Revision Resource

GCSE Maths
Cheat Sheet

Every formula, definition and key fact you need for GCSE Maths in one place. Covers all exam boards: AQA, Edexcel, OCR and WJEC.

56+ formulas7 topicsHigher & Foundation

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Area & Perimeter

Rectangle

A = l x w

Length times width

Triangle

A = 1/2 x b x h

Half base times height

Parallelogram

A = b x h

Base times perpendicular height

Trapezium

A = 1/2(a + b) x h

Half the sum of parallel sides times height

Circle area

A = pi x r^2

Pi times radius squared

Circle circumference

C = 2 x pi x r

Or C = pi x d (diameter)

Sector area

A = (theta/360) x pi x r^2

Fraction of full circle

Arc length

L = (theta/360) x 2 x pi x r

Fraction of circumference

Volume & Surface Area

Cuboid

V = l x w x h

Length x width x height

Prism

V = area of cross-section x length

Works for any uniform cross-section

Cylinder

V = pi x r^2 x h

Circle area times height

Cone

V = 1/3 x pi x r^2 x h

One-third of a cylinder

Sphere

V = 4/3 x pi x r^3

Given on the formula sheet

Sphere SA

SA = 4 x pi x r^2

Four times the area of a great circle

Cone SA

SA = pi x r x l + pi x r^2

Curved surface + base. l = slant height

Pyramid

V = 1/3 x base area x h

One-third of the prism

Pythagoras & Trigonometry

Pythagoras

a^2 + b^2 = c^2

c is the hypotenuse (longest side)

SOH

sin(x) = opposite / hypotenuse

Use when you have opp and hyp

CAH

cos(x) = adjacent / hypotenuse

Use when you have adj and hyp

TOA

tan(x) = opposite / adjacent

Use when you have opp and adj

Sine rule

a/sin(A) = b/sin(B) = c/sin(C)

Use with 2 sides and 2 angles

Cosine rule

a^2 = b^2 + c^2 - 2bc x cos(A)

Use with 3 sides or 2 sides + included angle

Area of triangle

A = 1/2 x a x b x sin(C)

When you know 2 sides and the included angle

Exact values

sin30=1/2, cos30=root3/2, tan30=1/root3

Also sin60, cos60, tan45=1

Algebra

Quadratic formula

x = (-b +/- sqrt(b^2 - 4ac)) / 2a

Solves ax^2 + bx + c = 0

Discriminant

D = b^2 - 4ac

D>0: 2 roots. D=0: 1 root. D<0: no real roots

Completing the square

(x + b/2)^2 - (b/2)^2 + c

Rewrite x^2 + bx + c

Difference of squares

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

Quick factorisation trick

nth term (linear)

an + b

a = common difference, b = zero term

nth term (quadratic)

an^2 + bn + c

Second difference / 2 = a

Simultaneous equations

Eliminate or substitute

Make coefficients equal, then add/subtract

Index laws

a^m x a^n = a^(m+n), a^m / a^n = a^(m-n)

(a^m)^n = a^(mn), a^0 = 1, a^(-n) = 1/a^n

Graphs & Coordinates

Straight line

y = mx + c

m = gradient, c = y-intercept

Gradient

m = (y2 - y1) / (x2 - x1)

Rise over run between two points

Midpoint

((x1+x2)/2, (y1+y2)/2)

Average of coordinates

Distance

d = sqrt((x2-x1)^2 + (y2-y1)^2)

Pythagoras on coordinates

Parallel lines

Same gradient

m1 = m2

Perpendicular lines

m1 x m2 = -1

Negative reciprocal gradients

Circle equation

x^2 + y^2 = r^2

Centre at origin, radius r

Quadratic graph

y = ax^2 + bx + c

U-shape (a>0) or n-shape (a<0)

Probability & Statistics

Probability

P = favourable / total

Always between 0 and 1

AND rule

P(A and B) = P(A) x P(B)

For independent events

OR rule

P(A or B) = P(A) + P(B) - P(A and B)

Subtract overlap

Mean

Sum of values / number of values

Affected by outliers

Median

(n+1)/2 th value

Middle value when ordered

Mode

Most frequent value

Can have more than one

Range

Highest - Lowest

Measure of spread

IQR

Q3 - Q1

Interquartile range, ignores extremes

Ratio, Proportion & Rates

Speed

speed = distance / time

DST triangle

Density

density = mass / volume

DMV triangle

Pressure

pressure = force / area

PFA triangle

Percentage change

((new - old) / old) x 100

Positive = increase, negative = decrease

Compound interest

A = P(1 + r/100)^n

P = principal, r = rate, n = years

Reverse percentage

Divide by multiplier

e.g. after 20% increase, divide by 1.2

Direct proportion

y = kx

k is the constant of proportionality

Inverse proportion

y = k/x

As x increases, y decreases

Common Exam Mistakes

Forgetting to square root after Pythagoras

a^2 + b^2 = c^2 gives c^2, not c. Always square root at the end.

Using diameter instead of radius

Circle formulas use radius. If given diameter, halve it first.

Not showing working on 'show that' questions

Write every step. Even obvious ones. Marks are for the working, not the answer.

Rounding too early

Keep full calculator values until the final answer. Only round at the end.

Confusing area and perimeter

Area = space inside (square units). Perimeter = distance around (linear units).

Forgetting negative solutions in quadratics

x^2 = 9 gives x = 3 AND x = -3. Always consider both roots.

Wrong order of operations with negatives

-3^2 = -9, but (-3)^2 = 9. Brackets matter.

Not converting units

1 m = 100 cm. 1 km = 1000 m. 1 litre = 1000 cm^3. Check units match.

Key Facts & Definitions

Integer — A whole number (positive, negative, or zero)

Prime number — Only divisible by 1 and itself (2, 3, 5, 7, 11...)

Factor — A number that divides exactly into another

Multiple — The result of multiplying a number by an integer

HCF — Highest Common Factor. Use prime factorisation.

LCM — Lowest Common Multiple. Use prime factorisation.

Rational number — Can be written as a fraction (includes terminating and recurring decimals)

Irrational number — Cannot be written as a fraction (e.g. pi, sqrt 2)

Polygon — A 2D shape with straight sides

Regular polygon — All sides and angles equal

Interior angle sum — (n-2) x 180 degrees

Exterior angle sum — Always 360 degrees

Similar shapes — Same shape, different size. Angles equal, sides in ratio.

Congruent shapes — Identical in shape and size

Bearing — Measured clockwise from North. Always 3 digits (e.g. 045).

Vector — Has magnitude and direction. Written as a column vector.

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