IGCSE Mathematics: Formula Booklet
1. NUMBER
 Order of Operations (BIDMAS)
Brackets
Indices
Division
Multiplication
Addition
Subtraction

Converting between Units
o Mass
o
1 leap year = 366 days
1 ordinary year = 365 days
1 week = 7 days
1 day = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds

Sequences (nth term)
o Arithmetic
 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑,
where 𝑎1 is the first
term of the sequence
and 𝑑 is the common
difference between
terms.
o Geometric
 𝑎𝑛 = 𝑎𝑟 𝑛−1 , where 𝑎
is the first term of the
sequence and 𝑟 is the
common
multiplication ratio
between terms.

Percentages
1 tonne = 1000 kilograms (kg)
1kg = 1000 grams (g)
1 gram = 1000 milligrams (mg)
o
Time
Volume
1m3 = 1000 litres (l)
1000 cm3 = 1 litre (l)
1000 millitres (ml) = 1 litre (l)
1cm3 = 1 milliltre (ml)
Percentage Change
o
Length
1 kilometre (km) = 1000 metres (m)
𝐹𝑖𝑛𝑎𝑙 − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙
× 100
𝐼𝑛𝑖𝑡𝑖𝑎𝑙
-
1 metre (m) = 100 centimetres (cm)
1 centimetre (cm) = 10 millimetres (mm)
-
If the calculated value is
negative, there has been a
percentage decrease.
If the calculated value is positive,
there has been a percentage
increase.
o

𝐴 = 𝑃(1 +
Interest
o Simple Interest
where 𝐼 is the interest earned, 𝑃 is
the principal amount (i.e. the initial
amount invested), 𝑅 is the rate of
interest (%) per period of time, and
𝑇 is the time, i.e. the number of
periods the money is invested for.

Speed, distance and time
Examples of periods of time
-
Distance
Per month / monthly
Per 3 months / quarterly
Per 6 months / semiannually
Per year / per annum
From the equation above (*),
𝐴=𝑃+
𝑃×𝑅×𝑇
100
where 𝐴 is the final value of the
investment after interest, 𝑃 is the
principal amount (i.e. the initial
amount invested), and the secondhalf of the RHS is the interest
earned from the investment (∗)
𝑟 𝑛𝑡
)
100𝑛
where 𝐴 is the final value of the
investment after interest, 𝑃 is the
principal amount (i.e. the initial
amount invested), 𝑟 is the rate of
interest (%) per period of time, 𝑛 is
the compound frequency (i.e. how
many times interest is compounded
per year, e.g. if it was 2% per month,
𝑛 would be 12 since there are 12
months in a year), and 𝑡 is the time,
i.e. the number of periods the
money is invested for.
𝑃×𝑅×𝑇
𝐼=
(∗)
100
Note: a ‘period of time’ does not
specifically have to be 1 year. If, for
example, the investment has an
interest of 1% per month, and the
money is invested for a year, T = 12
(because there are 12 months in a
year) and not 1, since we are dealing
with interest per month and not per
annum (per year).
Compound Interest
Speed
Time
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑇𝑖𝑚𝑒
o
𝑆𝑝𝑒𝑒𝑑 =
o
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑆𝑝𝑒𝑒𝑑 × 𝑇𝑖𝑚𝑒
o
𝑇𝑖𝑚𝑒 =
o
𝑘𝑚/ℎ × 18 = 𝑚/𝑠
o
𝑚/𝑠 × 5 = 𝑘𝑚/ℎ
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑆𝑝𝑒𝑒𝑑
5
18

2. Algebra
 The Quadratic Formula
Indices
𝑎0 = 1, for all 𝑎 > 0
2
Given an equation 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0,
𝑥=

