Complete
Mathematics
for Cambridge IGCSE®
Fifth Edition
Core
David Rayner
Ian Bettison
Mathew Taylor
Oxford excellence for Cambridge IGCSE®
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Contents
Syllabus matching grid
Introduction
iv-vii
viii
1
1–46
Shape and Space 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Algebra 1
2.1
2.2
2.3
2.4
3
Place value
Arithmetic without a calculator
Inverse operations
Decimals
Flow diagrams
Properties of numbers
Inequalities
Time
Long multiplication and division
Percentages
Map scales and ratio
Proportion
Speed, distance and time
Approximations
Metric units
Problems 1
Revision exercise 3A
Examination-style exercise 3B
Handling Data 1
4.1
4.2
4.3
4.4
5
Sequences
Solving equations
Drawing graphs
Gradient, y = mx + c
Revision exercise 2A
Examination-style exercise 2B
Number 1
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
4
Accurate drawing
Angle facts
Angles in polygons and circles
Symmetry
Circle calculations
Arc length and sector area
Area
Volume and surface area
Scale drawing
Revision exercise 1A
Examination-style exercise 1B
Displaying data
Questionnaires
Averages
Frequency polygons
Revision exercise 4A
Examination-style exercise 4B
Shape and Space 2
5.1
5.2
5.3
5.4
5.5
Transforming shapes
Quadrilaterals and other polygons
Bearings
Pythagoras’ theorem
Problems 2
Revision exercise 5A
Examination-style exercise 5B
1
5
11
14
16
24
26
31
37
40
41
6
7
7.3
7.4
7.5
7.6
8
72–137
72
74
75
77
85
87
92
94
97
101
107
111
115
118
121
123
132
134
9
10
138–178
138
160
165
169
173
174
179–220
179
198
203
209
212
214
216
11
12
257
259
263
265
275
288
290
292–311
292
297
299
301
304
308
309
312–327
312
315
316
322
323
328–351
Powers and roots
Standard form
Fractions
Negative numbers
Substituting into formulae
Problems 3
Revision exercise 10A
Examination-style exercise 10B
Using and Applying
Mathematics
11.1
11.2
253
Similar shapes
Congruent shapes
Trigonometry in right-angled triangles
Revision exercise 9A
Examination-style exercise 9B
Number 3
10.1
10.2
10.3
10.4
10.5
10.6
253–291
Probability of an event
Exclusive events
Relative frequency
Venn diagrams
Probability diagrams
Revision exercise 8A
Examination-style exercise 8B
Shape and Space 3
9.1
9.2
9.3
221
226
232
241
243
245
247
Percentage change
Fractions, ratio, decimals
and percentage
Estimating
Errors in measurement
Mental arithmetic
Using a calculator
Revision exercise 7A
Examination-style exercise 7B
Probability
8.1
8.2
8.3
8.4
8.5
221–252
Finding a rule
Simultaneous equations
Interpreting graphs
Brackets and factors
Changing the subject of a formula
Revision exercise 6A
Examination-style exercise 6B
Number 2
7.1
7.2
47–71
47
49
57
61
67
68
Algebra 2
6.1
6.2
6.3
6.4
6.5
328
333
336
339
343
346
348
349
352–370
Investigation tasks
Puzzles and games
352
358
Multiple choice tests
371–383
Examination-style Paper 1
Examination-style Paper 3
Answers
Index
379–380
381–383
384–417
418–422
Access your support website for extra homework
questions and exam revision material
www.oxfordsecondary.com/9780198425045
iii
Cambridge IGCSE® Mathematics 0580: Core
Syllabus topic
C1: Number
C1.1
Identify and use natural numbers, integers (positive, negative
and zero), prime numbers, square numbers, common factors and
common multiples, rational and irrational numbers (e.g. π, 2 ), real
numbers, reciprocals.
C1.2
Understand notation of Venn diagrams. Definition of sets e.g.
A = {x: x is a natural number}, B = {a, b, c, …}
C1.3
Calculate squares, square roots, cubes and cube roots of numbers.
C1.4
Use directed numbers in practical situations.
C1.5
Use the language and notation of simple vulgar and decimal
fractions and percentages in appropriate contexts. Recognise
equivalence and convert between these forms.
C1.6
Order quantities by magnitude and demonstrate familiarity with
the symbols =, ≠, >, <, ⩾, ⩽
C1.7
72–74, 88–92
301–302
328–333
339–343
102–103, 257–258
78, 92–94, 337
Understand the meaning and rules of indices (fractional, negative
and zero) and use the rules of indices. Use the standard form A ×
10n where n is a positive or negative integer, and 1 ⩽ A < 10.
