{base} \times \ \mbox{height}$ which here would be $\frac{1}{2} \times a \times h$ $= \frac{ah}{2}$

Straight Line

 y = mx + c which here would be y = 2x + 1

### Circles

 The two most important formulas for circles are Circumference = 2 p r    ( or alternatively Circumference = p d,   where d is the diameter) Area = p r2

Differentiate between these two formulas by remembering what we have said previously about recognizing the form of line or area formulas. So of the two expressions above

p r2

must represent the expression for the area of the circle.

For completeness, when you come to find the volume of a sphere, you might expect that the formula will contain three lengths (or variable terms) multiplied together. And this is indeed so : The formula for the volume of a sphere is

$\frac{4}{3} \pi r^3$

## Past Exam Questions

#### Two parents are making a round sandpit for the children. It has a radius of 1.5 metres and a depth of 0.2 metres.

1. What is the volume of sand needed to fill the sandpit?

• A     0.18m3
• B     1.35m3
• C     1.80m3
• D     6.75m3

2. Coloured warning tape will be put all the way round the top edge of the sandpit. How much tape will be needed?

• A     4.5m
• B     6m
• C     6.75m
• D     9m

#### She calculates next year's costs for each type of car, excluding insurance and maintenance, using the expression: (cost to buy) + (fuel cost per mile $\times$ number of miles) For the diesel car D, this gives the equation: D = 13 500 + 0.08M where M is the number of miles travelled.

a)   Write down a similar equation for the petrol car, P, using the information in the table.

b)   Find the value of D for next year.

c)   Find the value of P for next year.

d)   How many miles would Jane have to travel next year for P and D to have the same value?

e)   Describe briefly how you checked that the value of M in part (d) is correct.