−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛
(𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
𝑎−𝑛 =
Algebraic Expressions
1
2
(𝑎 + 𝑏) = 𝑎 + 2𝑎𝑏 + 𝑏
𝑚
2
𝑛
𝑛
𝑎 𝑛 = √𝑎𝑚 = ( √𝑎)
𝑚
(𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2
𝑎𝑚 × 𝑏 𝑚 = (𝑎𝑏)𝑚
(𝑎 + 𝑏)3 = 𝑎3 + 3𝑎2 𝑏 + 3𝑎𝑏 2 + 𝑏 2
𝑎𝑚
𝑎 𝑚
=
(
)
𝑏𝑚
𝑏
(𝑎 − 𝑏)3 = 𝑎3 − 3𝑎2 𝑏 + 3𝑎𝑏 2 + 𝑏 2
𝑎2 + 𝑏 2 = (𝑎 + 𝑏)2 − 2𝑎𝑏
𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

𝑛
𝑎𝑛 = √𝑎
𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐
2
1
𝑎𝑛
Variation
o Direct Variation
𝑦 is proportional to 𝑥
𝑦∝𝑥
𝑦 = 𝑘𝑥
o
Inverse Variation
𝑦 is inversely proportional to 𝑥
𝑦∝
1
𝑥
𝑦=
𝑘
𝑥
3. Mensuration
 Area and Perimeter
Figure
Square
Diagram
Perimeter
𝑠
Area
4𝑠
𝑠2
2(𝑙 + 𝑤)
𝑙×𝑤
𝑠
Rectangle
𝑤
𝑙
Triangle
1
𝑏ℎ
2
𝐵
𝑐
𝑎
ℎ
𝐴
𝑎+𝑏+𝑐
𝐶
𝑏
Parallelogram
𝑎
ℎ
2(𝑎 + 𝑏)
𝑎𝑏 sin 𝜃
where 𝜃 is the
included angle
between the sides
𝑎 and 𝑏
𝑑
𝑎+𝑏+𝑐+𝑑
ℎ(𝑎 + 𝑏)
2
𝜃
𝑏
Trapezium
1
𝑎𝑏 sin 𝐶
2
where 𝑐 is the
included angle
between the sides
𝑎 and 𝑏
𝑏ℎ
𝑎
𝑐
ℎ
𝑏

Figure
Cube
Surface Area and Volume
Diagram
Surface Area
𝑠
𝑠
𝑠
6𝑠 2
Volume
𝑠3
Cuboid
ℎ
2(𝑙𝑤 + 𝑤ℎ + 𝑙ℎ)
𝑙×𝑤×ℎ
𝑤
𝑙
Sphere
r
4𝜋𝑟
2
4 3
𝜋𝑟
3
𝑟 is the radius of the
sphere
Cylinder
ℎ
2𝜋𝑟(𝑟 + ℎ)
𝜋𝑟 2 ℎ
𝑟
𝑟 is the radius of the
circular base
Cone
Curved S.A.:
𝜋𝑟𝑙
𝑙
ℎ
𝑟
𝑟 is the radius of the
circular base
ℎ is the height which is
perpendicular to the base
𝑙 is the slant height
By Pythagoras’ Theorem,
𝑟 2 + ℎ2 = 𝑙 2
Total S.A.:
𝜋𝑟𝑙 (curved SA) +
𝜋𝑟 2 (area of the
circular base)
= 𝜋𝑟(𝑙 + 𝑟)
1 2
𝜋𝑟 ℎ
3
Pyramid (in general)
ℎ
𝐴
1
𝐴×ℎ
3
Base Area (A) + Area
of the remaining
sides
ℎ is the height which is
perpendicular to the base
𝐴 is the area of the base
of the pyramid (for e.g.
the area of the triangular
base in a triangular-based
pyramid or the square in a
square-based pyramid)

