Resources

IGCSE Mathematics (0580) formula sheet

Number & powers

Idea	Formula
Multiplying powers	aᵐ × aⁿ = aᵐ⁺ⁿ
Dividing powers	aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of a power	(aᵐ)ⁿ = aᵐⁿ
Zero index	a⁰ = 1
Negative index	a⁻ⁿ = 1 ÷ aⁿ
Fractional index	a^(m/n) = (ⁿ√a)ᵐ
Standard form	A × 10ⁿ, where 1 ≤ A < 10 and n is an integer
Speed	speed = distance ÷ time (distance = speed × time; time = distance ÷ speed)
Bounds	a value rounded to a given accuracy lies half a unit either side: lower bound ≤ value < upper bound

Percentages, interest & growth

Idea	Formula
Percentage change	(change ÷ original) × 100%
Percentage increase / decrease multiplier	(1 + r ÷ 100) for an increase; (1 − r ÷ 100) for a decrease
Reverse percentage	original = new value ÷ multiplier
Simple interest	I = (P × R × T) ÷ 100
Compound interest	final value = P × (1 + R ÷ 100)ⁿ; interest = final value − P
Exponential growth and decay	y = a × (1 ± r ÷ 100)ᵗ, with + for growth and − for decay
Multiplying surds	√(ab) = √a × √b
Dividing surds	√(a ÷ b) = √a ÷ √b
Rationalising a denominator	a ÷ √b = a√b ÷ b

Algebra

Idea	Formula
Difference of two squares	a² − b² = (a + b)(a − b)
Quadratic formula	x = (−b ± √(b² − 4ac)) ÷ 2a, for ax² + bx + c = 0
Completing the square	ax² + bx + c = a(x + b ÷ 2a)² + (c − b² ÷ 4a)
Turning point of a(x + p)² + q	(−p, q), with axis of symmetry x = −p
Linear sequence	Tₙ = dn + (a − d), where a is the first term and d the common difference
Quadratic sequence	Tₙ = an² + bn + c, where the second difference equals 2a
Direct proportion	y = kx, y = kx², y = k√x
Inverse proportion	y = k ÷ x, y = k ÷ x²

Graphs, calculus & functions

Idea	Formula
Differentiation	d/dx (axⁿ) = anxⁿ⁻¹
Stationary point	dy/dx = 0
Composite function	fg(x) = f(g(x)): apply g first, then f
Inverse function	if y = f(x) then f⁻¹(y) = x

Coordinate geometry

Idea	Formula
Equation of a straight line	y = mx + c
Gradient between two points	m = (y₂ − y₁) ÷ (x₂ − x₁)
Length of a line segment	d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint	((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Parallel lines	m₁ = m₂
Perpendicular lines	m₁ × m₂ = −1

Geometry

Idea	Formula
Sum of interior angles of an n-sided polygon	180° × (n − 2)
Interior angle of a regular polygon	180° × (n − 2) ÷ n
Exterior angle of a regular polygon	360° ÷ n
Scale factor	k = new length ÷ original length
Area of similar shapes	area ratio = k²
Volume of similar solids	volume ratio = k³

Mensuration

Shape	Formula
Rectangle	area = length × width
Parallelogram	area = base × perpendicular height
Triangle (given)	area = ½ × base × height
Trapezium	area = ½(a + b)h, where a and b are the parallel sides
Circle	circumference = 2πr; area = πr²
Arc length	θ ÷ 360 × 2πr
Sector area	θ ÷ 360 × πr²
Prism (given)	volume = cross-section area × length
Cylinder (given)	volume = πr²h; curved surface area = 2πrh
Pyramid (given)	volume = ⅓ × base area × height
Cone (given)	volume = ⅓πr²h; curved surface area = πrl, where l² = r² + h²
Sphere (given)	volume = 4⁄3 πr³; surface area = 4πr²
Hemisphere	volume = 2⁄3 πr³; curved surface area = 2πr²

Trigonometry

Idea	Formula
Pythagoras	a² + b² = c², where c is the hypotenuse
Sine, cosine, tangent	sin θ = opp ÷ hyp; cos θ = adj ÷ hyp; tan θ = opp ÷ adj
Sine rule (given)	a ÷ sin A = b ÷ sin B = c ÷ sin C
Cosine rule (given)	a² = b² + c² − 2bc cos A
Area of a triangle (given)	area = ½ab sin C
Second solution of sin x = k	180° − x
Second solution of cos x = k	360° − x

Transformations & vectors

Idea	Formula
Reflection in the x-axis	(x, y) → (x, −y)
Reflection in the y-axis	(x, y) → (−x, y)
Reflection in y = x	(x, y) → (y, x)
Reflection in y = −x	(x, y) → (−y, −x)
Enlargement	OP′ = k × OP, from the centre of enlargement O
Magnitude of a vector	\|a\| = √(x² + y²), for a = (x, y)
Vector from A to B	AB = b − a
Midpoint of AB	OM = ½(a + b)

Probability

Idea	Formula
Theoretical probability	P(A) = favourable outcomes ÷ total equally likely outcomes
Complement	P(A′) = 1 − P(A)
Relative frequency	times the event occurs ÷ total number of trials
Expected frequency	P(event) × number of trials
Mutually exclusive events	P(A ∪ B) = P(A) + P(B)
Independent events	P(A ∩ B) = P(A) × P(B)

Statistics

Idea	Formula
Mean	sum of values ÷ number of values
Range	highest value − lowest value
Estimated mean of grouped data	Σfx ÷ Σf, where x is the class midpoint
Interquartile range	IQR = Q₃ − Q₁
Pie chart sector angle	(frequency ÷ total) × 360°
Frequency density	frequency ÷ class width, so frequency = frequency density × class width

Print it, pin it, get the PDF

Get the Mathematics formula sheet PDF Get the Mathematics revision notes PDF