Maths formulae

aᵐ × aⁿ = aᵐ⁺ⁿ · aᵐ ÷ aⁿ = aᵐ⁻ⁿ · (aᵐ)ⁿ = aᵐⁿ

To multiply, add the powers; to divide, subtract them; for a power of a power, multiply them.

Study notes for index laws

x = (−b ± √(b² − 4ac)) ÷ 2a

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

Study notes for quadratic formula

nth term = dn + (a − d)

The common difference times n, plus the first term minus the common difference.

Study notes for nth term of a linear sequence

m = (y₂ − y₁) ÷ (x₂ − x₁)

gradient = change in y ÷ change in x

Study notes for gradient of a straight line

y = mx + c

y = gradient × x + y-intercept

Study notes for equation of a straight line

m₁ × m₂ = −1

The gradients of perpendicular lines multiply to give −1, so m₂ = −1/m₁.

Study notes for perpendicular gradients

x² + y² = r²

A circle with its centre at the origin, (0, 0).

Study notes for equation of a circle

percentage change = (change ÷ original) × 100

Divide the increase or decrease by the original amount, then multiply by 100.

Study notes for percentage change

I = P × r/100 × n

interest = original amount × rate × number of years

Study notes for simple interest

total accrued = P(1 + r/100)ⁿ

The principal times the multiplier, to the power of the number of times it's compounded.

Study notes for compound interest

s = d ÷ t

speed = distance ÷ time

Study notes for speed

D = m ÷ V

density = mass ÷ volume

Study notes for density

P = F ÷ A

pressure = force ÷ area

Study notes for pressure

a + b + c = 180°

The angles in a triangle add up to 180°.

Study notes for angles in a triangle

a + b + c + d = 360°

The angles in a quadrilateral add up to 360°.

Study notes for angles in a quadrilateral

sum of interior angles = (n − 2) × 180°

A polygon with n sides splits into n − 2 triangles.

Study notes for sum of interior angles

sum of exterior angles = 360°

True for any polygon. Each exterior angle of a regular polygon is 360° ÷ n.

Study notes for sum of exterior angles

A = lw

area = length × width

Study notes for area of a rectangle

A = ½bh

area = ½ × base × perpendicular height

Study notes for area of a triangle

A = bh

area = base × perpendicular height

Study notes for area of a parallelogram

A = ½(a + b)h

area = ½ × (sum of the parallel sides) × perpendicular height

Study notes for area of a trapezium

C = 2πr = πd

circumference = 2 × π × radius = π × diameter

Study notes for circumference of a circle

A = πr²

area = π × radius²

Study notes for area of a circle

V = lwh

volume = length × width × height

Study notes for volume of a cuboid

V = Al

volume = area of cross-section × length

Study notes for volume of a prism

A = 2πr² + 2πrh

Two circular ends, plus the curved surface (a rectangle 2πr long and h high).

Study notes for surface area of a cylinder

a² + b² = c²

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Study notes for pythagoras' theorem

sin A = a/c · cos A = b/c · tan A = a/b

sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent

Study notes for trigonometric ratios (soh cah toa)

sin 0° = 0, sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1. cos runs the other way: cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½, cos 90° = 0. tan 0° = 0, tan 30° = √3/3, tan 45° = 1, tan 60° = √3; tan 90° has no value.

Study notes for exact trig values

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

In any triangle, each side divided by the sine of the opposite angle gives the same value.

Study notes for sine rule

a² = b² + c² − 2bc cos A

Finds a side from the other two sides and the angle between them.

Study notes for cosine rule

cos A = (b² + c² − a²) ÷ 2bc

The cosine rule rearranged to find an angle from three sides. It isn't on the formulae sheet.

Study notes for cosine rule for an angle

area = ½ab sin C

Half the product of two sides, times the sine of the angle between them.

Study notes for area of a triangle using sine

P(not A) = 1 − P(A)

The probabilities of an event happening and not happening add up to 1.

Study notes for probability of an event not happening

P(A or B) = P(A) + P(B) − P(A and B)

Add the probabilities, then take away the overlap so it isn't counted twice.

Study notes for probability of a or b

P(A and B) = P(A given B) × P(B)

The probability of B, times the probability of A once B has happened.

Study notes for probability of a and b

mean = Σfx ÷ Σf

The total of (value × frequency), divided by the total frequency.

Study notes for mean from a frequency table

frequency density = frequency ÷ class width

The height of a histogram bar. The area of the bar is the frequency.

Study notes for frequency density