Algebra
Algebraic Formulas
(a + b)2 = a2 + 2ab +b2
(a –b)2 = a2 – 2ab +b2
(a + b)(a –b) = a2+ b2
Quadratic Formula
x = ( – b ± √ (b2 – 4ac ) )/ 2a
Percentage
Percentage Discount = Discount/ Marked Price × 100%
Percentage Increase and Decrease
Percentage Increase = Increase/ Original × 100%
Percentage Decrease = Decrease/ Original × 100%
Rate and Speed
Speed
Distance = Speed × Time
Time = Distance/ Speed
Speed = Distance/ Time
Average Speed is Total Distance ÷ Total Time
Angles and Polygons
Sum of interior angles = (n – 2) × 180°, where n is the number of sides of the polygon.
Sum of the exterior angles = 360°, regardless of the number of sides of the polygon.
For the complete list of angle properties, click here >> Angle Properties – O Level Exam Preparation Guide
Number Patttern
Arithmetic Sequences
Number Pattern Formula for Arithmetic Sequences: Tn = a + (n – 1)d,
where n is the ordinal numerical value of the term, a is the first term and d is the common difference between any two consecutive terms.
Geometric Sequences
Number Pattern Formula for Geometric Sequence: Tn = arn–1 where n is the ordinal numerical value of the term, a is the first term and r is the common ratio between any two consecutive terms.
Harmonic Sequences
Number Pattern Formula for Harmonic Sequences: Tn = 1/ ( a + (n – 1)d ) where n is the ordinal numerical value of the term, a is the denominator of the first term, and d is the common difference between the denominators of any two consecutive terms.
Sequence of Square Numbers
Number Pattern Formula for Square Numbers: Tn = n2 , where n is the ordinal numerical value of the term.
Sequence of Triangular Numbers
Number Pattern Formula for Triangular Numbers: Tn = n(n + 1)/2,where n is the ordinal numerical value of the term.
Fibonacci Sequence
Number Pattern Formula for Fibonacci Sequence: Tn = Tn–1 + Tn–2 , where n is the ordinal numerical value of the term.
For the complete explanation of each of these number pattern formulas, click here >> How to Derive a Number Pattern Formula
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O-Level Math Formula Sheet: Measurement
Area and circumference of a circle
Area of circle = π × r2
Circumference of circle = 2 πr
Area of a parallelogram = base × height
Area of a parallelogram = c × h
Area of a trapezium = 1/2 × (sum of parallel sides) × height
Area of a trapezium = 1/2 × (a + b) × h
Volume and Surface Area of Cylinder
Volume of cylinder = πr2h
Total surface area of a solid cylinder = area of curved surface + 2 × area of base = 2πrh + 2πr2
Pyramid
A pyramid is a solid that has a base with a perpendicular vertex and slant lateral faces.
The base can be a triangle, a square or a rectangle
Volume = 1/3 × base area × h
Base area = s2
Surface area = 1/2 × s × l × 4 sides
= 2sl
Surface area of pyramid = base area + total area of slant Δ faces
= s2 + 2sl
Cone
A cone is a solid with a circular base and a vertex.
volume = (1/3)(πr2h)
Arc length = circumference of circular base
Curved surface area = πrl
Surface area of a cone = Area of circle + Area of curved surface
= πr2+ πrl
r = radius
l = length of the slant height = √ (r2 + h2)
Sphere and Hemisphere (half-sphere)
Every point on the surface of a sphere is equidistance from the centre.
A hemisphere is half a sphere.
Volume = (4/3)(πr3)
Surface area of a sphere = 4πr2
Volume = 1/2 × volume of sphere = (2/3)(πr3)
Surface area of a sphere = 1/2 × spherical surface area + area of circle
= 2πr2+ πr2 = 3πr2
Radian Measure
Conversion from radians to degrees
2π radians = 360°
π radians = 360° /2 =180°
1 radians =180°/π
Conversion from degree to radians
360° = 2π radians
180° = π radians
1° = π/180 radians
Conversion between radians and degrees
Parts of a circle
Arc length
If θ is measured in degree then arc length = θ/ 360° × 2πr
If θ is measured in radian then arc length = θ/2π × 2πr = rθ
Area of Sector
Area of sector/Area of circle = Central angle,θ/360°
Area of circle = πr2
If θ is measured in degree then arc length = θ/ 360° × πr2
If θ is measured in radian then arc length = θ/2π ×πr2
= (1/2)r2θ
Area of Segment
Area of segment = Area of Sector – Area of Triangle
= θ°/ 360° × πr2 – (1/2)r2sinθ
= (1/2)r2θ – (1/2)r2sinθ (if θ is in radian)
Important: If angles are in radian, change the calculator to radian mode.
