O-Level E-Math formula sheet

The exam gives you a short formula list. Everything else you must know by heart. Both are below, each formula with a worked example.

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Given in the exam

Printed at the front of Paper 1 and Paper 2.

Compound interestTotal amount

= P\left(1 + \dfrac{r}{100}\right)^{n}

MensurationCurved surface area of a cone

= \pi r l

Surface area of a sphere

= 4\pi r^{2}

Volume of a cone

= \dfrac{1}{3}\pi r^{2} h

Volume of a sphere

= \dfrac{4}{3}\pi r^{3}

Area of triangle

ABC = \dfrac{1}{2}ab\sin C

Arc length

= r\theta

, where

\theta

is in radiansSector area

= \dfrac{1}{2}r^{2}\theta

, where

\theta

is in radians

Trigonometry

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

a^{2} = b^{2} + c^{2} - 2bc\cos A

StatisticsMean

= \dfrac{\sum fx}{\sum f}

Standard deviation

= \sqrt{\dfrac{\sum fx^{2}}{\sum f} - \left(\dfrac{\sum fx}{\sum f}\right)^{2}}

Formulas to memorise

Algebra

NameFormulaExample

Perfect square

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

(x + 3)^2 = x^2 + 6x + 9

Perfect square

(a - b)^2 = a^2 - 2ab + b^2

(x - 4)^2 = x^2 - 8x + 16

Difference of two squares

a^2 - b^2 = (a + b)(a - b)

x^2 - 9 = (x + 3)(x - 3)

Quadratic formula

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x^2 - 5x + 6 = 0

x = 2

x = 3

Laws of indices

NameFormulaExample

Multiplying

a^m \times a^n = a^{m+n}

2^3 \times 2^4 = 2^7

Dividing

a^m \div a^n = a^{m-n}

5^6 \div 5^2 = 5^4

Power of a power

(a^m)^n = a^{mn}

(3^2)^4 = 3^8

Power of a product

(ab)^n = a^n b^n

(2x)^3 = 8x^3

Power of a quotient

\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}

\left(\dfrac{2}{3}\right)^2 = \dfrac{4}{9}

Zero index

a^0 = 1

7^0 = 1

Negative index

a^{-n} = \dfrac{1}{a^n}

2^{-3} = \dfrac{1}{8}

Fractional index

a^{\frac{1}{n}} = \sqrt[n]{a}

27^{\frac{1}{3}} = 3

Fractional index

a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m

8^{\frac{2}{3}} = 4

Percentage and money

NameFormulaExample

Percentage change

\dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%

80 \to 92

\dfrac{12}{80} \times 100\% = 15\%

Simple interest

I = \dfrac{PRT}{100}

P = 2000

R = 3

T = 4

I = 240

Coordinate geometry

NameFormulaExample

Gradient

m = \dfrac{y_2 - y_1}{x_2 - x_1}

(1, 2)

and

(3, 8)

m = 3

Length of a line segment

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

(1, 2)

and

(4, 6)

: length

= 5

Equation of a straight line

y = mx + c

Gradient

2

y

-intercept

-3

y = 2x - 3

Parallel lines

m_1 = m_2

y = 2x + 1

is parallel to

y = 2x - 5

Mensuration (not given)

NameFormulaExample

Area of a triangle

\dfrac{1}{2} \times \text{base} \times \text{height}

Base

8

cm, height

5

cm:

20

cm²

Area of a parallelogram

\text{base} \times \text{height}

Base

9

cm, height

4

cm:

36

cm²

Area of a trapezium

\dfrac{1}{2}(a + b)h

a = 4

b = 6

h = 5

25

cm²

Area of a circle

\pi r^2

r = 7

cm:

154

cm² (3 s.f.)

Circumference of a circle

2\pi r

r = 7

cm:

44.0

cm (3 s.f.)

Volume of a prism

\text{area of cross-section} \times \text{length}

Cross-section

12

cm², length

10

cm:

120

cm³

Volume of a cylinder

\pi r^2 h

r = 3

cm,

h = 10

cm:

283

cm³ (3 s.f.)

Curved surface area of a cylinder

2\pi r h

r = 3

cm,

h = 10

cm:

188

cm² (3 s.f.)