Use the four rules for calculations with whole numbers, decimals
and vulgar (including mixed numbers and improper fractions),
including correct ordering of operations and use of brackets.
Make estimates of numbers, quantities and lengths, give
approximations to specified numbers of significant figures and
decimal places and round off answers to reasonable accuracy in the
context of a given problem.
Give appropriate upper and lower bounds for data given to a
specified accuracy.
331–336
C1.11
Demonstrate an understanding of ratio and proportion. Calculate
average speed. Use common measures of rate.
107–114, 115–118
C1.12
Calculate a given percentage of a quantity. Express one quantity as a 101–104, 253–256
percentage of another. Calculate percentage increase or decrease.
C1.13
C1.14
Use a calculator efficiently. Apply appropriate checks of accuracy.
Calculate times in terms of the 24-hour and 12-hour clock. Read
clocks, dials and timetables.
C1.8
C1.9
C1.10
iv
Page numbers in
student book
Syllabus matching grid
74–77, 80–88,
97–100, 275–276,
336–338
118–120, 259–260
263–264
276–278
94–97
C1.15
C1.16
Calculate using money and convert from one currency to another.
Use given data to solve problems on personal and household
finance involving earnings, simple interest and compound interest.
Extract data from tables and charts.
C2: Algebra and graphs
C2.1 Use letters to express generalised numbers and express basic
arithmetic processes algebraically. Substitute numbers for words and
letters in formulae. Transform simple formulae. Construct simple
expressions and set up simple equations.
C2.2 Manipulate directed numbers. Use brackets and extract common factors.
Expand products of algebraic expressions.
C2.4 Use and interpret positive, negative and zero indices. Use the rules
of indices.
C2.5 Derive and solve simple linear equations in one unknown. Derive
and solve simultaneous linear equations in two unknowns.
C2.7 Continue a given number sequence. Recognise patterns in sequences
including the term-to-term rule and relationships between different
sequences. Find the nth term of sequences.
C2.10 Interpret and use graphs in practical situations including travel
graphs and conversion graphs. Draw graphs from given data.
C2.11 Construct tables of values for functions of the form ax + b,
a
± x2 + ax + b, (x ≠ 0), where a and b are integral constants.
x
Draw and interpret these graphs. Solve linear and quadratic
equations approximately, including finding and interpreting roots
by graphical methods. Recognise, sketch and interpret graphs of
functions.
C3: Coordinate geometry
C3.1 Demonstrate familiarity with Cartesian coordinates in two
dimensions.
C3.2 Find the gradient of a straight line.
C3.4 Interpret and obtain the equation of a straight-line graph in the form
y = mx + c.
C3.5 Determine the equation of a straight line parallel to a given line.
C4: Geometry
C4.1 Use and interpret the geometrical terms: point, line, parallel,
bearing, right angle, acute, obtuse and reflex angles, perpendicular,
similarity and congruence. Use and interpret vocabulary of triangles,
quadrilaterals, circles, polygons and simple solid figures including nets.
C4.2 Measure and draw lines and angles. Construct a triangle given the
three sides using ruler and pair of compasses only.
114–115
104–106
49–57, 243–245,
343–346
241–242
331–333
226–232
47–49, 221–226
57–60, 150–151,
232–240
65–67
61–65, 196–198
61–62
62–65
65
4–5
1–3
Syllabus matching grid
v
C4.3
C4.4
C4.5
C4.6
C4.7
C3.7
Read and make scale drawings.
Calculate lengths of similar figures.
Recognise congruent shapes.
Recognise rotational and line symmetry (including order of
rotational symmetry) in two dimensions.
Calculate unknown angles using the following geometrical properties:
• angles at a point
• angles at a point on a straight line and intersecting straight lines
• angles formed within parallel lines
• angle properties of triangles and quadrilaterals
• angle properties of regular polygons
• angle in a semicircle
• angle between tangent and radius of a circle.
Use the following loci and the method of intersecting loci for sets of
points in two dimensions which are:
37–39
312–315
315–316
14–16
5–14
200–203
• at a given distance from a given point
• at a given distance from a given straight line
• equidistant from two given points
• equidistant from two given intersecting straight lines.
C5: Mensuration
C5.1 Use current units of mass, length, area, volume and capacity in
practical situations and express quantities in terms of larger or
smaller units.
C5.2 Carry out calculations involving the perimeter and area of a
rectangle, triangle, parallelogram and trapezium and compound
shapes derived from these.
C5.3 Carry out calculations involving the circumference and area of a circle.
Solve problems involving the arc length and sector area as fractions of
the circumference and area of a circle.
