Heinemann Maths Zone 9

F
or centuries ships navigators had used the

stars to determine latitude, the north
south position. The eastwest position,
longitude, was much harder to work out, and
prize money was offered to whoever could first
solve the problem. The key was knowing
exactly what time it was in Greenwich,
England, when it was midday at the ships
location. The time difference was then used in
a quick calculation to find the longitude of the
ships position. Clocks in the 1770s were not
very accurate, however, and they relied on a
pendulum action which was affected by the
motion of a ship. If a clock lost even 6 seconds
a day, this could mean that the ships position
would be out by over 700 km. After 60 years
of trying, watchmaker John Harrison was
recognised as the person who finally solved
the longitude problem.
Starter 2
65
Prepare for this chapter by attempting the following questions. If you have
difficulty with a question, click on its Replay Worksheet icon on your Student
DVD or ask your teacher for the Replay Worksheet.
Worksheet R2.1
Worksheet R2.2
Worksheet R2.3
1 Solve the following.

(a) 7.21 10
(c) 1571.23 100
(b) 18.21 1000

(d) 0.74 10 000
2 If x = 2, y = 4 and z = 5, find the value of the following.

(a) x + y
(b) xy2
(c) 4z 5y
3 (a) What is the length of AB in each of the following diagrams?

(i) A
(ii)
B
3m
5m
(iii)
B
A
1.6 mm
D 16 cm A
4.8 cm
2.9 mm
C
D
(d) z(4x + 5y)
(b) Find the perimeter of each shape in part (a).

Worksheet R2.4
Worksheet R2.5
4 Match the name to its shape.

A rectangle
B scalene triangle
D parallelogram
E trapezium
(a)
(b)
(c)
C isosceles triangle
(d)
(e)
5 Calculate the area of each of these shapes:

(a)
14 cm
(b)
20 cm
(c)
12 cm
8 cm
area
capacity
diagonal
error
kite
66
14 cm
16 cm
net
percentage error
perimeter
prism
quadrilateral
HEINEMANN MATHS ZONE 9
relative error
rhombus
trapezium
volume
02HMZVELS9EN_text Page 67 Monday, June 30, 2008 10:57 AM
Hardly a day goes by when we are not required to express

a measurement of some sort. How far is it to school? How
tall are you? What time is it? What is the distance around
the edge of an area (the perimeter)? These questions, and
many more like them, need to be answered accurately.
Consequently, a number of different measurement
systems have been developed over the years. Australia
now uses the metric system, which it adopted in the 1960s.
One of the biggest benefits of the metric system is that
it is based around the decimal system, i.e. multiples of 10.
This makes it simpler and more efficient to use than the
previous imperial system, which used many different
factors and multiples. This is very desirable as it is used in
such a wide variety of jobs ranging from dietitians needing
to measure quantities of foods in grams through to
decorators needing to know wall measurements in metres
so that the correct amount of paint can be bought.
Metric units of length

The basic unit of length is the metre. Although there are more than half a dozen
possible multiples of a metre that can be used to measure length, there are four
that are used most often:
millimetre (mm)
centimetre (cm)
metre (m)
kilometre (km)
very small measurements, such as the length of an ant.

small to medium measurements, such as the width of this page.
most standard measurements, such as the length of the room.
large distances, such as the length of the journey to the city.
The use of appropriate units avoids very large or very small numbers.
It is important to be able to convert one unit to another. This is easy,
because all metric units are based around multiples of 10. The basic conversion
rates are:
10 millimetres = 1 centimetre
100 centimetres = 1 metre
1000 metres = 1 kilometre
1000
km
100
10
m cm mm
1000
100
10
MEASUREMENT
67
worked example 1
Fill in the following expressions:
(a) 40 mm = ____ cm
(b) 2.73 m = ____ mm
Steps
Solutions
(a) 1. Decide on the type of conversion.

Smaller to larger.
(a)
cm mm
>
>
exercise 2.1
>
(b)
2. Multiply by 1000 and state the answer. Note

that the decimal point moves three places to
the right.
mm
40 10
=4
2. Divide by 10 and state the answer. Note that

the decimal point moves one place to the left.
(b) 1. Decide on the type of conversion.
Larger to smaller.
cm
2.73 1000
= 2730 mm
Units of length
Skills
1 Fill in the gaps:

(a) 30 mm = ____ cm
(d) 14 cm = ____ mm
(g) 78 cm = ____ m
(b) 85 m = ____ cm
(e) 6400 m = ____ cm
(h) 6.71 m = ____ cm
(c) 7.5 km = ____ m

(f) 2.8 cm = ____ mm
(i) 9.5 km = ____ m
e
e
e
e
Worked Example 1
Hint
Interactive
Worksheet C2.1
2 Suggest the most appropriate units for measuring each of the following
e Worksheet C2.2
lengths.
(a) diameter of a compact disc (b) height of a Christmas tree (c) height of the Eiffel Tower
68
HEINEMANN MATHS ZONE
(d) distance to Sydney

from Melbourne
(e) width of a fingernail
(f) distance you can kick a

football
Sydney
Melbourne
3 All metric units have a prefix that indicates size. Match the prefix with the
correct number.
(a) kilo
(b) centi
(c) deci
(d) mega
(e) milli
(f) deca
(g) hecto
(h) giga
1
A 100
B -----C 1000
D 1 000 000 000
10
1
1
--------E
F 10
G -----------H 1 000 000
100
1000
Applications
4 An ultramarathon runner is completing 20 laps per hour of a track 220 m
long. The distance that he will run in one day (24 hours) is:
A 52.8 km
B 74.3 km
C 105.6 km D 10.56 km E 1056 km
5 A rectangular swimming pool is
surrounded by a path which is 1 m wide.
It is 15 m long and 10 m wide.
The perimeter of the swimming pool is
A 36 m
B 42 m
C 46 m
D 50 m
E 52 m
6 The perimeter of this composite
figure is
A 90 mm
B 152 mm
C 180 mm
D 212 mm
E 252 mm
Hint
15 m
10 m
26 mm
16 mm
3 cm
20 mm
6 cm
7 Two pieces of wood are 2.4 m and 50 cm long. Their average length is
A 37 cm
B 1.2 m
C 1.45 m
D 2.62 m
E 2.9 m
8 How many laps of a 50 m pool would you need to swim in order to
cover 2.5 km?
A 20 laps
B 30 laps
C 40 laps
D 50 laps
E 60 laps
2
MEASUREMENT
69
9 Elizabeth is 1.75 m tall and her friend Sharon is 187 cm tall. How much
taller is Sharon than Elizabeth?
Hint
10 The maker claimed that my new pen could write an unbroken line 3 km
long. I tested this and wrote 15 m per minute until it stopped working after
2 1--2- hours. Is their advertising claim correct?
Hint
Hint
11 The local fabric shop is selling a new fabric for $18.50 per metre.
How much will it cost to buy 140 cm?
12 What is 35 mm film and why is it called this?
12
Analysis
13 An electric light pole is to be