Arc Length and Sector Area
𝜃
Arc length, 𝑙 = 360 × 2𝜋𝑟
𝜃
is the fraction of the circumference the arc
360
length is, depending on the angle at the centre of
the circle, 𝜃; 2𝜋𝑟 is the formula for calculating the
circumference of the entire circle, where 𝑟 is the
radius of the circle.
𝑟
𝑙
sector
𝜃
𝜃
Sector Area, 𝐴 = 360 × 𝜋𝑟 2
𝜃
is the fraction of the area of the circle the
360
sector area occupies, depending on the angle at the
centre of the circle, 𝜃; 𝜋𝑟 2 is the formula for
calculating the area of the entire circle, where 𝑟 is
the radius of the circle.
𝜃
Perimeter of a Sector, 𝑃 = (360 × 2𝜋𝑟) + 2𝑟
𝜃
× 2𝜋𝑟 is the length of the arc (𝑙), and 2𝑟 is the
360
combined length of the 2 radii that make up the
rest of the sector.
4. Geometry
 Pythagoras’ Theorem
For any right-angled triangle, the square of the length of the
hypotenuse (the side opposite the right angle; the longest
side) is equal to the sum of the squares of the lengths of the
other two sides.
𝑐
𝑎
Algebraically,
𝑏
𝑎2 + 𝑏 2 = 𝑐 2 , where 𝑐 is the hypotenuse.

Similarity
o 2D Shapes
×𝑘
If two shapes are similar and the ratio
of corresponding sides (i.e. the linear
𝑎𝑘
𝑎𝑘
scale factor) is 𝑘 { 𝑎 = 𝑘}, the ratio of
𝑎
𝑎2 𝑘 2
𝑎𝑘
𝑎
𝑎
areas is 𝑘 2 {𝑎𝑠
2
Note: this applies to all similar shapes,
including regular and irregular
polygons.
× 𝑘2
o
𝑎2 𝑘2
= 𝑘 2}
𝑎2
3D Shapes
×𝑘
𝑎𝑘
If two objects are similar and the
ratio of corresponding sides (i.e. the
𝑎
linear scale factor) is 𝑘 {
𝑎𝑘
𝑎
𝑎𝑘
= 𝑘},
𝑎
the ratio of surface areas is
6𝑎 2 𝑘 2
𝑎
Surface Area: 6𝑎2
Volume: 𝑎3
𝑎𝑘
×𝑘
× 𝑘3
2
Surface Area: 6𝑎2 𝑘 2
𝑘 2 {𝑎𝑠 6𝑎2 = 𝑘 2 }, and the ratio of
𝑎3 𝑘 3
volumes is 𝑘 3 {𝑎𝑠 𝑎3 = 𝑘 3 }
Volume: 𝑎3 𝑘 3
𝐴
= 𝑘. Then, the
𝐵
𝐴 2
𝐴2
𝐴 3
𝐴3
ratio for areas is (𝐵) = 𝐵2 = 𝑘 2 and for 3D shapes, the ratio for volumes is (𝐵) = 𝐵3 = 𝑘 3
In general, for similar shapes, suppose the ratio for corresponding sides is

Polygons
𝐸𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒
𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒
Interior Angles
Exterior Angles
The sum of the interior angles of a
polygon is (𝑛 − 2) × 180°, where 𝑛 is the
number of sides of the polygon
The sum of the exterior angles of a polygon
is 360°
If the polygon is regular (it has sides of
equal length and all the interior angles
are equal), then each interior angle is
equal to
180(𝑛−2)
𝑛
If the polygon is regular, then each exterior
360°
angle is 𝑛
𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒 + 𝐸𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒 = 180°
5. Trigonometry
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
Given a right-angled triangle:
𝑂
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝐴
𝑠𝑖𝑛 𝜃 = 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝐻

𝑐𝑜𝑠 𝜃 = 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝐻

𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑂
𝑡𝑎𝑛 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 = 𝐴
𝑂
𝜃
𝑠𝑖𝑛𝜃
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑺𝑶𝑯 𝑪𝑨𝑯 𝑻𝑶𝑨
𝐴
𝐻
𝑐𝑜𝑠𝜃
𝑂
𝐻
𝑡𝑎𝑛𝜃
𝐴
Sine Rule
1. For calculating the length of an
unknown side:
𝐵
𝑐
𝑎
𝑏
𝑐
=
=
𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵 𝑠𝑖𝑛 𝐶
𝑎
2. For calculating an unknown angle:
𝐴