C5.4 Carry out calculations involving the surface area and volume of a cuboid,
prism and cylinder. Carry out calculations involving the surface area and
volume of a sphere, pyramid and cone.
C5.5 Carry out calculations involving the areas and volumes of
compound shapes.
C6: Trigonometry
C6.1 Interpret and use three-figure bearings.
C6.2 Apply Pythagoras’ theorem and the sine, cosine and tangent ratios
for acute angles to the calculation of a side or of an angle of a
right-angled triangle.
vi
Syllabus matching grid
121–123
26–29
16–26
31–36
17, 20, 21, 27, 32, 35
203–208
209–211, 316–322
C7: Vectors and transformations
C7.1
x Describe a translation by using a vector represented by e.g.   , AB
193–195
 y
or a. Add and subtract vectors. Multiply a vector by a scalar.
C7.2 Reflect simple plane figures in horizontal or vertical lines. Rotate
simple plane figures about the origin, vertices or midpoints of edges
of the figures, through multiples of 90°. Construct given translations
and enlargements of simple plane figures. Recognise and describe
reflections, rotations, translations and enlargements.
C8: Probability
C8.1 Calculate the probability of a single event as either a fraction,
decimal or percentage.
C8.2 Understand and use the probability scale from 0 to 1.
C8.3 Understand that the probability of an event occurring = 1 − the
probability of the event not occurring.
C8.4 Understand relative frequency as an estimate of probability.
Expected frequency of occurrences.
C8.5 Calculate the probability of simple combined events, using
possibility diagrams, tree diagrams and Venn diagrams.
C9: Statistics
C9.1 Collect, classify and tabulate statistical data.
C9.2 Read, interpret and draw simple inferences from tables and
statistical diagrams. Compare sets of data using tables, graphs and
statistical measures. Appreciate restrictions on drawing conclusions
from given data.
C9.3 Construct and interpret bar charts, pie charts, pictograms,
stem-and-leaf diagrams, simple frequency distributions, histograms
with equal intervals and scatter diagrams.
C9.4 Calculate the mean, median, mode and range for individual and
discrete data and distinguish between the purposes for which they
are used.
C9.7 Understand what is meant by positive, negative and zero correlation
with reference to a scatter diagram.
C9.8 Draw, interpret and use lines of best fit by eye.
179–193
293–307
292–293
293–307
296–297, 299–300
294–307
138–150, 153–154
138–150
138–150, 169–173
165–168
151–154
155–156
Syllabus matching grid
vii
Introduction
About this book
This revised 5th edition is designed to provide the best preparation for your Cambridge IGCSE
examination, and has been completely updated for the latest Mathematics 0580 and 0980 Core
syllabus.
Finding your way around
To get the most out of this book when studying or revising, use the:
●
Contents list to help you find the appropriate units.
●
Index to find any key concept straight away.
Exercises and exam-style questions
There are thousands of questions in this book, providing ample opportunities to practise the skills
and techniques required in the exam.
●
●
●
●
●
●
Worked examples and comprehensive exercises are one of the main features of the book. The
examples show you the important skills and techniques required. The exercises are carefully
graded, starting from the basics and going up to exam standard, allowing you to practise the skills
and techniques.
Revision exercises at the end of each unit allow you to bring together all your knowledge on a
particular topic.
Examination-style exercises at the end of each unit consist of questions from past Cambridge
IGCSE papers.
Examination-style papers: there are two papers, corresponding to the papers you will take at the
end of your course: Paper 1 and Paper 3. They give you the opportunity to practise for the real thing.
Revision section: Unit 12 contains multiple-choice questions to provide an extra opportunity to
revise.
Answers to numerical problems are at the end of the book so you can check your progress.
Investigations
Unit 11 provides many opportunities for you to explore the world of mathematical problem-solving
through investigations, puzzles and games.
Links to curriculum content
At the start of each unit you will find a list of objectives that are covered in the unit. These objectives
are drawn from the Core sections of the Cambridge IGCSE syllabus.
What’s on the website?
The support website contains a wealth of material to help solidify your understanding of the
Cambridge IGCSE Mathematics course, and to aid revision for your examinations.
All this material can be found online, at www.oxfordsecondary.com/9780198425045
viii
1
Shape and Space 1
C4.1 Use and interpret the geometrical terms: point, line, parallel, bearing, right angle, acute, obtuse
and reflex angles, perpendicular, similarity and congruence. Use and interpret vocabulary of
triangles, quadrilaterals, circles, polygons and simple solid figures including nets.
C4.2 Measure and draw lines and angles. Construct a triangle given the three sides using ruler
and pair of compasses only.
C4.3 Read and make scale drawings.