erected. To make sure that it
does not fall over, a hole is
1
dug and --- of the pole is
4
concreted below ground
3
level, leaving --- of it standing
4
above the ground.
(a) If the part above the
ground is 9 m tall, what
is the total length of the
pole?
(b) How many centimetres
of the pole is below the
ground?
9m
xm
Pamelas fence
Pamela has 100 m of fencing material with which to build a fence around a rectangular or
square paddock. She does not know what area the paddock should be.
1. Work in groups to find the areas of five different paddocks Pamela could enclose.
2. What is the largest area Pamela could enclose? Give the dimensions of this paddock.
3. The fence requires a post every 5 m. Assuming the posts are evenly spaced, how many fence
posts will Pamela need?
4. Pamela is thinking of using an existing fence along one side of her paddock. Give new
answers for parts 1, 2 and 3 that take this into account.
70
From fingers to feet to metres
The Colosseum, the most famous monument of ancient Rome, was constructed between 70
and 80 AD. It was the largest of many arenas built to stage fights to the death between wild
animals and between slaves and criminals trained to be gladiators. For centuries men lost
their lives and animals were slaughtered, all for entertainment.
The Colosseum is extremely large and is elliptical in shape, 86 metres long and 54 metres
wide. It has a perimeter of 527 metres and is 50 metres high. Only skilled architects and
engineers could have drawn up the detailed plans required for such a building. But in 70 AD
the metric system did not exist.
Units of measurement have changed greatly over time. We now use a metric system, but
older books sometimes refer to imperial measurements, a system still used in some Englishspeaking countries including the USA. As far back as 2500 BC the Babylonians, who lived in
Mesopotamia, an area that is now shared between eastern Syria, south-eastern Turkey and
most of Iraq, standardised their weights and measures. This was possibly a move designed
to eliminate tension and conflict in the marketplace.
In the Babylonian system, the smallest unit of length was the finger, about 1.7
centimetres. Other units were the cubit, which was 30 fingers, the cord (surveyors rope) of
MEASUREMENT
71
120 cubits or 3600 fingers and the league, of 180 cords. The smallest unit of weight was the
grain, about 45 mg. The shekel was 180 grains and the talent was about 3600 shekels.
Most of these units come from body parts or other things that occur in nature. The talent
is meant to be the weight that a man can ordinarily carry at each end of a carrying yoke. It
is the mass of about 30 litres of water, about 30 kilograms.
The diagram shows some Babylonian and imperial units of measurement.
b
e
a = 1 cubit = 1 1--2- feet

b = 1 hand = 4 inches
c = 1 digit = --34- inch
d = 1 fathom = 6 feet
e = 1 palm = 3 inches
The commonly used British (imperial) foot is derived originally from Egypt. But even the
foot is not as uniform a unit as we would expect.
For each of the following countries the foot represents a different length.
British foot
304.8 mm
Canadian (French) foot
325.1 mm
Amsterdam foot
283.1 mm
South African foot
313.8 mm
The metric system was designed to simplify calculations but there is constant debate over
whether it is actually a better system. One major argument for the metric system is that
some calculations are easier, especially those dealing with extremely small or extremely
large values. Metric relationships are also much easier to remember.
An argument against the metric system is that unlike units in the imperial system, which
had some physical origins and meaning, the metre is an arbitrary length that doesnt relate
to any real measurement. But most countries are gradually adopting the metric system.
When you are describing yourself to a penpal overseas, it would be nice to be able to tell
them your height and have your penpal understand what you mean.
72
Questions
1 Define a talent and its relation to a natural measurement.
2 Which nation of those listed has the longest foot?
3 Convert the following units. Use the British foot. Round to two decimal places where
necessary.
(a) 1 cord = _____ feet
(b) 2 leagues = _____ cubits
(c) 2 talents = _____ shekels
(d) 4 feet = _____ cm
(e) 1 fathom = _____ m
(f) 5 cubits = _____ m
4 Ignoring the arches, what would the area of the outside surface of the Colosseum have
been?
5 The area of an ellipse is given by ab, where a is the distance from the centre of the
ellipse to an end of the ellipse, called a semi-major axis of length, and b is half of the
distance across the ellipse, the semi-minor axis of length.
b
a
Calculate the area of land taken up by the Colosseum, in square metres correct to two
decimal places.
6 How many:
(a) (i) cubits in a cord
(ii) fingers in a cord
(iii) cords in a league
(iv) grains in a shekel
(v) shekels in a talent?
(b) What is the highest common factor of your answers to part (a)?
(c) Why might it be more convenient to work with multiples of this number than
multiples of ten, as in the metric system?
(d) Give at least one example of a measuring system we use that deals with multiples
of this number.
Research
Prepare a poster, a web page using software such as FrontPage or Netscape Communicator
or a computer presentation using software such as PowerPoint or Presentation about the
Babylonians. In particular research and report on their number system and/or measurement
system. Discuss the advantages and disadvantages of the systems used.
MEASUREMENT
73
The accuracy of any measurement that is made depends on a number of

factors, such as any assumptions made, the preciseness of any equipment used,
and human error. Some error will always be present. An important
consideration when recording measurements is the appropriateness of the
units used in the answer.
Approximation and rounding

