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Formulas

IGCSE Maths Complete Formula Sheet

Every formula you need for Cambridge IGCSE Mathematics 0580, organised by topic for quick reference.

Trigonometry

Sine ratio

Key formula for trigonometry in IGCSE Maths.

sin⁡θ=OppositeHypotenuse\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}

Cosine ratio

Key formula for trigonometry in IGCSE Maths.

cos⁡θ=AdjacentHypotenuse\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}

Tangent ratio

Key formula for trigonometry in IGCSE Maths.

tan⁡θ=OppositeAdjacent\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}

Sine rule

Key formula for trigonometry in IGCSE Maths.

asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

Cosine rule

Key formula for trigonometry in IGCSE Maths.

a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A

Area = ½ab·sinC

Key formula for trigonometry in IGCSE Maths.

Area=12absin⁡C\text{Area} = \tfrac{1}{2}ab\sin C

Algebra and Graphs

Quadratic formula

Key formula for algebra in IGCSE Maths.

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

Square of sum

Key formula for algebra in IGCSE Maths.

(a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2

Difference of squares

Key formula for algebra in IGCSE Maths.

a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)

Completing the square

Key formula for algebra in IGCSE Maths.

x2+bx=(x+b2)2−(b2)2x^2 + bx = \left(x + \dfrac{b}{2}\right)^2 - \left(\dfrac{b}{2}\right)^2

Change of subject

Key formula for algebra in IGCSE Maths.

Rearrange to isolate the variable\text{Rearrange to isolate the variable}

Transformations and Vectors

Magnitude

Key formula for vectors in IGCSE Maths.

∣v∣=x2+y2|\mathbf{v}| = \sqrt{x^2 + y^2}

Addition

Key formula for vectors in IGCSE Maths.

(a1,a2)+(b1,b2)=(a1+b1,  a2+b2)(a_1, a_2) + (b_1, b_2) = (a_1+b_1,\; a_2+b_2)

Scalar multiplication

Key formula for vectors in IGCSE Maths.

k(a,b)=(ka,  kb)k(a, b) = (ka,\; kb)

Parallel vectors

Key formula for vectors in IGCSE Maths.

a=kb for some scalar k\mathbf{a} = k\mathbf{b} \text{ for some scalar } k

Position vector

Key formula for vectors in IGCSE Maths.

OA→ represents point A from origin\overrightarrow{OA} \text{ represents point } A \text{ from origin}

Algebra and Graphs

Composite

Key formula for functions in IGCSE Maths.

fg(x)=f ⁣(g(x))fg(x) = f\!\big(g(x)\big)

Inverse

Key formula for functions in IGCSE Maths.

Swap x and y, then solve for y\text{Swap } x \text{ and } y\text{, then solve for } y

Domain

Key formula for functions in IGCSE Maths.

Set of valid input values\text{Set of valid input values}

Range

Key formula for functions in IGCSE Maths.

Set of possible output values\text{Set of possible output values}

Statistics

Mean = Σx / n

Key formula for statistics & probability in IGCSE Maths.

Mean=Σxn\text{Mean} = \dfrac{\Sigma x}{n}

Estimated mean = Σfx / Σf

Key formula for statistics & probability in IGCSE Maths.

Estimated mean=ΣfxΣf\text{Estimated mean} = \dfrac{\Sigma fx}{\Sigma f}

P(A or B) = P(A) + P(B) − P(A∩B)

Key formula for statistics & probability in IGCSE Maths.

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Independent events

Key formula for statistics & probability in IGCSE Maths.

P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)

Interquartile range = Q3 − Q1

Key formula for statistics & probability in IGCSE Maths.

IQR=Q3−Q1\text{IQR} = Q_3 - Q_1

Geometry

Triangle angles sum to 180°

Key formula for geometry in IGCSE Maths.

Triangle angles sum to 180°\text{Triangle angles sum to } 180°

Quadrilateral angles sum to 360°

Key formula for geometry in IGCSE Maths.

Quadrilateral angles sum to 360°\text{Quadrilateral angles sum to } 360°

Polygon interior angles = (n−2) × 180°

Key formula for geometry in IGCSE Maths.

Interior angles=(n−2)×180°\text{Interior angles} = (n-2) \times 180°

Exterior angle = 360° / n

Key formula for geometry in IGCSE Maths.

Exterior angle=360°n\text{Exterior angle} = \dfrac{360°}{n}

Similar triangles

Key formula for geometry in IGCSE Maths.

Corresponding sides in the same ratio\text{Corresponding sides in the same ratio}

Number

Percentage change = (change / original) × 100

Key formula for number in IGCSE Maths.

Percentage change=changeoriginal×100\text{Percentage change} = \dfrac{\text{change}}{\text{original}} \times 100

Compound interest

Key formula for number in IGCSE Maths.

A=P ⁣(1+r100)nA = P\!\left(1 + \dfrac{r}{100}\right)^n

Standard form

Key formula for number in IGCSE Maths.

a×10n where 1≤a<10a \times 10^n \text{ where } 1 \leq a < 10

Upper bound = value + ½ unit

Key formula for number in IGCSE Maths.

Upper bound=value+12  unit\text{Upper bound} = \text{value} + \tfrac{1}{2}\;\text{unit}

HCF and LCM by prime factorisation

Key formula for number in IGCSE Maths.