C4.6 Recognise rotational and line symmetry (including order of rotational symmetry) in
two dimensions.
C4.7 Calculate unknown angles using the following geometrical properties:
• angles at a point
• angles at a point on a straight line and intersecting straight lines
• angles formed within parallel lines
• angle properties of triangles and quadrilaterals
• angle properties of regular polygons
• angle in a semicircle
• angle between tangent and radius of a circle.
C5.2 Carry out calculations involving the perimeter and area of a rectangle, triangle,
parallelogram and trapezium and compound shapes derived from these.
C5.3 Carry out calculations involving the circumference and area of a circle. Solve simple
problems involving the arc length and sector area as fractions of the circumference and
area of a circle.
C5.4 Carry out calculations involving the surface area and volume of a cuboid, prism and
cylinder. Carry out calculations involving the surface area and volume of a sphere,
pyramid and cone.
C5.5 Carry out calculations involving the areas and volumes of compound shapes.
1.1 Accurate drawing
Some questions involving bearings or irregular shapes are easy to
solve by drawing an accurate diagram.
Navigators on ships use scale drawings to work out their position
or their course.
1
To improve the accuracy of your work, follow these guidelines.
●
Use a sharp HB pencil.
●
Don’t press too hard.
●
If drawing an acute angle make sure your angle is less than 90°.
●
If you use a pair of compasses make sure they are fairly stiff so
the radius does not change accidently.
Example
C
Draw the triangle ABC full size and measure the length x.
a) Draw a base line longer than 8.5 cm.
x
b) Put the centre of the protractor on A and measure an
angle 64°. Draw line AP.
64°
c) Similarly draw line BQ at an angle 40° to AB.
d) The triangle is formed.
Measure x = 5.6 cm.
A
P
Q
40°
B
8.5 cm
C
70
60
50
40
90
100
80
110
70
120
60
14
30
0
10
1177 0
0
64°
40°
1180
80
20
180 170 1
60
20
160
15
30
10
150
40
0
13
0
50
0
100
0
14
0
13
120
110
80
8·5 cm
A
B
Exercise 1
Use a protractor and ruler to draw full size diagrams and measure the sides marked with letters.
1.
2.
3.
4.
7 cm
43°
x
78°
z
b
x
42°
62°
80°
8 cm
7 cm
5.
6.
63°
c
7.5 cm
7.
7 cm
x
8.
6.2 cm
y
48°
51°
2
44°
6 cm
39°
9 cm
101°
5 cm
122°
z
6 cm
71°
8.8 cm
Shape and Space 1
7.8 cm
7 cm
9 cm
51°
9.
8.8 cm
10.
40°
30°
70°
c
48°
31°
9.2 cm
42°
40°
70°
4.8 cm
b
88°
8 cm
12.
7.8 cm
46°
54°
70°
a
11.
7 cm
d
9 cm
72°
57°
Example
C
Draw the triangle ABC full size and measure the angle x.
4 cm
A
5 cm
x
B
6 cm
a) Draw a base line AB exactly 6 cm long.
b) Set a pair of compasses to 4 cm and draw an arc centred on
A above the base line.
A
B
c) Similarly, set a pair of compasses to 5 cm and draw
another arc centred on B intersecting the first.
d) Join this crossing point to A and B.
The triangle is formed.
e) Measure the angle marked x = 56°.
x
Exercise 2
Use a ruler and pair of compasses to make accurate drawings of these triangles and measure the
angles marked with x.
1.
2.
3 cm
3.
4.5 cm
6 cm
x
4 cm
x
5 cm
8 cm
5 cm
7 cm
x
9 cm
4.
5.
x
6.
6.3 cm
10 cm
x
x
4 cm
7 cm
5.5 cm
6 cm
6.3 cm
12 cm
3.5 cm
Accurate drawing
3
Nets
If the cube here was made of cardboard, and you cut
along some of the edges and laid it out flat, you would
have the net of the cube.
Here is the net for a square-based pyramid.
vertex
Exercise 3
1. Which of the nets below can be used to make a cube?
a)
b)
c)
d)
2. The numbers on opposite faces of a dice add up to 7. Take one
of the possible nets for a cube from Question 1 and show the
number of dots on each face.
3. Here we have started to draw the net of a cuboid (a closed
rectangular box) measuring 4 cm × 3 cm × 1 cm.
Copy and then complete the net.
4. This diagram shows the net of a solid.
a) Use a ruler and pair of compasses to draw the net
accurately on paper or card.
b) Draw on some flaps.
c) Cut out the net, fold and glue it to make the solid.
d) What is the name of the solid?
