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Mathematical Formulae - GCSE Maths

The AQA GCSE Mathematics (8300) specification includes a set of mathematical formulae that students need to be familiar with. These are categorised into formulae that students must know, formulae they should be able to derive, and formulae that will be provided in the exam. Refer to the Subject content section to determine the tier at which these formulae could be used.

The quadratic formula

The solutions of ax² + bx + c = 0, where a ≠ 0

The quadratic formula

Quadratic formula
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Circumference and area of a circle

Where r is the radius and d is the diameter:

Circumference and area of a circle

Circumference of a circle
C=2πr=πdC = 2\pi r = \pi d
Area of a circle
A=πr2A = \pi r^2

Pythagoras' theorem

In any right-angled triangle where a, b and c are lengths of the sides and c is the hypotenuse:

Pythagoras' theorem

Pythagoras' theorem
a2+b2=c2a^2 + b^2 = c^2

Trigonometry formulae

In any right-angled triangle ABC where a, b and c are lengths of the sides and c is the hypotenuse:

Right-angled triangle ratios

Sine
sin⁡A=ac\sin A = \frac{a}{c}
Cosine
cos⁡A=bc\cos A = \frac{b}{c}
Tangent
tan⁡A=ab\tan A = \frac{a}{b}

Any triangle ABC (a, b and c are lengths of the sides)

Sine rule
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
Cosine rule
a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A
Area of a triangle
Area=12absin⁡C\text{Area} = \frac{1}{2}ab \sin C

Compound interest

Where P is the principal amount, r is the interest rate over a given period and n is number of times that the interest is compounded:

Compound interest

Total accrued
Total accrued=P ⁣(1+r100)n\text{Total accrued} = P\!\left(1 + \frac{r}{100}\right)^{n}

Probability

Where P(A) is the probability of outcome A and P(B) is the probability of outcome B:

Probability

Addition rule
P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
Conditional probability
P(A and B)=P(A given B)×P(B)P(A \text{ and } B) = P(A \text{ given } B) \times P(B)

Perimeter, area, surface area and volume formulae (to know)

Students are expected to know the following formulae or be able to derive them; they will not be given in the exam. Refer to the Subject content section to determine the tier at which these formulae could be used.

Where a and b are the lengths of the parallel sides and h is their perpendicular separation:

Perimeter, area, surface area and volume (to know)

Area of a trapezium
Area=12(a+b)h\text{Area} = \frac{1}{2}(a + b)h
Volume of a prism
Volume=area of cross section×length\text{Volume} = \text{area of cross section} \times \text{length}

Perimeter, area, surface area and volume formulae (given in exam)

Students are not expected to memorise the following formulae; they will be given in the exam in the relevant question. Refer to the Subject content section to determine the tier at which these formulae could be used.

Where r is the radius of the sphere or cone, l is the slant height of a cone and h is the perpendicular height of a cone:

Perimeter, area, surface area and volume (given in exam)

Curved surface area of a cone
CSA=πrl\text{CSA} = \pi r l
Surface area of a sphere
SA=4πr2\text{SA} = 4\pi r^2
Volume of a sphere
V=43πr3V = \frac{4}{3}\pi r^3
Volume of a cone
V=13πr2hV = \frac{1}{3}\pi r^2 h

Kinematics formulae

Where a is constant acceleration, u is initial velocity, v is final velocity, s is displacement from the position when t = 0 and t is time taken:

Kinematics formulae

Velocity
v=u+atv = u + at
Displacement
s=ut+12at2s = ut + \frac{1}{2}at^2
Velocity-displacement
v2=u2+2asv^2 = u^2 + 2as