Volume of a pyramid

\dfrac{1}{3} \times \text{base area} \times \text{height}

Base

36

cm², height

10

cm:

120

cm³

Radians

\pi \text{ rad} = 180^\circ

60^\circ = \dfrac{\pi}{3}

rad

Geometry and trigonometry

NameFormulaExample

Pythagoras' theorem

a^2 + b^2 = c^2

5

and

12

c = 13

Sine

\sin \theta = \dfrac{\text{opp}}{\text{hyp}}

opp

3

, hyp

5

\sin \theta = \dfrac{3}{5}

\theta = 36.9^\circ

Cosine

\cos \theta = \dfrac{\text{adj}}{\text{hyp}}

adj

4

, hyp

5

\cos \theta = \dfrac{4}{5}

\theta = 36.9^\circ

Tangent

\tan \theta = \dfrac{\text{opp}}{\text{adj}}

opp

3

, adj

4

\tan \theta = \dfrac{3}{4}

\theta = 36.9^\circ

Obtuse angles: sine

\sin(180^\circ - \theta) = \sin \theta

\sin 150^\circ = \sin 30^\circ = \dfrac{1}{2}

Obtuse angles: cosine

\cos(180^\circ - \theta) = -\cos \theta

\cos 120^\circ = -\cos 60^\circ = -\dfrac{1}{2}

Angles on a straight line

\text{sum} = 180^\circ

x + 130^\circ = 180^\circ

x = 50^\circ

Angles in a triangle

\text{sum} = 180^\circ

50^\circ

and

60^\circ

: third angle

70^\circ

Sum of interior angles

(n - 2) \times 180^\circ

Hexagon,

n = 6

720^\circ

Exterior angle of a regular polygon

\dfrac{360^\circ}{n}

n = 8

45^\circ

Similar figures: areas

\dfrac{A_1}{A_2} = \left(\dfrac{l_1}{l_2}\right)^2

Lengths

2 : 3

: areas

4 : 9

Similar solids: volumes

\dfrac{V_1}{V_2} = \left(\dfrac{l_1}{l_2}\right)^3

Lengths

2 : 3

: volumes

8 : 27

Rate and speed

NameFormulaExample

Speed

\text{speed} = \dfrac{\text{distance}}{\text{time}}

150

km in

2

75

km/h

Average speed

\dfrac{\text{total distance}}{\text{total time}}

60

km in

1

h, then

90

km in

2

50

km/h

Statistics, probability and sets

NameFormulaExample

Probability

P(A) = \dfrac{n(A)}{n(S)}

6

on a fair die:

\dfrac{1}{6}

Not happening

P(\text{not } A) = 1 - P(A)

Not a

6

1 - \dfrac{1}{6} = \dfrac{5}{6}

Interquartile range

Q_3 - Q_1

Q_3 = 70

Q_1 = 45

25

Sets

n(A \cup B) = n(A) + n(B) - n(A \cap B)

18 + 17 - 7 = 28

Vectors

NameFormulaExample

Magnitude

\left|\begin{pmatrix} x \\ y \end{pmatrix}\right| = \sqrt{x^2 + y^2}

\left|\begin{pmatrix} 3 \\ 4 \end{pmatrix}\right| = 5

Practise using them

Given in the exam

Printed at the front of Paper 1 and Paper 2.

Compound interestTotal amount

= P\left(1 + \dfrac{r}{100}\right)^{n}

MensurationCurved surface area of a cone

= \pi r l

Surface area of a sphere

= 4\pi r^{2}

Volume of a cone

= \dfrac{1}{3}\pi r^{2} h

Volume of a sphere

= \dfrac{4}{3}\pi r^{3}

Area of triangle

ABC = \dfrac{1}{2}ab\sin C

Arc length

= r\theta

, where

\theta

is in radiansSector area

= \dfrac{1}{2}r^{2}\theta

, where

\theta

is in radians

Trigonometry

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

a^{2} = b^{2} + c^{2} - 2bc\cos A

StatisticsMean

= \dfrac{\sum fx}{\sum f}

Standard deviation

= \sqrt{\dfrac{\sum fx^{2}}{\sum f} - \left(\dfrac{\sum fx}{\sum f}\right)^{2}}

Formulas to memorise

Algebra

NameFormulaExample

Perfect square

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

(x + 3)^2 = x^2 + 6x + 9

Perfect square

(a - b)^2 = a^2 - 2ab + b^2

(x - 4)^2 = x^2 - 8x + 16

Difference of two squares

a^2 - b^2 = (a + b)(a - b)

x^2 - 9 = (x + 3)(x - 3)

Quadratic formula

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x^2 - 5x + 6 = 0

x = 2

x = 3

Laws of indices

NameFormulaExample

Multiplying

a^m \times a^n = a^{m+n}

2^3 \times 2^4 = 2^7

Dividing

a^m \div a^n = a^{m-n}

5^6 \div 5^2 = 5^4

Power of a power

(a^m)^n = a^{mn}

(3^2)^4 = 3^8

Power of a product

(ab)^n = a^n b^n

(2x)^3 = 8x^3

Power of a quotient

\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}

\left(\dfrac{2}{3}\right)^2 = \dfrac{4}{9}

Zero index

a^0 = 1

7^0 = 1

Negative index

a^{-n} = \dfrac{1}{a^n}

2^{-3} = \dfrac{1}{8}

Fractional index

a^{\frac{1}{n}} = \sqrt[n]{a}

27^{\frac{1}{3}} = 3

Fractional index

a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m

8^{\frac{2}{3}} = 4

Percentage and money

NameFormulaExample

Percentage change

\dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%

80 \to 92

\dfrac{12}{80} \times 100\% = 15\%

Simple interest

I = \dfrac{PRT}{100}

P = 2000

R = 3

T = 4

I = 240

Coordinate geometry

NameFormulaExample

Gradient

m = \dfrac{y_2 - y_1}{x_2 - x_1}

(1, 2)

and

(3, 8)

m = 3

Length of a line segment

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

(1, 2)

and

(4, 6)