4
Shape and Space 1
6 cm
5 cm
5 cm
6 cm
5. A cube can be made from three equal pyramids.
Make three solids from the net shown and fit them together
to make a cube. All lengths are in cm.
7.1
5
7.1
5
5
5
7.1
5
6. Sketch a possible net for each of the following:
8.7
5
a) a cuboid measuring 5 cm by 2 cm by 8 cm
7.1
b) a cuboid with sides 3 cm, 4 cm and 5 cm.
8.7
1.2 Angle facts
●
The angles at a point add up to 360°.
●
The angles on a straight line add up to 180°.
●
An angle of 90° is called a right angle.
●
An acute angle is less than 90°.
●
An obtuse angle is between 90° and 180°.
●
A reflex angle is greater than 180°.
Example
Find the missing angles:
a)
b)
x
x
150°
a
a
a
x + x + 150° + 100° = 360°
100°
3a + 90° = 180°
∴2x = 360° − 250°
3a = 90°
x = 55°
a = 30°
Exercise 4
Find the angles marked with letters. The lines AB and CD are straight.
1.
150°
5.
2.
c
A
120°
140°
71°
y
65°
B
3.
x
a
140°
40°
40°
90°
60°
7.
58°
e
42°
y
160°
160°
6.
A
4.
B
A
8.
100°
x
A
h
x
B
84°
h
B
Angle facts
5
9.
10.
11.
x
60°
A
y y
75°
B
72°
C
b
40°
A
A
B
a
2x
y
12.
D
x
D
y
B
C
Triangles
The angles in a triangle add up to 180°.
Example
R
Find the missing angles:
x
a = 180° − 150° = 30°
150°
a
P
The triangle is isosceles so angle RQP = x
Q
∴ 2x + 30° = 180°
2x = 150°
x = 75°
Exercise 5
Find the angles marked with letters. For the more difficult questions it is helpful to draw a diagram.
1.
2.
x
70°
45°
65°
60°
5.
6.
30°
10.
4.
114°
x
70°
7.
40°
8.
55°
40°
a
9.
y
37°
y
75°
145°
3.
y
x
b
72°
11.
x
12.
a
e
90°
140°
a
122°
6
Shape and Space 1
y
a
100°
13.
14.
77°
15.
w
a
x
2x
150°
x
16.
3x
38°
Parallel lines
When a line cuts a pair of parallel lines all the acute angles are
equal and all the obtuse angles are equal.
Some people remember:
‘F angles’
b
a
b
a
and ‘Z angles’
a
b
a
b
Exercise 6
Find the angles marked with letters.
1.
2.
a
3.
b
100°
t
82°
72°
4.
5.
92°
86°
7.
8.
9.
74°
y
x
10.
77°
a
50°
a
a
b
93°
95°
x
y
e
74°
6.
115°
c
11.
b
65°
130°
12.
c
50°
68°
42°
y
z
125°
a
b
Angle facts
7
Quadrilaterals and regular polygons
P
The sum of the angles in a quadrilateral is 360°.
f
Proof: The quadrilateral PQRS has been split into two triangles.
We know that
a + e + f = 180°
and that
b + c + d = 180°
e
a
∴ a + b + c + d + e + f = 360°
S
Q
d
b
c
But the angles of the quadrilateral are (a + b), c, (d + e) and f.
R
∴ The sum of the angles in a quadrilateral is 360°.
Two straight lines meet at a vertex.
A shape with straight sides is called a polygon.
A regular polygon has equal sides and angles.
For example a square is a regular quadrilateral.
Example
x
x
Polygons
Name
Number of sides
Quadrilateral
4
Pentagon
5
Hexagon
6
Heptagon
7
Octagon
8
Nonagon
9
Decagon
10
x
x
x
Regular pentagon:
x + x + x + x + x = 360°
∴ x = 72°
Exercise 7
Find the angles marked with letters.
1.
2.
82°
3.
75°
100°
88°
x
Shape and Space 1
p
83°
30°
°
8
y
z
20
70°
145°
40°
113°
90°
4.
5.
6.
7.
b
d
a
c
88°
O
a
130°
86°
b
67°
50°
regular octagon
O is the centre
70°
Mixed questions
The next exercise contains questions which summarise the work
of the last five exercises.
Exercise 8
Find the angles marked with letters.
1.
2.
3.
68°
x
4.
42°
70°
41°
110°
y
z
59°
e
73°
5.
6.
n
7.
g
n
60°
x
52°
n
9.
10.
x
8.
a
x
70°
11.
12.
71°
60°
40°
132°
y
b
e
58°
x
e
y
50°
f 75°
Angle facts
9
13.
14.
x
15.