Does it really matter if we report that 34 000 people attended
the Australian Tennis Open at the Rod Laver Arena if in fact
there were 34 185 people? Rounding numbers like this is used
to conveniently present information to people. We use
rounding all the time when measuring. For example, saying
that it is approximately 12 km from the city to home is
considered reasonable. People would consider you quite
strange if you said that you lived 11 753 m from the city! Julie
is about 170 cm tall and Hilul weighs about 65 kg are
considered quite reasonable statements.
Errors
The accuracy or exactness of a measurement is indicated by the number of
significant figures. All the figures in a measurement are accurate, or reliable,
until the last decimal place. This last figure is uncertain; that is, we dont know
what comes after it and the real value of the quantity being measured could be
a little above or below the value indicated by this last decimal place.
In really precise scientific work, measurements need to be accurate, and are
given, to many decimal places. For example, the metre is defined as the
distance light travels in 1/299 792 458 second.
Most everyday measurements are not this accurate. For example, Rene
may say that she is 182 cm tall. She has rounded off her height to the nearest
centimetre. This is really saying that she is between 181.5 and 182.5 cm tall.
This accuracy is reasonable.
However, suppose Rene had said she was 182.4 cm tall. This means that
her height was between 182.35 and 182.45 cm, and that she had measured it
accurately to the nearest millimetre. Does this sound reasonable? Could you
accurately measure your height to the nearest millimetre?
When we said that Renes height of 182 cm meant that she was really
between 181.5 and 182.5 cm tall, we added or subtracted 5 from one more
decimal place than was given in the measurement. That is the size of the error
or uncertainty implied when no information to the contrary is given.
74
Sometimes it is important to specify the size of the uncertainty in the last

decimal place. For Rene we can say that her height is 182 0.5 cm. The 182
represents the measurement and the 0.5 is the error.
Relative errors
Relative and percentage errors allow the significance of errors to be
calculated by comparing the size of the error to the actual measurement.
Error
Relative error = -------------------------------------------------Measurement value
Error
100
Percentage error = -------------------------------------------------- --------- %
Measurement value
1
worked example 2
(a) Calculate the relative and percentage errors for the following measurements, correct to
one decimal place.
(i) 20 1 cm
(ii) 18.2 0.4 m
(b) Which is the proportionally larger error?
Steps
Solutions
(a) (i) 1. State the formula.
Error
(a) (i) --------------------------------------------Measurement value
2. Substitute the error and measurement

values into the formula. This is the relative
error.
1
= ----20
3. Convert to percentage error.
1
100
= ----- -------- %
20
1
= 5.0%
(ii) 1. State the formula.

2. Substitute the error and measurement
values into the formula. This is the relative
error.
3. Convert to percentage error.
(b) Compare the percentage errors.
Error
(ii) --------------------------------------------Measurement value
0.4
= ---------18.2
0.4
100
= ---------- -------- %
18.2
1
= 2.2%
(b) Error in part (i) is 5.0%, error in
part (ii) is 2.2%. Therefore the
error in (i) is proportionally
larger.
MEASUREMENT
75
Addition of quantities involving

errors
When measurements with errors are added, it makes sense that the resulting
answer will also be subject to error.
To find the range of results possible when 11 0.2 cm is added to
15 0.4 cm, find the maximum result first by adding the measurements with
their errors added.
11.2 +15.4 = 26.6
Then find the minimum result possible.
10.8 + 14.6 = 25.4
The range of results is 25.4 cm to 26.6 cm.
25.4 + 26.6
Note that the average result is --------------------------- or 26. This can be obtained
2
quickly by just adding together the two measurement results from the
question. The error is 0.6, which could also be found just by adding together
the respective errors from the question.
worked example 3
Find the result when 11 0.2 cm is added to 15 0.4 cm.
Steps
1. Set up a sum adding the two measurements,
with their errors, together.
Solution
2. Rearrange so the errors are grouped together.

3. Add the measurements together and the
errors together.
= (11 + 15) (0.2 + 0.4)

= 26 0.6 cm
(11 0.2) + (15 0.4)
When measurements are added the errors are added together.

(a b) + (c d) = (a + c) (b + d)
Multiplication of quantities with

errors
When multiplying quantities involving errors, you need to calculate both a
maximum and minimum answer and, from there, determine an average result.
You will then be able to work out the degree of error from your answers.
worked example 4
Calculate the area of a rectangle with a length of 12.5 0.1 cm and a width of
6.3 0.1 cm. Give your answer correct to two decimal places.
76
Steps
1. State the formula.
2. Add the respective errors to the measurements.
3. Multiply the values for l and w obtained in step 2.
This represents the largest possible area.
4. Subtract the respective errors from the
measurements.
5. Multiply the values for l and w obtained in step 4.
This represents the smallest possible area.
6. Calculate the average result.
7. Calculate the error by finding the difference
between the smallest and largest areas and the
average result.
8. State the answer.
Solution
A =lw
l = 12.5 + 0.1 = 12.6
w = 6.3 + 0.1 = 6.4
A = 12.6 6.4
= 80.64 cm2
l = 12.5 0.1 = 12.4
w = 6.3 0.1 = 6.2
A = 12.4 6.2
= 76.88 cm2
80.64 + 76.88
------------------------------- = 78.76 cm2
2
80.64 78.76 = 1.88
78.76 76.88 = 1.88
The area of the rectangle is
78.76 1.88 cm2.
When multiplying quantities with errors, the size of the error cant be worked out until
the average of the maximum and minimum values is known.
In Worked Example 4, we were asked to give the answer to two decimal places,
but is this sensible? It is impractical for a person to measure such a minute area.
Imagine how small 0.01 cm2 is! It would be more sensible to round the answer
to 78.8 1.9 cm2 (which means that the area lies between 76.9 and 80.7 cm2)
or even to 79 2 cm2 (which means that the answer lies between 77 and
81 cm2).
If the question doesnt specify the accuracy or number of decimal places
required in the answer, try to consider how many decimal places would be
meaningful. Recall that all measurements contain errors, even if these are not
stated. When measurements are added, answers should not contain more
decimal places than the original measurements. When measurements are
given in whole numbers we usually give the answer in whole numbers, even
if this accuracy is not really justified.
Do not round off answers until all the calculations are complete.
MEASUREMENT
77
exercise 2.2
Errors and approximation
Skills
1 State the range of values possible for the following measurements.

