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GCSE Resit Maths: Formulae and Methods

Key formulae and methods for GCSE Maths at Foundation level, for students resitting to reach grade 4: area and volume, trigonometry, percentages, algebra, ratio and probability.

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Area of a rectangle and a triangle

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Area of a rectangle and a triangle

Rectangle:

length \times width

. Triangle:

\frac{1}{2} \times base \times perpendicular height

Area of a parallelogram and a trapezium

Parallelogram:

base \times perpendicular height

. Trapezium:

\frac{1}{2} (a + b) h

where

a

and

b

are the parallel sides.

Area and circumference of a circle

Area

= π r^{2}

. Circumference

= π d = 2 π r

Volume of a prism and a cylinder

Prism: area of cross-section

\times

length. Cylinder:

π r^{2} h

Pythagoras' theorem

In a right-angled triangle

a^{2} + b^{2} = c^{2}

, where

c

is the hypotenuse (the longest side, opposite the right angle).

SOH CAH TOA

sin = \frac{opp}{hyp}

cos = \frac{adj}{hyp}

tan = \frac{opp}{adj}

Exact trigonometric values

sin 3 0^{\circ} = \frac{1}{2}

cos 6 0^{\circ} = \frac{1}{2}

tan 4 5^{\circ} = 1

sin 9 0^{\circ} = 1

cos 0^{\circ} = 1

Sum of interior angles of a polygon

(n - 2) \times 18 0^{\circ}

, where

n

is the number of sides.

Exterior angles of a regular polygon

Each exterior angle

= 36 0^{\circ} \div n

. Interior angle

= 18 0^{\circ} - exterior angle

Angle facts on parallel lines

Alternate angles are equal, corresponding angles are equal, co-interior angles add up to

18 0^{\circ}

Bearings

Measure clockwise from north using three figures. The back bearing is the bearing

\pm 18 0^{\circ}

Percentage multipliers

Increase by

r %

: multiply by

1 + \frac{r}{100}

. Decrease by

r %

: multiply by

1 - \frac{r}{100}

Percentage change

\frac{change}{original} \times 100

Reverse percentages

Divide the new amount by the multiplier: if a price is up

20%

the original is the new price

\div 1.2

Compound interest

Final amount

= P \times (1 + \frac{r}{100})^{n}

, where

n

is the number of years.

Simple interest

Interest

= P \times \frac{r}{100} \times n

. The same amount is added every year.

Speed, distance and time

Speed

= \frac{distance}{time}

. Distance

= speed \times time

. Time

= \frac{distance}{speed}

Density

= \frac{mass}{volume}

Sharing in a ratio

Add the parts, divide the total by the number of parts to find one part, then multiply by each share.

Direct and inverse proportion

Direct:

y = k x

(both increase together). Inverse:

y = \frac{k}{x}

(one rises as the other falls).

Index laws

a^{m} \times a^{n} = a^{m + n}

a^{m} \div a^{n} = a^{m - n}

(a^{m})^{n} = a^{mn}

a^{0} = 1

a^{- n} = \frac{1}{a ^{n}}

Standard form

A number between

1

and

10

multiplied by a power of

10

, for example

4.2 \times 1 0^{- 3} = 0.0042

Dividing fractions

Keep, change, flip:

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Expanding double brackets

(x + a) (x + b) = x^{2} + (a + b) x + ab

Factorising a quadratic

Find two numbers that multiply to the constant term and add to the coefficient of

x

x^{2} + 7 x + 12 = (x + 3) (x + 4)

Difference of two squares

a^{2} - b^{2} = (a - b) (a + b)

Equation of a straight line

y = m x + c

, where

m

is the gradient and

c

is the

y

-intercept.

Gradient between two points

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

. Parallel lines have equal gradients.

nth term of a linear sequence

Find the common difference

d

; the nth term is

d n + (first term - d)

Midpoint of a line segment

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Mean, median, mode and range

Mean

=

total

\div

number of values. Median

=

middle value when ordered. Mode

=

most common. Range

=

largest

-

smallest.

Mean from a frequency table

Multiply each value by its frequency, add the results, then divide by the total frequency.

Probability rules

Probabilities add up to

1

P (not A) = 1 - P (A)

. For independent events

P (A and B) = P (A) \times P (B)

Expected number of outcomes

Probability

\times

number of trials.

Area of a triangle using sine

Area

= \frac{1}{2} ab sin C

, where

C

is the angle between sides

a

and

b

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