Sec 1 and Sec 2 Maths formulas

School exams in Sec 1 and Sec 2 usually give no formula list, so these are the ones to know by heart, each with a worked example.

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Formulas

Number, percentage and rate

NameFormulaExample

Percentage change

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

80 \to 92

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

Percentage of a quantity

\dfrac{p}{100} \times \text{quantity}

15\%

80 = 12

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

Direct proportion

y = kx

y = 12

when

x = 3

k = 4

y = 4x

Inverse proportion

y = \dfrac{k}{x}

y = 6

when

x = 2

k = 12

y = \dfrac{12}{x}

Algebra and graphs

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)

Gradient

\dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}

(1, 2)

and

(3, 8)

3

Equation of a straight line

y = mx + c

Gradient

2

y

-intercept

-3

y = 2x - 3

Angles and polygons

NameFormulaExample

Angles on a straight line

\text{sum} = 180^\circ

x + 130^\circ = 180^\circ

x = 50^\circ

Angles at a point

\text{sum} = 360^\circ

100^\circ

120^\circ

x

x = 140^\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

Pythagoras' theorem 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

Area and perimeter

NameFormulaExample

Area of a rectangle

\text{length} \times \text{breadth}

8

cm by

5

cm:

40

cm²

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 and surface area

NameFormulaExample

Volume of a cuboid

l \times b \times h

5 \times 4 \times 3 = 60

cm³

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

Volume of a cone

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

r = 3

cm,

h = 4

cm:

37.7

cm³ (3 s.f.)

Curved surface area of a cone

\pi r l

r = 3

cm,

l = 5

cm:

47.1

cm² (3 s.f.)

Volume of a sphere

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

r = 3

cm:

113

cm³ (3 s.f.)

Surface area of a sphere

4\pi r^2

r = 3

cm:

113

cm² (3 s.f.)

Statistics and probability

NameFormulaExample

Mean

\dfrac{\text{sum of values}}{\text{number of values}}

4, 7, 10

: mean

= 7

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}

Practise using them

Formulas

Number, percentage and rate

NameFormulaExample

Percentage change

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

80 \to 92

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

Percentage of a quantity

\dfrac{p}{100} \times \text{quantity}

15\%

80 = 12

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

Direct proportion

y = kx

y = 12

when

x = 3

k = 4

y = 4x

Inverse proportion

y = \dfrac{k}{x}

y = 6

when

x = 2

k = 12

y = \dfrac{12}{x}

Algebra and graphs

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)

Gradient

\dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}

(1, 2)

and

(3, 8)

3

Equation of a straight line

y = mx + c

Gradient

2

y

-intercept

-3

y = 2x - 3

Angles and polygons

NameFormulaExample

Angles on a straight line

\text{sum} = 180^\circ

x + 130^\circ = 180^\circ

x = 50^\circ

Angles at a point

\text{sum} = 360^\circ

100^\circ

120^\circ

x

x = 140^\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

Pythagoras' theorem 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

Area and perimeter

NameFormulaExample

Area of a rectangle

\text{length} \times \text{breadth}

8

cm by

5

cm:

40

cm²

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 and surface area

NameFormulaExample

Volume of a cuboid

l \times b \times h

5 \times 4 \times 3 = 60

cm³

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

Volume of a cone

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

r = 3

cm,

h = 4

cm:

37.7

cm³ (3 s.f.)

Curved surface area of a cone

\pi r l

r = 3

cm,

l = 5

cm:

47.1

cm² (3 s.f.)

Volume of a sphere

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

r = 3

cm:

113

cm³ (3 s.f.)

Surface area of a sphere

4\pi r^2

r = 3

cm:

113

cm² (3 s.f.)

Statistics and probability

NameFormulaExample

Mean

\dfrac{\text{sum of values}}{\text{number of values}}

4, 7, 10

: mean

= 7

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}