(a) 76 0.4 cm
(b) 1.57 0.04 km
(c) 19.6 0.3 m
(d) 7.64 0.05 cm
(e) 12.8 0.5 m
(f) 1.09 0.001 km
2 Find (i) the relative error, to three decimal places, and (ii) the percentage
error to two decimal places, for the following measurements.
(a) 15.12 0.007
(b) 2518 3.5
(c) 0.0432 0.000 05
(d) 578 5
Hint
Worked Example 2
Hint
3 Find the range of results possible when the following lengths are added. e
(a) 29 1 m and 55 0.5 m
(b) 15 0.5 cm and 33 0.4 cm
e
(c) 485 7 cm and 1.71 0.09 m
(d) 1750 10 m and 3.01 0.005 km
4 Find the distance around each of the following shapes, giving your
answers in the form a b.
(a) A square whose length is 25.6 0.4 cm.
(b) A rectangle with length 40 0.3 m and width 35 0.2 m.
(c) An equilateral triangle with side length 235 2.5 mm.
5 A rectangle has a length 50 0.4 cm and width 40 0.8 cm. Find
its area.
6 Fill in the missing amounts.
(a) A boat that is said to be 15 m long could be between 14.5 m
and _____m.
(b) A pencil that is said to be 8 cm long could be between
7.5 cm and _____cm.
(c) The diameter of a circle written as 75 mm long could be between
_____mm and 75.5 mm.
(d) An electrical cable said to be 76.5 m long could be between
______m and _____m.
7 A potato chip manufacturer produces 125 g bags of chips. If they will
accept all bags subject to an error of 4%, the range of weights possible is:
A 120 g130 g
B 125 g130 g
C 120 g125 g
D 122.5 g127.5 g
E 118.5 g131.5 g
8 (a) A rectangular lawn measures 26 m long and 18 m wide. The error in
measuring the length is 0.6 m, the widths error being 0.5 m. The
distance around the lawn could be stated as:
A 44 1.1 m
B 88 2.2 m
C 88 0.1 m
D 90 0.1 m
E 90 0.2 m
(b) This same rectangle as in (a) has an area of:
A 468.3 23.8 m2
B 467.7 2.2 m2
C 499.7 39.7 m2
2
2
D 468.0 0.3 m
E 468.0 2.2 m
78
e
e
Worked Example 3
Hint
Worked Example 4
Hint
Applications
9 A rectangular sheet of silver foil with

the guaranteed dimensions shown is
used in a jewellery shop. Find:
(a) the relative error of each side
(b) the percentage error of each side
(c) the area of the sheet of foil in the
form a b.
Worked Example 4
Hint
Homework 2.1
3.2 0.2 cm
5 0.2 cm
10 Marnie and Cam have just finished making

8.4 0.1 m
a new garden bed with the measurements
shown.
5.8 0.1 m
(a) What is the maximum and minimum
perimeter of this garden bed?
(b) Correct to one decimal place, what is
the percentage error in the perimeter?
(c) What is the maximum and minimum area of the garden bed?
(d) Marnie decided that she needed to put better quality soil onto the
garden. Green Thumb Nursery said that they could provide bags of
soil that would give her the required depth and would cover
approximately 0.6 m2. How many bags should she buy to be sure of
having enough?
11 A can of paint can cover 54 m2 of wall space.
(a) How many cans are needed to paint a square windowless shed with
floor dimensions of 6.2 0.1 m and a height of 3.5 0.1 m? Assume
that the door and the four walls are to be painted.
(b) After finishing the shed, there was one tin of paint left. If it was to
be used on fences that are 1.6 0.1 m high, what is the maximum
and minimum length of the fence that could be painted?
12 A rectangular lawn has a length of 45 1.1 m and width 29 0.9 m.
(a) Calculate its area.
(b) How many 10 kg boxes of lawn seed are required to grass this lawn
if 1 kg of seed covers 4 m2?
13 Measure the width of the pen that you are using. Give your answer in
13
millimetres correct to two decimal places.
Analysis
14 The true speed of a car is 80 km/h but the speedometer is inaccurate and
shows it as 78 km/h.
(a) Find the percentage error of this reading.
(b) What speed would the speedometer show if this percentage error
was added to 80 km/h?
(c) Is it better for the speedometer to register too high a reading or too
low? Explain why. The Australian standard is 10%.
MEASUREMENT
79
Area is the amount of space contained inside a plane (flat) shape. Many areas
that you are asked to find are based on rectangles, triangles and circles.
Sometimes you are also required to find the area of a semicircle or a quadrant
(quarter of a circle).
The necessary formulae to remember are:
Shape
Rectangle
Diagram
Formula
A = lw
w
Triangle
A = 1--2- bh
h
b
Parallelogram
A = bh
h
b
Circle
r
A = r 2
When solving an area problem, follow this sequence of steps:

1 Draw a picture of the situation (if not provided).
2 Draw a diagram using the correct shapes.
3 Divide the diagram into its component rectangles, triangles and circles.
4 Write out the formula appropriate to each part.
5 Find the area of each part then find the total or difference to obtain
the answer.
6 Give your answer in the context of the question.
80
worked example 5
Find the area of each shape. Where any shading is shown, find the shaded area. Give
answers to the nearest cm2.
(a)
(b)
15 cm
8 cm
30 cm
15 cm
40 cm
Steps
(a) 1. Break the shape up into parts. Because
areas 2 and 3 are both semicircles with
the same radius, we can calculate them
as one circle.
2. Write out the appropriate formula for
each part and find the areas.
3. Find the total area.
(b) 1. Identify the shapes.

2. Write out the appropriate formula for
each part and find the areas.
Solutions
(a)
15 cm
2
8 cm
Area = lw
= 15 8
= 120
Area 2 + Area 3 = r 2
= 42
= 50.265 48 ...
Total area = 120 + 50.265 48 ...
= 170.265 48 ... cm2
170 cm2
(b) The shape is made of a triangle and a
semicircle.
Area of the triangle = 1-2- bh
= 1-2- 40 30
= 600
Area of the semicircle = 1-2- r2
= 1-2- 152
3. Find the difference between the areas.
= 353.429 17...
Shaded area = 600 353.429
= 246.571 cm2
= 247 cm2
Do not round your answers until after you have added or subtracted the areas. Otherwise
your result will be less accurate.
2
MEASUREMENT
81
worked example 6
Andrea is painting the wall (not the door) of her bedroom.
If each litre of paint covers eight square metres, how much
paint will she need?
4m
1m
3.5 m
2m
Steps
1. Draw a diagram of the area to be found
and divide it into sections.
Solution
4m
1.5 m
1
1m
3.5 m
2
3m
2. Find the area of each part.
Area 1: A = lw
= 1 1.5
= 1.5 m2
Area 2: A = lw
= 3 3.5
= 10.5 m2
3. Find the total area.
Total area = 1.5 + 10.5 = 12 m2

