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Every key formula organised by topic. Exam Sheet formulas are provided in the official GCE O-Level exam.

Official GCE O-Level Exam Formula Sheet

View the exact sheet provided in the E Math (4048) and A Math (4049) papers

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Total Amount
Exam Sheet
A=P(1+r100)nA = P\left(1 + \dfrac{r}{100}\right)^n

P = principal, r = rate per period (%), n = number of periods

Simple Interest
I=P×r×t100I = \dfrac{P \times r \times t}{100}

P = principal, r = rate per year (%), t = time in years

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

For ax² + bx + c = 0

Difference of Two Squares
a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
Perfect Square (sum)
(a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2
Perfect Square (difference)
(a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2
Laws of Indices
am×an=am+n,am÷an=am−n,(am)n=amna^m \times a^n = a^{m+n}, \quad a^m \div a^n = a^{m-n}, \quad (a^m)^n = a^{mn}
Negative Index
a−n=1ana^{-n} = \dfrac{1}{a^n}
Fractional Index
a1n=an,amn=amna^{\frac{1}{n}} = \sqrt[n]{a}, \quad a^{\frac{m}{n}} = \sqrt[n]{a^m}
Zero and One Index
a0=1,a1=aa^0 = 1, \quad a^1 = a
Angle Sum in Triangle
A+B+C=180∘A + B + C = 180^\circ
Sum of Interior Angles of Polygon
S=(n−2)×180∘S = (n-2) \times 180^\circ

n = number of sides

Each Interior Angle (regular polygon)
θ=(n−2)×180∘n\theta = \dfrac{(n-2) \times 180^\circ}{n}
Sum of Exterior Angles
∑exterior angles=360∘\sum \text{exterior angles} = 360^\circ
Area of Triangle
A=12bhA = \dfrac{1}{2}bh

b = base, h = perpendicular height

Area of Triangle (SAS)
Exam Sheet
A=12absin⁡CA = \dfrac{1}{2}ab\sin C

a, b = two sides, C = included angle

Area of Parallelogram
A=bhA = bh
Area of Trapezium
A=12(a+b)hA = \dfrac{1}{2}(a+b)h

a, b = parallel sides, h = height

Circumference of Circle
C=2πr=πdC = 2\pi r = \pi d
Area of Circle
A=πr2A = \pi r^2
Arc Length
Exam Sheet
l=rθl = r\theta

θ in radians

Sector Area
Exam Sheet
A=12r2θA = \dfrac{1}{2}r^2\theta

θ in radians

Area of Segment
A=12r2(θ−sin⁡θ)A = \dfrac{1}{2}r^2(\theta - \sin\theta)

θ in radians

Curved Surface Area of Cone
Exam Sheet
A=πrlA = \pi r l

l = slant height

Total Surface Area of Cone
A=πrl+πr2A = \pi r l + \pi r^2
Surface Area of Sphere
Exam Sheet
A=4πr2A = 4\pi r^2
Volume of Prism / Cylinder
V=Abase×hV = A_{\text{base}} \times h
Volume of Cone
Exam Sheet
V=13πr2hV = \dfrac{1}{3}\pi r^2 h
Volume of Sphere
Exam Sheet
V=43πr3V = \dfrac{4}{3}\pi r^3
Volume of Pyramid
V=13×Abase×hV = \dfrac{1}{3} \times A_{\text{base}} \times h
Gradient
m=y2−y1x2−x1m = \dfrac{y_2 - y_1}{x_2 - x_1}
Length of Line Segment
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
Equation of Straight Line
y=mx+cy = mx + c
Perpendicular Lines
m1×m2=−1m_1 \times m_2 = -1

Two lines are perpendicular if the product of their gradients is −1

Parallel Lines
m1=m2m_1 = m_2

Parallel lines have equal gradients

Basic Trig Ratios (SOH-CAH-TOA)
sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \dfrac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \dfrac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \dfrac{\text{opp}}{\text{adj}}
Pythagoras' Theorem
a2+b2=c2a^2 + b^2 = c^2

c = hypotenuse

Exact Values
sin⁡30∘=12, cos⁡30∘=32, tan⁡30∘=13\sin 30^\circ = \tfrac{1}{2},\ \cos 30^\circ = \tfrac{\sqrt{3}}{2},\ \tan 30^\circ = \tfrac{1}{\sqrt{3}}
Exact Values (45°)
sin⁡45∘=12, cos⁡45∘=12, tan⁡45∘=1\sin 45^\circ = \tfrac{1}{\sqrt{2}},\ \cos 45^\circ = \tfrac{1}{\sqrt{2}},\ \tan 45^\circ = 1
Exact Values (60°)
sin⁡60∘=32, cos⁡60∘=12, tan⁡60∘=3\sin 60^\circ = \tfrac{\sqrt{3}}{2},\ \cos 60^\circ = \tfrac{1}{2},\ \tan 60^\circ = \sqrt{3}
Sine Rule
Exam Sheet
asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}
Cosine Rule
Exam Sheet
a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A
Radian Conversion
θrad=θdeg×π180\theta_{\text{rad}} = \theta_{\text{deg}} \times \dfrac{\pi}{180}
Magnitude of Vector
∣a∣=∣(xy)∣=x2+y2|\mathbf{a}| = \left|\begin{pmatrix}x\\y\end{pmatrix}\right| = \sqrt{x^2 + y^2}
Unit Vector
a^=a∣a∣\hat{\mathbf{a}} = \dfrac{\mathbf{a}}{|\mathbf{a}|}
Vector Addition
a+b=(a1+b1a2+b2)\mathbf{a} + \mathbf{b} = \begin{pmatrix}a_1 + b_1\\ a_2 + b_2\end{pmatrix}
Mean (ungrouped)
xˉ=∑xn\bar{x} = \dfrac{\sum x}{n}
Mean (grouped data)
Exam Sheet
xˉ=∑fx∑f\bar{x} = \dfrac{\sum fx}{\sum f}
Standard Deviation (ungrouped)
σ=∑x2n−xˉ2\sigma = \sqrt{\dfrac{\sum x^2}{n} - \bar{x}^2}
Standard Deviation (grouped)
Exam Sheet
σ=∑fx2∑f−(∑fx∑f)2\sigma = \sqrt{\dfrac{\sum fx^2}{\sum f} - \left(\dfrac{\sum fx}{\sum f}\right)^2}
Interquartile Range
IQR=Q3−Q1\text{IQR} = Q_3 - Q_1
Basic Probability
P(A)=number of favourable outcomestotal number of outcomesP(A) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}
Complement Rule
P(A′)=1−P(A)P(A') = 1 - P(A)
Addition Rule
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
Mutually Exclusive Events
P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)

When A and B cannot both occur

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

When A and B do not affect each other