16.
a
36°
e
c
72°
72°
b
44°
40°
y
a
17.
18.
19.
y
50°
20.
x
58°
90°
70°
20°
85°
62°
c
66°
102°
z
x
21.
22.
y
23.
a
70°
50°
2x
72°
24.
b
x
100°
b
49°
x
y
110°
a
Questions 25 to 28 are more difficult.
68°
25. The diagram shows two equal squares joined to a triangle.
Find the angle x.
26. Find the angle a between the diagonals of the parallelogram.
x
56°
a
x
24°
27. The diagram shows the cross-section of a roof.
PQ and RS are horizontal and ST is vertical.
80°
P
Shape and Space 1
S
y
z
Work out angles x, y and z.
10
R
T
35°
Q
28. Given AB = AC and DA is parallel to EC, find x.
D
A
35°
x
B
42°
E
C
1.3 Angles in polygons and circles
Exterior angles of a polygon
The exterior angle of a polygon is the angle between a produced
side and the adjacent side of the polygon. The word ‘produced’ in
this context means ‘extended’.
p
t
q
If we put all the exterior angles together we can see that the sum
of the angles is 360°. This is true for any polygon.
r
s
The sum of the exterior angles of a polygon = 360°.
Note:
a) In a regular polygon all exterior angles are equal.
360°
b) For a regular polygon with n sides, each exterior angle =
n
t
s
r
q
p
Example
The diagram shows a regular octagon (8 sides).
a) Calculate the size of each exterior angle (marked e).
b) Calculate the size of each interior angle (marked i).
i
e
e
a) There are 8 exterior angles and the sum of these angles is 360°.
360°
∴ angle e =
= 45°
8
b) e + i = 180° (angles on a straight line)
∴ i = 135°
Angles in polygons and circles
11
Exercise 9
c
1. Look at the polygon shown.
115°
110°
a) Calculate each exterior angle.
b) Check that the total of the exterior angles is 360°.
d
2. The diagram shows a regular decagon.
a) Calculate the angle a.
94°
100° a
121°
b) Calculate the interior angle of a regular decagon.
e
a
3. Find:
a) the exterior angle
b) the interior angle of a regular polygon with
i) 9 sides
ii) 18 sides
iii) 45 sides
4. Find the angles marked with letters.
42°
78°
44°
110°
m
95°
q
p
n
50°
140°
x
5. Each exterior angle of a regular polygon is 15°.
How many sides has the polygon?
6. Each interior angle of a regular polygon is 140°.
How many sides has the polygon?
7. Each exterior angle of a regular polygon is 18°.
How many sides has the polygon?
12
Shape and Space 1
b
iv) 60 sides.
8. The sides of a regular polygon subtend angles of 18° at the
centre of the polygon.
How many sides has the polygon?
18° 18°
18°
Angles in circles
A
C
O
O
B
tangent
A
B
C
The tangent ABC touches the circle at B.
AB is a diameter.
OB is a radius of the circle.
The angle at the circumference, angle ACB, is 90°.
We can write AĈB = 90°.
Angle OBA = 90°.
Exercise 10
1. a) Draw a circle with radius 5 cm and draw any diameter AB.
b) Draw triangles ABC, ABD, ABE and measure the angles at the circumference.
D
C
?
?
A
B
?
E
In Questions 2 to 13 find the angles marked with letters. Point O is the centre of the circle.
2.
3.
4.
c
a
O
5.
O
25°
b
O
c
d
Angles in polygons and circles
13
6.
7.
50°
8.
2g
g
O
e
9.
h
h
3
O
f
O
O
i
j
41°
10.
11.
k
O
12.
30°
O
40°
13.
l
m
n + 10°
O
O
130°
n
p
1.4 Symmetry
a) Line symmetry
The letter M has one
line of symmetry,
shown dotted.
b) Rotational symmetry
The shape may be turned about O into three identical
positions. It has rotational symmetry of order 3.
O
Note:
14
●
Isosceles triangles have one line of symmetry but no rotational symmetry.
●
Equilateral triangles have three lines of symmetry and rotational symmetry of order 3.
●
Circles have an infinite number of lines of symmetry and an infinite order of rotational symmetry.
Shape and Space 1
Exercise 11
For each shape state:
a) the number of lines of symmetry
b) the order of rotational symmetry.
1.
2.
5.
6. 5
5
5
5
9.
10.
3.
4.
7.
8.
11.
12.
Exercise 12
In Questions 1 to 8, the broken lines are axes of symmetry. In each question only part of the shape is
given. Copy what is given onto squared paper and then carefully complete the shape.
1.
2.
3.
4.
5.
6.
Symmetry
15
7.
8.