HCF and LCM by prime factorisation\text{HCF and LCM by prime factorisation}

Coordinate Geometry

Gradient

Key formula for coordinate geometry in IGCSE Maths.

m=y2−y1x2−x1m = \dfrac{y_2 - y_1}{x_2 - x_1}

Midpoint

Key formula for coordinate geometry in IGCSE Maths.

(x1+x22,  y1+y22)\left(\dfrac{x_1+x_2}{2},\; \dfrac{y_1+y_2}{2}\right)

Distance

Key formula for coordinate geometry in IGCSE Maths.

(x2−x1)2+(y2−y1)2\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Equation of line

Key formula for coordinate geometry in IGCSE Maths.

y=mx+cy = mx + c

Perpendicular gradients

Key formula for coordinate geometry in IGCSE Maths.

m1×m2=−1m_1 \times m_2 = -1

Algebra and Graphs

Linear nth term

Key formula for sequences in IGCSE Maths.

a+(n−1)da + (n-1)d

Quadratic nth term

Key formula for sequences in IGCSE Maths.

an2+bn+can^2 + bn + c

Geometric nth term

Key formula for sequences in IGCSE Maths.

arn−1ar^{n-1}

Arithmetic sum

Key formula for sequences in IGCSE Maths.

Sn=n2(first+last)S_n = \dfrac{n}{2}(\text{first} + \text{last})

Algebra and Graphs

Solve like equations but flip sign when ×/÷ by negative

Key formula for inequalities in IGCSE Maths.

Solve like equations, but flip sign when ×/÷ by negative\text{Solve like equations, but flip sign when } \times / \div \text{ by negative}

Quadratic

Key formula for inequalities in IGCSE Maths.

Factorise, test regions on number line\text{Factorise, test regions on number line}

Graph

Key formula for inequalities in IGCSE Maths.

Shade the region that satisfies the inequality\text{Shade the region that satisfies the inequality}

Integer solutions

Key formula for inequalities in IGCSE Maths.

List whole numbers in range\text{List whole numbers in range}

Transformations and Vectors

Rotation

Key formula for transformations in IGCSE Maths.

Centre, angle, direction\text{Centre, angle, direction}

Reflection

Key formula for transformations in IGCSE Maths.

Equation of mirror line\text{Equation of mirror line}

Translation

Key formula for transformations in IGCSE Maths.

Column vector (xy)\text{Column vector } \begin{pmatrix} x \\ y \end{pmatrix}

Enlargement

Key formula for transformations in IGCSE Maths.

Centre and scale factor\text{Centre and scale factor}

Combined

Key formula for transformations in IGCSE Maths.

Apply in order given\text{Apply in order given}

Mensuration

Circle

Key formula for mensuration in IGCSE Maths.

A=πr2,C=2πrA = \pi r^2, \quad C = 2\pi r

Sphere

Key formula for mensuration in IGCSE Maths.

V=43πr3,SA=4πr2V = \dfrac{4}{3}\pi r^3, \quad SA = 4\pi r^2

Cone

Key formula for mensuration in IGCSE Maths.

V=13πr2h,Curved SA=πrlV = \dfrac{1}{3}\pi r^2 h, \quad \text{Curved SA} = \pi r l

Cylinder

Key formula for mensuration in IGCSE Maths.

V=πr2h,SA=2πrh+2πr2V = \pi r^2 h, \quad SA = 2\pi r h + 2\pi r^2

Arc length = (θ/360) × 2πr

Key formula for mensuration in IGCSE Maths.

Arc length=θ360×2πr\text{Arc length} = \dfrac{\theta}{360} \times 2\pi r

Sector area = (θ/360) × πr²

Key formula for mensuration in IGCSE Maths.

Sector area=θ360×πr2\text{Sector area} = \dfrac{\theta}{360} \times \pi r^2

Number

n(A∪B) = n(A) + n(B) − n(A∩B)

Key formula for set theory in IGCSE Maths.

n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)

Complement

Key formula for set theory in IGCSE Maths.

A′=elements not in AA' = \text{elements not in } A

De Morgan

Key formula for set theory in IGCSE Maths.

(A∪B)′=A′∩B′(A \cup B)' = A' \cap B'

Venn diagram

Key formula for set theory in IGCSE Maths.

Overlapping circles show relationships\text{Overlapping circles show relationships}

Algebra and Graphs

Power rule

Key formula for differentiation in IGCSE Maths.

ddx(axn)=naxn−1\dfrac{d}{dx}(ax^n) = nax^{n-1}

Gradient at point

Key formula for differentiation in IGCSE Maths.

Substitute x into dydx\text{Substitute } x \text{ into } \dfrac{dy}{dx}

Stationary point

Key formula for differentiation in IGCSE Maths.

dydx=0\dfrac{dy}{dx} = 0

Nature test

Key formula for differentiation in IGCSE Maths.

d2ydx2>0 (minimum),<0 (maximum)\dfrac{d^2y}{dx^2} > 0 \text{ (minimum)}, \quad < 0 \text{ (maximum)}

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Frequently Asked Questions

Which formulas are given in the IGCSE exam?

You are given: quadratic formula, sine rule, cosine rule, area of triangle formula, and volumes of cone, sphere, and pyramid.

Should I memorise all these formulas?

Focus on the ones not provided in the exam. SOH-CAH-TOA, circle formulas, gradient, and basic algebra identities must be memorised.

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