𝐶
𝑏
𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵 𝑠𝑖𝑛 𝐶
=
=
𝑎
𝑏
𝑐
Cosine Rule
1. For calculating the length of an
unknown side:
𝐵
𝑐
𝐴
𝑎2 = 𝑏 2 + 𝑐 2 − (2𝑏𝑐 × 𝑐𝑜𝑠𝐴)
∴ 𝑎 = √𝑏 2 + 𝑐 2 − (2𝑏𝑐 × 𝑐𝑜𝑠𝐴)
𝑎
𝑏
𝐶
2. For calculating an unknown angle:
𝑏 2 + 𝑐 2 − 𝑎2
𝑐𝑜𝑠𝐴 =
2𝑏𝑐
6. Graphs
 Coordinate Geometry
𝑦
𝐵 (𝑥2 , 𝑦2 )
𝐴(𝑥1 , 𝑦1 )
𝑥
𝑦 −𝑦
1. Gradient of line 𝐴𝐵 = 𝑥2 −𝑥1
2
1
2. Gradient of a parallel line is same as that of the line AB
1
3. Gradient of a perpendicular line is − 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝐴𝐵
4. By the Pythagorean Theorem, distance between the points 𝐴 and 𝐵 is
√(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
𝑥 +𝑥
5. Midpoint of 𝐴𝐵 = ( 1 2 2 ,

𝑦1 +𝑦2
)
2
Differentiation
Given the equation of a line 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑, we differentiate it term by term:
𝑑𝑦
= (3𝑎)𝑥 2 + (2𝑏)𝑥 + 𝑐
𝑑𝑥
In general, given a term 𝑎𝑥 𝑛 , its differentiated form is (𝑛𝑎)𝑥 𝑛−1
7. Statistics
 Histograms
Grouped Frequency Table
Ages (years)
0 ≤ 𝑥 < 20
20 ≤ 𝑥 < 40
40 ≤ 𝑥 < 50
50 ≤ 𝑥 < 70
70 ≤ 𝑥 < 100
Frequency (𝒇)
28
36
17
24
12
To construct a histogram from the table above, we need to calculate the frequency density,
𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 =
𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚
𝒄𝒍𝒂𝒔𝒔 𝒘𝒊𝒅𝒕𝒉, 𝒊. 𝒆. 𝒖𝒑𝒑𝒆𝒓 𝒃𝒐𝒖𝒏𝒅 − 𝒍𝒐𝒘𝒆𝒓 𝒃𝒐𝒖𝒏𝒅
28
28
The frequency density for the first row is therefore 20−0 = 20 = 1.4. Continuing this process
gives the frequency densities for the remaining columns as: 1.8, 1.7, 1.2 and 0.4,
repspectively. The following chart shows a histogram constructed for this data:
Age Distribution Histogram
Frequency Density
2
1.5
1
0.5
0
0 ≤ x < 20
20 ≤ x < 40
40 ≤ x < 50
50 ≤ x < 70
70 ≤ x < 100
Ages (years)