9. Fold a piece of paper twice and cut out any shape from the
corner. What are the number of lines of symmetry and the
order of rotational symmetry of your shape?
cut here
1.5 Circle calculations
Circumference of a circle
The circumference of a circle is given by C = πd.
Example
Find the circumference of this circle.
A
arc
B
12 cm
O
C = π × 12 cm
C = 37.7 cm (to 3 s.f.)
We have used the π button on a calculator.
The value of π (pi) is 3.142 approximately.
X
Y
AB is an arc of the circle.
AOB is a sector of the circle.
The shaded area cut off by the
chord XY is a segment of the circle.
A semicircle is half a circle.
The plural of radius is radii.
16
Shape and Space 1
Exercise 13
In Questions 1 to 8, find the circumference. Use the π button on a calculator or take π = 3.142.
Give the answers correct to 3 significant figures.
1.
2.
3.
4.
5c
8 cm
11 cm
m
6 cm
5.
6.
7.
17 m
4.5
8.
7.1
23 m
m
cm
9. A circular pond has a diameter of 2.7 m.
Calculate the length of the perimeter of the pond.
10. How many complete revolutions does a cycle wheel of
diameter 60 cm make in travelling 400 m?
11. A running track has two semicircular ends of radius 34 m
and two straights of 93.2 m as shown.
Calculate the total distance around the track to the nearest
metre.
34 m
93.2 m
12. The minute hand of a clock is 14.4 cm long.
How far does the tip of the minute hand move between 12:00 and 12:15?
Circle calculations
17
13. An old-fashioned type of bicycle is shown.
In a journey the front wheel rotates completely 156 times.
How far does the bicycle travel?
radius
0.84 m
radius
0.2 m
14. The diagram shows a framework for a target. The radius of the
outer circle is 30 cm and the radius of the inner circle is 15 cm.
Calculate the total length of wire needed for the whole
framework.
Area of a circle
The area of a circle of radius r is given by A = πr2.
Example
Find the area of this circle.
In this circle r = 4.5 cm
9 cm
∴ Area of circle = π × 4.5
2
= 63.6 cm2 (to 3 s.f.)
Remember the formula is π (r2) not (πr)2.
On a calculator, work out the answer like this:
4.5 × 4.5 × π =
Exercise 14
In Questions 1 to 8 find the area of the circle. Use the π button on a calculator or use
π = 3.142. Give the answers correct to three significant figures.
1.
2.
11 cm
18
Shape and Space 1
3.
5 cm
4.
7m
3m
5.
6.
7.
8.
8 cm
11 cm
5m
m
12 c
9. A spinner of radius 7.5 cm is divided into six equal sectors.
Calculate the area of each sector.
10. A circular swimming pool of diameter 12.6 m is to be covered
by a plastic sheet.
3
5 1
6 2
4
Work out the surface area it must cover.
11. A circle of radius 5 cm is inscribed inside a square as shown.
Find the area shaded.
5 cm
12. Each square metre of a lawn requires 2 g of weedkiller. How
much weedkiller is needed for a circular lawn of radius 27 m?
13. Discs of radius 4 cm are cut from a rectangular plastic sheet of
length 84 cm and width 24 cm.
4 cm
a) How many complete discs can be cut out?
Find
b) the total area of the discs cut
c) the area of the sheet wasted.
14. A circular pond of radius 6 m is surrounded by a path of
width 1 m.
Find the area of the path.
Circle calculations
19
15. The diagram below shows a lawn (unshaded)
surrounded by a path of uniform width (shaded).
The curved end of the lawn is a semicircle of
diameter 10 m.
10 m
14 m
25 m
Calculate the total area of the path.
More complicated shapes
Example
For the shape below find:
a) the perimeter
b) the area.
 π × 11 
a) Perimeter = 
+ 11 + 3 + 3
 2 
= 34.3 cm (3 s.f .)
b) Area
3 cm
 π × 5.52 
= 
+ (11 × 3)
2 

= 80.5 cm 2 (3 s.f .)
11 cm
Exercise 15
Use the π button on a calculator or take π = 3.142. Give the answers correct to 3 s.f. For each shape
find the perimeter.
1.
2.
3.
4.
8m
15 cm
9 cm
5.
6.
7.
8.
8.5 m
12 m
4 cm
8 cm
20
Shape and Space 1
6m
5 cm
3.1 m
3.2 cm
9.
10.
11.
8 cm
3 cm
3 cm
7m
6 cm
4 cm
11 m
Exercise 16
Find the shaded area for each shape. All lengths are in cm.
1.
2.
6
3.
6
4
6
10
4
4
5
4.
5.
6.
