# 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} = *a*^{2} + 2*ab* + *b*^{2}

(*a* – *b*)^{2} = *a*^{2} – 2*ab* + *b*^{2}

(*a* + *b*)(*a* – *b*) = *a*^{2} + *b*^{2}

**Quadratic Formula**

*x* = ( – *b ± √ (**b*^{2} – 4*ac ) )/ 2*a

## 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: *T*_{n} = *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: *T _{n}* =

*ar*

^{n–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: *T _{n}* = 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: *T _{n}* =

*n*

^{2 }, where

*n*is the ordinal numerical value of the term.

**Sequence of Triangular Numbers**

Number Pattern Formula for Triangular Numbers: *T _{n}* =

*n(n*+ 1

*)*/2, where

*n*is the ordinal numerical value of the term.

**Fibonacci Sequence**

Number Pattern Formula for Fibonacci Sequence: *T*_{n} = *T*_{n–1} + *T*_{n–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?

**Download our A-Math Formula sheet below**

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 × r^{2 }

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 = π*r ^{2}h*

Total surface area of a solid cylinder = area of curved surface + 2 × area of base = 2π*rh + 2*π*r ^{2}*

**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* = s^{2}

Surface area = 1/2 ×* s* × *l* × 4 *sides*

= 2*sl*

Surface area of pyramid = base area + total area of slant Δ faces

= s^{2} + 2*sl*

**Cone**

A cone is a solid with a circular base and a vertex.

volume = (1/3)(π*r ^{2}h) *

*Arc length = circumference of circular base *

Curved surface area = π*rl*

Surface area of a cone = Area of circle + Area of curved surface

= π*r ^{2 }+ *π

*rl*

*r* = radius

*l* = length of the slant height = √ (*r*^{2} + *h*^{2})

**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)(π*r ^{3}) *

Surface area of a sphere = 4π*r ^{2}*

Volume = 1/2 × volume of sphere = (2/3)(π*r ^{3}) *

Surface area of a sphere = 1/2 × spherical surface area + area of circle

= 2π*r ^{2 }+ *π

*r*3π

^{2}=*r*

^{2}

**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 *

* 18*0° *= *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° × 2π*r*

If θ is measured in radian then arc length = θ/2π × 2π*r = r*θ

**Area of Sector**

Area of sector/Area of circle = Central angle,θ/360°

Area of circle = π*r ^{2}*

If θ is measured in degree then arc length = θ/ 360° × π*r ^{2}*

If θ is measured in radian then arc length = θ/2π ×π*r ^{2}*

* = (1/2)r ^{2}*θ

**Area of Segment**

Area of segment = Area of Sector – Area of Triangle

= θ°/ 360° × π*r ^{2} – (1/2)r^{2}sin*θ

= (1/2)*r ^{2}*θ

*–*

*(1/2)r*θ (

^{2}sin*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

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

**Download our A-Math Formula sheet below**

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

**Calculation of Gradient**

Gradient = (*y*_{2} – *y*_{1})/(*x*_{2} – *x*_{1}) = 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=√( (*y*_{2} – *y*_{1})^{2} + (*x*_{2} – *x*_{1})^{2}) units

**Coordinate Geometry Formula: Equation of Line**

For a line with gradient m and passing through the point (x_{1}, y_{1}), the equation of the line is given by: y – y_{1 }= *m*(x – x_{1}).

**You may still use y = mx + c and substitute (x_{1}, y_{1}) 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)*ab*sin *C*

= (1/2)*ac*sin B

= (1/2)*bc*sin *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,

*a*^{2} = *b*^{2} +* c*^{2} – 2*bc* cos *A * cos *A* is an included angle

*b*^{2} = *a*^{2} +* c*^{2} – 2a*c* cos *B * cos B is an included angle

*c*^{2} = *a*^{2} +* b*^{2} – 2a*b* cos *C * cos C is an included angle

or

cos *A* = (*b*^{2} +* c*^{2} – *a*^{2})/2*bc*

cos *B* = (*a*^{2} +* c*^{2} – *b*^{2})/2a*c*

cos *C* = (*a*^{2} +* b*^{2} – *c*^{2})/2a*b*

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 {*x*_{1}, *x*_{2}, *x*_{3}, …., *x*_{n}} , 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:

## Last Minute Revision for O Level Math?

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

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