: length

= 5

Equation of a straight line

y = mx + c

Gradient

2

y

-intercept

-3

y = 2x - 3

Parallel lines

m_1 = m_2

y = 2x + 1

is parallel to

y = 2x - 5

Mensuration (not given)

NameFormulaExample

Area of a triangle

\dfrac{1}{2} \times \text{base} \times \text{height}

Base

8

cm, height

5

cm:

20

cm²

Area of a parallelogram

\text{base} \times \text{height}

Base

9

cm, height

4

cm:

36

cm²

Area of a trapezium

\dfrac{1}{2}(a + b)h

a = 4

b = 6

h = 5

25

cm²

Area of a circle

\pi r^2

r = 7

cm:

154

cm² (3 s.f.)

Circumference of a circle

2\pi r

r = 7

cm:

44.0

cm (3 s.f.)

Volume of a prism

\text{area of cross-section} \times \text{length}

Cross-section

12

cm², length

10

cm:

120

cm³

Volume of a cylinder

\pi r^2 h

r = 3

cm,

h = 10

cm:

283

cm³ (3 s.f.)

Curved surface area of a cylinder

2\pi r h

r = 3

cm,

h = 10

cm:

188

cm² (3 s.f.)

Volume of a pyramid

\dfrac{1}{3} \times \text{base area} \times \text{height}

Base

36

cm², height

10

cm:

120

cm³

Radians

\pi \text{ rad} = 180^\circ

60^\circ = \dfrac{\pi}{3}

rad

Geometry and trigonometry

NameFormulaExample

Pythagoras' theorem

a^2 + b^2 = c^2

5

and

12

c = 13

Sine

\sin \theta = \dfrac{\text{opp}}{\text{hyp}}

opp

3

, hyp

5

\sin \theta = \dfrac{3}{5}

\theta = 36.9^\circ

Cosine

\cos \theta = \dfrac{\text{adj}}{\text{hyp}}

adj

4

, hyp

5

\cos \theta = \dfrac{4}{5}

\theta = 36.9^\circ

Tangent

\tan \theta = \dfrac{\text{opp}}{\text{adj}}

opp

3

, adj

4

\tan \theta = \dfrac{3}{4}

\theta = 36.9^\circ

Obtuse angles: sine

\sin(180^\circ - \theta) = \sin \theta

\sin 150^\circ = \sin 30^\circ = \dfrac{1}{2}

Obtuse angles: cosine

\cos(180^\circ - \theta) = -\cos \theta

\cos 120^\circ = -\cos 60^\circ = -\dfrac{1}{2}

Angles on a straight line

\text{sum} = 180^\circ

x + 130^\circ = 180^\circ

x = 50^\circ

Angles in a triangle

\text{sum} = 180^\circ

50^\circ

and

60^\circ

: third angle

70^\circ

Sum of interior angles

(n - 2) \times 180^\circ

Hexagon,

n = 6

720^\circ

Exterior angle of a regular polygon

\dfrac{360^\circ}{n}

n = 8

45^\circ

Similar figures: areas

\dfrac{A_1}{A_2} = \left(\dfrac{l_1}{l_2}\right)^2

Lengths

2 : 3

: areas

4 : 9

Similar solids: volumes

\dfrac{V_1}{V_2} = \left(\dfrac{l_1}{l_2}\right)^3

Lengths

2 : 3

: volumes

8 : 27

Rate and speed

NameFormulaExample

Speed

\text{speed} = \dfrac{\text{distance}}{\text{time}}

150

km in

2

75

km/h

Average speed

\dfrac{\text{total distance}}{\text{total time}}

60

km in

1

h, then

90

km in

2

50

km/h

Statistics, probability and sets

NameFormulaExample

Probability

P(A) = \dfrac{n(A)}{n(S)}

6

on a fair die:

\dfrac{1}{6}

Not happening

P(\text{not } A) = 1 - P(A)

Not a

6

1 - \dfrac{1}{6} = \dfrac{5}{6}

Interquartile range

Q_3 - Q_1

Q_3 = 70

Q_1 = 45

25

Sets

n(A \cup B) = n(A) + n(B) - n(A \cap B)

18 + 17 - 7 = 28

Vectors

NameFormulaExample

Magnitude

\left|\begin{pmatrix} x \\ y \end{pmatrix}\right| = \sqrt{x^2 + y^2}

\left|\begin{pmatrix} 3 \\ 4 \end{pmatrix}\right| = 5