Mean
To find the mean of the grouped frequency table above, we firstly need to calculate the
midpoints of each group,
𝒎𝒊𝒅𝒑𝒐𝒊𝒏𝒕 =
𝒖𝒑𝒑𝒆𝒓 𝒃𝒐𝒖𝒏𝒅 − 𝒍𝒐𝒘𝒆𝒓 𝒃𝒐𝒖𝒏𝒅
𝟐
Adding this as an extra column to the table gives:
Ages (years)
0 ≤ 𝑥 < 20
20 ≤ 𝑥 < 40
40 ≤ 𝑥 < 50
50 ≤ 𝑥 < 70
70 ≤ 𝑥 < 100
Frequency (𝒇)
28
36
17
24
12
Midpoint (𝒙)
10
30
45
60
85
The mean is now calculated using the following formula:
𝒎𝒆𝒂𝒏 =
𝚺𝒇𝒙
,
𝚺𝒇
where Σ𝑓𝑥 is the sum of the products of the frequencies and respective midpoints, and
Σ𝑓 is the sum of all frequencies. Therefore, for the table above,
𝑚𝑒𝑎𝑛 =
(28 × 10) + (36 × 30) + (17 × 45) + (24 × 60) + (12 × 85)
28 + 36 + 17 + 24 + 12
=
4585
117
= 39.2 (𝑡𝑜 3 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑓𝑖𝑔𝑢𝑟𝑒𝑠)
Therefore, the mean age for the grouped frequency table above is 39.2 years.
o
Cumulative Frequency Tables
The table above could be represented as a cumulative frequency table:
Ages (years)
0 ≤ 𝑥 < 20
20 ≤ 𝑥 < 40
40 ≤ 𝑥 < 50
50 ≤ 𝑥 < 70
70 ≤ 𝑥 < 100
Cumulative Frequency (𝒄𝒇)
28
64
81
105
117
A cumulative frequency table shows the total frequency up to a certain group. Start
with the frequency of the first group and then add the frequencies of subsequent
groups to get the cumulative total.
Since the cumulative frequency up to group 2 is 64, the individual frequency for that
group is 64 − 28 = 36. In general, for all 𝑛 > 1 (𝑤ℎ𝑒𝑟𝑒 𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑝 𝑛𝑢𝑚𝑏𝑒𝑟),
the individual frequency (𝑓) for the group can be calculated as:
𝒇 = 𝒄𝒇𝒏 − 𝒄𝒇𝒏−𝟏
Once all the individual frequencies have been calculated, proceed with the steps
above to obtain the mean of the cumulative frequency table.
8. Probability
𝑷(𝑨) =
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒊𝒎𝒆𝒔 𝑨 𝒐𝒄𝒄𝒖𝒓𝒔
𝑻𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒆𝒒𝒖𝒂𝒍𝒍𝒚 𝒍𝒊𝒌𝒆𝒍𝒚 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔
𝑃(𝑛𝑜𝑡 𝐴) = 𝑃(𝐴′ ) = 1 − 𝑃(𝐴)
For mutually exclusive events (i.e. two events
that cannot occur at the same time),
For independent events (i.e. events whose
occurrence is not dependent on any other
event),
𝑷(𝑨 𝒐𝒓 𝑩) = 𝑷(𝑨) + 𝑷(𝑩)
𝑷(𝑨 𝒂𝒏𝒅 𝑩) = 𝑷(𝑨) × 𝑷(𝑩)

Conditional Probability
To work out 𝑃(𝐵|𝐴), i.e. the probability that event B occurs, given that event A has already
occurred, we can use the following formula:
𝑷(𝑩|𝑨) =
𝑷(𝑨 𝒂𝒏𝒅 𝑩)
𝑷(𝑨)
This concept can be illustrated on a Venn Diagram:
No. of Maths students
15
5
No. of English Students
10
From the Venn diagram above, if we want to calculate the probability that a student studies
Maths, given that they study English, we can use the formula for conditional probability:
𝑃(𝑀|𝐸) =
𝑃(𝑀 𝑎𝑛𝑑 𝐸)
𝑃(𝐸)
5
5
𝑃(𝑀 𝑎𝑛𝑑 𝐸) = 15+5+10 = 35
and
5+10
15
𝑃(𝐸) = 15+5+10 = 35
5
5
1
35
∴ 𝑃(𝑀|𝐸) =
=
=
15 15 3
35
Hence, we see that the probability that a student studies Maths, given that they study
1
3
English, is , which is clear to see on the Venn diagram; of the 15 students that study
5
1
English, 5 of them also study Maths, which means that 𝑃(𝑀|𝐸) must be equal to 15 = 3.
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