Paint required = 12 8 = 1.5
Andrea needs 1.5 litres of paint.
4. Calculate the amount of paint required.
Can you see another way of working out the required area? Try it.
exercise 2.3
Area of composite figures
Skills
1 Find the area of each of the following shapes. Where a shaded area is
shown, find the shaded area. Give answer correct to the nearest whole
number.
(a)
(b)
20 cm
50 cm
30 cm
25 cm
50 mm
5 cm
e
e
e
(c)
40 mm
30 cm
82
eTutorial
30 mm
eTester
Worked Example 5
Hint
(e)
(f)
5 cm
15
cm
(d)
8 cm
20 cm
15 cm
(g)
20 cm
10 cm 15 cm
(h)
16 m
6m
20 cm
45 cm
16 m
8m
15 m
2 (a) The shaded area of this shape is:

A 827 cm2
B 1293 cm2
C 1827 cm2
D 293 cm2
E 1707 cm2
(b) The area of this shape is:
A 5398 cm2
B 8827 cm2
C 6514 cm2
D 9727 cm2
E 7414 cm2
(c) The shaded area of this shape is:
A 218 cm2
B 273 cm2
C 350 cm2
D 231 cm2
E 616 cm2
30 cm
Hint
Worked Example 6
40 cm
50 cm
70 cm
60 cm
100 cm
14 cm
Applications
3 A garden lawn has two ponds within its

boundary.
(a) Find the area of the lawn.
(b) How much seed is needed to grass
the lawn if 1 kg of seed covers 8 m2?
60 m
32 m
8m
8m
MEASUREMENT
83
4 Circular plastic drink coasters 8 cm wide are manufactured by

feeding a rectangular strip of plastic 9 cm wide into a machine
that cuts out the coasters. The machine starts one centimetre
from the end and leaves one centimetre between coasters. For
this machine, answer the following questions. Remember that
one centimetre is left at the end.
(a) How long would a strip of plastic have to be to cut out
50 coasters?
(b) What would be the total area of wasted plastic?
(c) What percentage of the strip is used for making coasters?
Hint
Hint
5 An architect is planning a decorative circular courtyard, 22 m in

diameter. The shaded area is to be a garden and the rest
pavement.
(a) Find the area of the pavement, correct to one decimal
place.
(b) Find the area of the garden, correct to one decimal place.
(c) What percentage of the courtyard will be pavement?
66 Steves Pizza shop sells pizza which are 14 cm in radius. He decides to stop
making them circular and instead make them rectangular. He uses
approximately the same amount of ingredients and makes them the same
thickness. Assuming that the length and width of the rectangular pizza are
in whole centimetres, write down at least three pairs of dimensions that
are practical.
Analysis
7 Touching circles are drawn in two

squares with side length 12 cm. Four
identical circles (called a 4-Pak) just fit
into the first and nine (called a 9-Pak)
fit into the second.
(a) What is the radius of the circle in:
(i) the 4-Pak (ii) the 9-Pak?
(b) Which shaded area in the background is the bigger?
12 cm
12 cm
e
e
eQuestions
Homework 2.2
The area of Australia

Calculate the area of Australia using a map from an atlas. Compare your answer
with the actual area.
Break it into a series of smaller steps
84
We have seen that shapes based on combinations of rectangles, circles and

triangles can have their area found by breaking them down into parts and
working out the areas of the parts separately. Some shapes have their own
formula which has been derived from those for rectangles and triangles.
Three such shapes are the trapezium, kite and rhombus, all quadrilaterals
(four-sided figures).
Trapezium
A trapezium is a quadrilateral with exactly one pair of parallel sides. If we label
the length of the top a, the length of the base b and the height h, then the
trapezium can be shown to be made up of two triangles. Each triangle has
height h, one triangle has a base length of a and the other of b.
Area of top triangle =
1
--2
ah
Area of bottom triangle =
1
--2
bh
Area of trapezium = 1--2- ah + 1--2- bh

= 1--2- h (a + b)
The high
trapezium
Kite
A kite consists of two congruent triangles. The pairs of equal sides are
adjacent to each other. The area of the kite will be twice the area of one of
the triangles. Remember also that the diagonals of a kite are
perpendicular.
Area of one triangle =
1
--2
base height of one triangle
C
y units
x
y --2
xy
= ----4
=
1
--2
xy
Area of kite = 2 ----4
xy
= ----2
D
x units
Rhombus
x units
Because a rhombus has all four sides equal, we can think of it as a

special kite. This means that we can use the same formula for the
area of a rhombus as we do for a kite.
xy
Area of rhombus = ----2
y units
MEASUREMENT
85
worked example 7
Find the areas of these shapes.
(a)
6 cm
(b)
(c)
8 cm
9m
14 cm
Steps
(a) 1. Write the appropriate formula.
2. Substitute the measurements and
calculate the area.
Solutions
(a) A = -12- h ( a + b )
=
=
1
--2
1
--2
8 (6 + 12)
8 18
= 72 cm2
(b) 1. Write down the appropriate formula.
calculate the area.
(c) 1. Write down the appropriate formula.
calculate the area.
xy
(b) A = ----2
14 21
A = ---------------2
= 147 cm2
xy
(c) A = ----2
18 9
A = ------------2
= 81 cm2
Area of a trapezium:
A = 1-2 h (a + b), where a and b are the lengths of the parallel sides and h
is the perpendicular height.
Area of a kite or rhombus:
xy
A = ------ , where x and y are the lengths of the two diagonals.
2
86
18 m
21 cm
12 cm
exercise 2.4
Areas of quadrilaterals
Skills
1 Find the area of each of these shapes.

(a)
(b)
(c)
Worked Example 7
Hint
10 mm
16 mm
18 cm
e
e
10 mm
eTester
18 mm
15 mm
13 cm
(d)
(e)
7 cm
(f)
20 cm
6 cm
5m
3m
15 cm
24 cm
(g)
(h)
(i)
7 cm
4 cm
20 mm
5 cm
4 km
6 km
13 mm
(j)
(k)
5 mm
12 mm
9 mm
(l)
11 m
24 mm
20 m
24 m
30 mm
2 (a) A trapezium with perpendicular height 8 cm and parallel sides

measuring 10 cm and 18 cm has an area of:
A 162 cm2 B 72 cm2
C 224 cm2 D 180 cm2 E 112 cm2
(b) A rhombus with side length 20 cm has diagonals 24 cm and 32 cm
long. Its area is:
A 384 cm2 B 400 cm2 C 240 cm2 D 560 cm2 E 768 cm2
2
MEASUREMENT
87
(c) A kite with diagonals 85 cm and 115 cm long has an area of:
A 200 cm2
B 10 000 cm2
C 9775 cm2
2
2
D 4887.5 cm
E 40 000 cm
Applications
3 Anthea is laying slate tiles which

are generally shaped as shown.
The last line of tiles would
require cutting so that they
could fit.
(Note: 1 m = 100 cm, but 1 m2 =
100 cm 100 cm = 10 000 cm2.)
(a) How much area does each
tile cover?
(b) How many tiles will she
need to pave a rectangular
area with dimensions 3 m
by 2 m?
20 cm
25 cm
30 cm
4 A square has diagonals that are 20 cm long.

(a) By treating the square as a rhombus, find its area.
(b) Calculate the length of the sides of the square from the area you
found, correct to one decimal place.
Hint
5 The doors at the local church are shaped as

shown. What is the area of each door?
Hint
Hint
1.0 m
4.6 m
6 A trapezium with an area of 900 cm2 has

parallel sides 40 cm apart. If one of these sides
is twice as long as the other, the lengths of the
4.1 m
parallel sides would be:
A 20 cm, 10 cm B 30 cm, 15 cm
C 24 cm, 12 cm D 40 cm, 20 cm
2.0 m
E 50 cm, 25 cm
77 A kite has an area of 160 cm2. Write down three possible pairs of lengths
(in whole centimetres) for the two diagonals.
Analysis
8 Omar has to design a kite that has

an area of close to 2 m2. He knows
that for the kite to fly well the
length of one diagonal has to be
2.5 times the length of the other
diagonal. How long does he need
to make each of the diagonals of
his kite? Write answers in metres
correct to two decimal places.
88
9 Sian has designed a garden for a trapezium-shaped block of land, as

shown in the diagram. She has decided to put in four circular flower beds,
each with a diameter of 4 m, around a central bed in the shape of a
rhombus with diagonals of 6 m and 8 m. She wants to put turf in the
areas not covered by garden beds. Calculate the area of turf Sian needs
for the job.
Hint
25 m
8m
6m 4m
22 m
35 m
10 Maria designs swimming pools.

50 m
One of her tasks is to calculate how
1.2 m
much it costs to tile the sides of a
pool. Her latest job is to calculate
2.4 m
the cost of tiling one side of an
Olympic-sized pool that is 1.2 m
deep at one end and 2.4 m deep at
the other. The length of an
Olympic pool is 50 m.
(a) Calculate the area of one side of the pool.
(b) If it costs $23 to tile 1 m2, how much will it cost to tile one side of
the pool?
e
e
Worksheet C2.3
Restarter 2
MEASUREMENT
89
Answer the questions, showing your working, then arrange the letters in the order shown by the
corresponding answers to find the cartoon caption.
The area, in m2, of a rhombus with diagonals of 4.5 m and 3.6 m.
A
The area, correct to one decimal place, of a circle with a radius of 7 cm.
B
The circumference of a circle of radius 15 cm to one decimal place.
E
2
Another diagonal of a rhombus with area 2400 cm and one diagonal 80 cm.
H
The area of the kite in this diagram. M
The perimeter, correct to one decimal
O
place, of a semicircle of diameter 20 cm.
23.5 cm
40.2 cm
The area of the trapezium in this

1.5 m
diagram.
2.7 m
3.4 m
8.1
8.1
90
4.575
4.575
94.2
60
51.4
8.1
The radius of a circle whose

circumference is 550 cm.
472.35 153.9
The area of a square that has diagonals

of 35 cm.
87.5
612.5
Do-it-yourself netball court

Investigating and designing
A school decides to make a large asphalt area that is 30.2 m by 32 m into a netball court.
However, the school cannot afford to pay a tradesperson for the line marking, so they decide to
do it themselves using the following diagram as a basis.
30.6 m
0.9 m
goal
circle
9.8 m
15.2 m
centre
circle
1 What is the total length of the lines that they need to paint (correct to one decimal place)?
2 A tradesperson at the school worked out that a small can of paint could cover approximately
4.4 m2. If each line is 7 cm wide, how many cans of paint would they need?
A physical education teacher really wants two equal sized courts, with dimensions adjusted
slightly to make them fit.
3 Design the courts with 1 m between the edge of the asphalt and the courts and 2 m between
the actual courts. Leave the centre circle the same and reduce the goal circles to 9 m diameter.
Producing
4 How many cans of paint would now be needed to do the line marking?
5 A person who plays goal attack is allowed to go anywhere in zones 1, 2 and 3. What area is
this? A centre player can go to any position on the court except zones 1 and 5. Find this area.
Who covers more area, and by how much?
Analysing and evaluating
6 Assuming that all netball courts must be divided into three equal parts, how much less space
would be available to the goal attack and centre players with the new court dimensions?
7 To assist the school council in deciding whether to have one or two courts, discuss the
positives, negatives and other considerations of the options.
MEASUREMENT
91
The total surface area (TSA) of a solid is the combined area of all external
surfaces.
Some solids have surfaces that are combinations of rectangles only. Others
have surfaces that are the combination of triangles, trapeziums or other
shapes.
To find the total surface area of a solid you find the areas of
each separate face and then add them together. On many
occasions opposite faces have the same area. This is
especially so with prisms. A prism is a three-dimensional
object that has a uniform polygon cross-section.
All of the figures above are examples of prisms. Prisms are
named according to the cross-sectional shape. The left-hand
figure above is an example of a rectangular prism (also called
a cuboid). The middle figure is a triangular prism (also called
a wedge) and the right-hand figure is a trapezoidal prism.
worked example 8
Find the total surface area of these prisms.
(a)
(b)
2
8 cm
3
15 cm
4m
5m
20 cm
3.5 m
3m
92
Steps
(a) 1. Find the area of each face.
2. Add areas to find total surface area.

Each of areas 1, 2 and 3 is repeated on
the opposite side of the solid, therefore
you double each area.
(b) 1. This triangular solid has been folded out
to form the net.
Solutions
(a) Area 1: Front face
A = lw
= 20 8
= 160 cm2
Area 2: Top face
A = lw
= 20 15
= 300 cm2
Area 3: Side face
A = lw
= 15 8
= 120 cm2
TSA = 2 160 + 2 300 + 2 120
= 320 + 600 + 240
= 1160 cm2
(b)
4m
4m
3m
5m
3.5 m
2. Write the formula and calculate each

part separately.
Triangles:
A = 1-2- bh
=
1
-2
43
= 6 cm2
For two triangles area, 2 6 = 12 cm2
Rectangles:
There are three rectangles. One for the
base, the vertical side and the slant
top.
A = lw (for each separate rectangle)
= 4 3.5 + 3 3.5 + 5 3.5
= 14 + 10.5 + 17.5
= 42 cm2
3. Add the areas together.
TSA = 12 + 42
= 54 cm2
MEASUREMENT
93
exercise 2.5
Total surface area
Skills
e
e
1 Find the surface area of these rectangular prisms.

(a) length = 8 m; width = 6 m; height = 3 m
(b) length = 25 cm; width = 10 cm; height = 4 cm
(c) length = 2 m; width = 85 cm; height = 90 mm
Worked Example 8
Hint
2 (a) Accurately draw the net of a cube of side length 3 cm.

(b) What is its total surface area?
3 Find the total surface area of the following shapes.
(a)
(b)
e
e
9cm
5 cm
Hint
Worksheet C2.4
10 cm
20 cm
3cm
1 cm
(c)
(d)
26 cm
(e)
11 mm
17 mm
5 cm
8 mm
24 cm
15 mm
10 cm
(f)
22 cm
12 cm
(g)
(h)
13 cm
20 cm
5 cm
40 cm
12 cm
15 cm
10 cm
5 cm
8 cm
24 cm
20 cm
15 cm
10 cm
4 (a) The total surface area of a cube with length 10 cm is:

(b) The total surface area of a closed rectangular box with dimensions
40 cm, 30 cm and 20 cm is:
(c) The area of the tarpaulin needed to
cover the exposed faces of a
rectangular haystack that is 5 m
wide, 12 m long and 10 m high is:
A 460 m2
B 600 m2
C 380 m2
D 445 m2
E 400 m2
94
12 cm
30 cm
Applications
5 What is the minimum amount of paper

needed to wrap up a present shaped as
shown? (Assume no overlapping.)
6 Joshua is trying to sell a new brand of
chocolate and he has to make a new
box to put it in. Assuming there is no
overlap, which of these designs uses
the least cardboard?
Type 1
40 cm
25 cm
Animation
Hint
Hint
Hint
35 cm
Type 2
5 cm
4 cm
20 cm
30 cm
3 cm
4 cm
7 The rectangular piece of cardboard on the left has four squares cut from
the corners to make an open box whose dimensions are
30 cm 20 cm 15 cm.
15 cm
20 cm
30 cm
(a) What are the length and width of the piece of cardboard?
(b) How long is the side of the square?
8 What is the effect on the total surface area of a cube if you double its
length?
9 A closed rectangular box has a total surface area
960 cm2. It has a square end, as shown. Each side
is a whole number of centimetres. Find two
different sets of dimensions that satisfy these
conditions.
Analysis
10 Leonardo has been hired to paint the room of a house, represented

by the figure. The room has dimensions 8 m 5 m 3 m and it has
one window 2 m 1.5 m as well as two doors 0.9 m 2.3 m each.
Find each of the following.
(a) The total surface area inside the room.
(b) The surface area of the region to be painted (exclude doors).
(c) The time taken for Leonardo to paint the room if he works at
the rate of 12 m2 per hour.
3m
5m
8m
MEASUREMENT
95
11 A rectangular prism measuring 16 cm 12 cm 8 cm is painted red. It is to

be cut up into cubes 4 cm long. Find each of the following.
(a) The surface area of the original prism.
(b) How many cubes will exist after the dissection.
(c) The total surface area of all of these cubes.
(d) What percentage of the total surface area of all the cubes will be red.
16 cm
12 cm
8 cm
Homework 2.3
In mathematics we use a stricter definition of volume than we use in daily

conversations.
The volume of a solid is the amount of space that the solid occupies. It is
measured in cubic units. The most frequently used metric units of volume are
cubic millimetres (mm3), cubic centimetres (cm3) and cubic metres (m3).
The following is useful for converting between each of these units:
1 cm3 = 1 cm 1 cm 1 cm = 10 mm 10 mm 10 mm = 1000 mm3
1 m3 = 1 m 1 m 1 m
= 100 cm 100 cm 100 cm = 1 000 000 cm3
Capacity is similar to volume. It refers to the amount that a container can hold.
It is also used in reference to the volume of gases and liquids. When measuring
capacity, the metric unit used is the litre (L). Other units used that are based
on the litre are millilitres (mL) and kilolitres (kL).
For converting between units, remember that
1000 millilitres = 1 litre or 1000 mL = 1 L
1000 litres = 1 kilolitre or 1000 L = 1 kL
There is a great deal of similarity between volume and capacity. Their units are
connected through knowing that a litre takes a space of 1000 cm3, or fills a cube
of side length 10 cm.
1 litre = 1000 cm3
and so
1 mL = 1 cm3
To find the volume of a prism, you use the formula

V = AH
where
A is the area of the base and
H is the height.
H
H
This formula can still be used when a prism is not sitting on

its base, as long as the base is identified correctly.
96
A
A
You have identified the base correctly if cutting

parallel to it would leave the prism with a new
base identical in shape and size to the original
base. In each of these diagrams, the base is
shaded.
Note: Although a cylinder is not strictly a prism,
its volume is found in the same way.
worked example 9
Find the volume of these figures in cm3.
(a)
(b)
7 cm
9 mm
12 cm
Steps
(a) 1. Write the formula.
2. Calculate the area of the base.

calculate the volume to the nearest cm3.
(b) 1. Write the formula.
2. Calculate the area of the base.
Note that the triangular face is used.
12 mm
15 mm
Solutions
(a) V = AH
A = Area of base
A = r2
= 72
= 153.9 cm2
V = 153.9 12
= 1847 cm3
(b) V = AH
A = Area of base
A = -12- bh
=
1
-2
12 9
= 54 mm2
calculate the volume to the nearest cm3.
V = 54 15
= 810 mm3
4. Convert mm3 to cm3.
V = 810 1000
= 0.81 cm3
MEASUREMENT
97
exercise 2.6
Volume and capacity
Skills
1 Which of the following solids are prisms?