8
7
4
4
12
7. a) Find the area of triangle OAD.
b) Hence find the area of the square ABCD.
D
5
c) Find the area of the circle.
d) Hence find the shaded area.
Finding the radius of a circle
Sometimes it is difficult to measure the diameter of a circle but it
is fairly easy to measure the circumference.
A
5
O
C
B
Circle calculations
21
Example
a) The circumference of a circle is 60 cm.
Find the radius of the circle.
C = πd
∴ 60 = π d
b) The area of a circle is 18 m2.
Find the radius of the circle.
π r 2 = 18
18
r2 =
π
60
=d
π
(60/π )
∴ r=
= 9.55 cm (to 3 s.f .)
2
∴
r =  18  = 2.39 m (to 3 s.f .)
π 
Exercise 17
In Questions 1 to 10 use the information given to calculate the radius of the circle. Use the π button
on a calculator or take π = 3.142.
1. The circumference is 15 cm.
2. The circumference is 28 m.
3. The circumference is 7 m.
4. The area is 54 cm2.
5. The area is 38 cm2.
6. The area is 49 m2.
7. The circumference is 16 m.
8. The area is 60 cm2.
9. The circumference is 29 cm.
10. The area is 104 cm2.
11. An odometer is a wheel used for measuring long distances.
The circumference of the wheel is exactly one metre.
Find the radius of the wheel.
12. A sheet of paper is 32 cm by 20 cm. It is made into a hollow
cylinder of height 20 cm with no overlap.
20
20
32
Find the radius of the cylinder.
13. The area of the centre circle on a football pitch is 265 m2. Calculate the radius of the circle to the
nearest 0.1 m.
14. Eight sections of curved railway track can be joined
to make a circular track. Each section is 23 cm long.
Calculate the diameter of the circle.
22
Shape and Space 1
23 cm
3
15. Calculate the radius of a circle whose
area is equal to the sum of the areas
of three circles of radii 2 cm, 3 cm and
4 cm.
2
4
16. The handle of a paint tin is a
semicircle of wire which is 28 cm long.
Calculate the diameter of the tin.
17. A television transmitter is designed so that people living inside
a circle of area 120 000 km2 can receive pictures. What is the
radius of this circle? Give your answer to the nearest km.
18. The circle and the square have the same area. Find the radius
of the circle.
r?
7 cm
19. The circumference of this circle is 52 m. Find its area.
Area ?
20. The area of a circular target is 1.2 m2. Find the circumference
of the target.
21. The perimeter of a circular pond is 85 m long. Work out the
area of the pond.
22. The sector shown is one quarter of a circle and has an area of
23 cm2. Find the radius of the circle.
23. Grass seed is sown at a rate of 40 grams per square metre
and one box contains 2.5 kg. The seed is just enough to sow a
circular lawn. Calculate the radius of this lawn to the nearest
0.1 metre.
23 cm2
r
Circle calculations
23
1.6 Arc length and sector area
r
arc l sector θ
O
θ
× 2π r
360
We take a fraction of the whole circumference depending on the
angle at the centre of the circle.
Arc length, l =
θ
× π r2
Sector area, A =
360
We take a fraction of the whole area depending on the angle at the
centre of the circle.
r
θ
O
Sector A
Example
a) Find the length of an arc which subtends
an angle of 140° at the centre of a circle of
radius 12 cm.
b) A sector of a circle of radius 10 cm
has an area of 25 cm2. Find the
angle at the centre of the circle.
12 cm
10 cm
θ
140°
25 cm2
140
× 2 × π × 12
360
28
=
π
3
= 29.3 cm (1 d.p.)
Arc length =
Let the angle at the centre of
the circle be θ.
θ
× π × 102 = 25
360
25 × 360
∴ θ=
π × 100
θ = 28.6° (3 s.f .)
The angle at the centre of the circle is 28.6°.
24
Shape and Space 1
Complete
Mathematics
for Cambridge IGCSE® Fifth Edition
David Rayner
Ian Bettison
Mathew Taylor
Core
Complete Mathematics for Cambridge IGCSE: Core directly matches the latest Cambridge IGCSE
Mathematics syllabus for first examination from 2020. The stretching, skills-focused approach
progressively strengthens student ability, enabling confident exam performance. Worked examples
and plenty of practice exercises develop thorough understanding of key concepts.
•
•
•
Fully prepare for exams – comprehensive coverage of the course
Develop advanced skills – extensive graduated practice extends performance
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Support learning with additional content
on the accompanying support site:
www.oxfordsecondary.com/9780198425045
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9780198428114
Complete Cambridge syllabus match
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ISBN 978-0-19-842504-5
9 780198 425045