Direct and Inverse Proportion
Direct Proportion
2 quantities x and y are said to be directly proportional to each other if x = ky , where k is a constant.
Inverse Proportion
2 quantities X and Y are said to be inversely proportional to each other if x = k/y, where k is a constant.
Financial Mathematics
Simple Interest
For a sum of money (Principal sum), P, deposited in a bank at R% interest per annum for T years, the simple interest (I) is given by:
I = PRT/100
Compound Interest
Amount = P (1 + R/100)n
P is the principal sum, R% is the interest rate and n is the number of times compounded.
Income Tax
Chargeable Income = Assessment Income – Personal Relief
Assessment Income = Annual Income – Donations
Hire Purchase
Total Interest = Loan × Flat Rate x Loan Period (in years)
Repayment Amount = Loan + Total Interest
Monthly Repayment (Instalment) = Repayment Amount ÷ Loan Period (in months)
Pythagoras Theorem
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You will find the complete list of formulas needed for O-Level Additional Math exams. Click the link below:
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O-Level Math Formula Sheet: Coordinate Geometry
Calculation of Gradient
Gradient = (y2 – y1)/(x2 – x1) = Change in y/Change in x
Equation of Straight Line
y = mx + c
m = gradient
c = y-intercept (The point when the graph cuts the y-axis)
Coordinate Geometry Formula: Length of Line Segment
Length of AB=√( (y2 – y1)2 + (x2 – x1)2) units
Coordinate Geometry Formula: Equation of Line
For a line with gradient m and passing through the point (x1, y1), the equation of the line is given by: y – y1 = m(x – x1).
**You may still use y = mx + c and substitute (x1, y1) into the equation to find the value of c. But may involve more steps.
Trigonometry
Trigonometric Ratios of Acute Angles
For right‒angled triangle,
1. Sine sin θ = opposite/hypotenuse SOH
2. Cosine cos θ = adjacent/hypotenuse CAH
3. Tangent tanθ = opposite/adjacent TOA
Sine and Cosine of Obtuse Angle
For any acute angle θ,
sin ( 180° – θ) = sin θ
cos ( 180° – θ) = – cos θ
Area of Triangle
For non‒right angled triangle with any 2 given sides and an included angle,
Area of ΔABC = (1/2)absin C
= (1/2)acsin B
= (1/2)bcsin A
Sine rule
For any triangle ABC,
a/sin A = b/sin B = c/sin C
where A, B and C are the interior angles
a, b and c are length of their opposite sides respectively.
Cosine rule
For any tiangle ABC,
a2 = b2 + c2 – 2bc cos A cos A is an included angle
b2 = a2 + c2 – 2ac cos B cos B is an included angle
c2 = a2 + b2 – 2ab cos C cos C is an included angle
or
cos A = (b2 + c2 – a2)/2bc
cos B = (a2 + c2 – b2)/2ac
cos C = (a2 + b2 – c2)/2ab
where A, B and C are the interior angles
a, b and c are length of their opposite sides respectively.
Probability
Probability is a measure of chance.
The probability of an event, A is:
P(A) = k/m
Where k is the number of outcomes of A while m is the total number of possible outcomes.
Mutually Exclusive Events and Addition Law
Two events are called mutually exclusive events if they cannot occur at the same time.
Eg. In tossing a coin, the event A of getting 1 head and the event of event B of getting 1 tail are exclusive events. They cannot happen at the same time.
Addition Law
If A and B are exclusive events, then the probability that either A or B occurring is given by
P(A or B) = P(A) + P(B)
If events A and B are not mutually exclusive then P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Independent Events and Multiplication Law
Two events are called independent events if the occurrence of one event does not affect the probability of occurrence of the other event.
Eg. In tossing a coin and a die, the event A that the coin is a head and the event B that the number on the die is even are independent events.
Multiplication Law
If A and B are independent events, then the probability that both A and B occurring is given by
P(A and B) = P(A) × P(B)
Statistics
Standard Deviation
To find the standard deviation of an ungrouped data set {x1, x2, x3, …., xn} , where n is the number of data in the set:
Standard Deviation for Grouped Data
To find the standard deviation of a grouped data set:
