# Your Complete O-Level Math Formula Sheet

In this O-level Math formula sheet, you will learn all the formulas needed for O-level math exams.

Keep this formula sheet with you as a useful companion when solving O-level Math questions.

All the formulas are sorted according to the following chapters so you can find them easily when required.

• Algebra
• Percentage
• Rate and Speed
• Angles and Polygons
• Number Patterns
• Measurement
• Direct and Inverse Proportion
• Financial Mathematics
• Pythagoras Theorem
• Coordinate Geometry
• Trigonometry
• Probability
• Statistics

Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
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## Algebra

Algebraic Formulas

(a + b)2 = a2 + 2abb2

(a – b)2 = a2 – 2abb2

(a + b)(a – b) = a2 + b2

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

## Do you find this O-level Math formula sheet useful?

You will find the complete list of formulas needed for O-Level Additional Math exams. Click the link below:

## 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 ×

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

= πr+ πrl

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

=  2πr+ πr2 = r2

π radians = 360° /2 =180°

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)

## Do you find this O-level Math formula sheet useful?

You will find the complete list of formulas needed for O-Level Additional Math exams. Click the link below:

## O-Level Math Formula Sheet: Coordinate Geometry

Gradient = (y2y1)/(x2x1) = Change in y/Change in x

Equation of Straight Line

y = mx + c

c = y-intercept (The point when the graph cuts the y-axis)

Coordinate Geometry Formula: Length of Line Segment

Length of AB=√( (y2y1)2 + (x2x1)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                    cos A is an included angle

b2 = a2 + c2 – 2ac cos                    cos B is an included angle

c2 = a2 + b2 – 2ab cos                    cos C is an included angle

or

cos A = (b2 + c2a2)/2bc

cos B = (a2 + c2b2)/2ac

cos C = (a2 + b2c2)/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.

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:

## Check out our exam guide on other topics here!

Secondary Math Revision Notes

Before you go, you might want to download this entire revision notes in PDF format to print it out, or to read it later.

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