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
eTester
Worksheet C2.5
e
e
2 Find the volume of these regular solids.

(a)
(b)
Worked Example 9
Hint
12 mm
10 cm
5 cm
12 cm
14 mm
15 mm
(c)
(d)
(e)
30 cm
5 cm
14 cm
6 cm
(f)
8m
(g)
2m
40 cm
(h)
6 cm
10 mm
20 mm
5m
9 cm
15 mm
12 cm
30 mm
14 cm
3 Find the volume of these prisms.

(a)
(b)
area = 45.2 m2
12.4 m
area = 30 cm2
98
20 cm
18 mm
Hint
4 All the lengths are in metres. The volume of each figure is:
(a)
(b)
A 8 m3
B
C
D
E
2.5
2
3.2
m3
11.4
18.6 m3
15.4 m3
16 m3
24
(c)
2.4
1.8
3.6
A
B
C
D
E
27 m3
12.8 m3
54 m3
16 m3
86.4 m3
14
20
16
(d)
A
B
C
D
E
A
B
C
D
E
30
28
1408 m3
6528 m3
1320 m3
7680 m3
13056 m3
9236 m3
36945 m3
4618 m3
79 168 m3
11 760 m3
Applications
5 Simon is installing a swimming pool in his backyard. The pool will be

rectangular and will be 2 m wide, 4 m long and 1.5 m deep.
(a) What volume of soil must be removed for the pool?
(b) What is the capacity of the pool in litres?
6 The council is digging a trench
to lay some new pipe. Find the
volume of soil that will have
to be removed.
Hint
Hint
Hint
950 m
2m
1.6 m
7 Jenny is making a fruit punch

for a party. She is using the cylindrical bowl
shown and will fill it up to 5 cm from the top.
If each of the cups hold 500 mL, find the
number of drinks that will be held in the
bowl.
40 cm
25 cm
8 A cylindrical water tank has a capacity of

37.7 kL and a height of 3 m.
(a) Find its radius to the nearest centimetre.
(b) Water starts leaking out of a full tank at
the rate of 45 litres per day. How long
will it be before the tank is half-empty?
9 Find the volume left after a wedge of 72 has
been cut from a cheese of diameter 30 cm and
height 10 cm.
MEASUREMENT
99
10 Write down the dimensions of three rectangular prisms which have a

volume of 36 m3.
Analysis
11 (a) Find the volume of a three tier

cake if each tier is 7 cm high and
the radii are 20 cm, 15 cm and 10
cm. Give your answer correct to
the nearest cubic centimetre.
(b) The baker decides to use the
same amount of ingredients to
make a cake with a square base
16 cm long. How high will it be,
correct to the nearest centimetre?
10 cm
15 cm
20 cm
12 A steel pipe has an outer radius of 12 cm and is 1 cm thick. It is made in

lengths of 2 m.
(a) What is the capacity of the pipe?
(b) How many cubic centimetres of steel in a length of pipe?
(c) How many litres of water would pass through one length of pipe
in an hour if the water is flowing at a rate of 4 metres per second?
e
e
Worksheet C2.6
Homework 2.4
The properties of cylinders

One of the great features of a CAS is that it enables us to define a function. In this Investigation
we will define a function to find the volume of a cylinder and then input various values for the
radius and the height.
TI-Nspire CAS
Press
ClassPad
b > Actions > Dene and then ll
Tap Action > Command > Dene and ll in
in the rest of the denition as shown. Press
the rest of the denition as shown. The v
when you are nished. Then, to nd
needs to come from the 0 menu while
the volume for r = 10 and h = 5 we simple
the r and h need to come from
. To get the
approximate answer press / .
Press E when you are
enter v(10,5) and press
nished. Then, to nd
the volume for r = 10 and
h = 5 we simple enter
v(10,5) and tap E . To
get the approximate
answer tap K
100
..
V.
1 Copy the table shown on the right into your

workbook. Use your CAS to find the volumes
of the cylinders in the table.
Cylinder
Radius
Height
1
2
3
4
5
6
7
8
9
5
10
20
2
6
18
4
16
64
10
10
10
10
10
10
10
10
10
Volume
For each of the questions below round your answer to the nearest whole number.
2 In all of the cylinders in Question 1, the height has been kept constant at 10 units. In the first
three cylinders the radius doubles, in the next three cylinders the radius triples, and in the
last three cylinders the radius quadruples.
(a) What do the mathematical terms double, triple and quadruple mean?
(b) Use the value obtained using the CYL program to calculate the ratios V3 : V2 and
V2 : V1. Complete the statement: Doubling the radius of a cylinder increases its volume
by a factor of ____.
(c) Calculate the ratios V6 : V5 and V5 : V4. Complete the statement: Multiplying the radius
of a cylinder by 3 increases its volume by a factor of ____.
(d) Calculate the ratios V9 : V8 and V8 : V7. Complete the statement: Multiplying the radius
of a cylinder by 4 increases its volume by a factor of ____.
(e) Can you generalise the pattern above? If we multiply the radius of a cylinder by a
number k, larger than 1, how many times is the volume increased?
(f) Use algebra to prove the statements you wrote for parts (b) to (e) above. Your teacher
will assist you with this.
Extension
3 Use cardboard to make models of two cylinders of the same height but one with radius twice
the other. Fill the smaller cylinder with sand and use this cylinder to fill the larger cylinder.
How many loads of the smaller cylinder are needed to fill the larger cylinder? How does this
compare to the answer to Question 2(b) above?
4 Write and test a CAS function to calculate the volume of a rectangular prism.
5 Investigate similar cylinders. Similar cylinders have
corresponding radii and heights in the same ratio. For
example, consider the cylinders shown. The radii are
respectively 6 cm and 12 cm, and the heights are
respectively 10 cm and 20 cm.
(a) Verify the ratios of the radii and heights are the same
r
h
(i.e. show that ---2- = ----2- ).
r1 h1
(b) Calculate the ratio V2 : V1. What is the relationship between the ratios of the
corresponding sides and the volumes of the cylinders?
(c) Repeat parts (a) and (b) for other pairs of similar cylinders.
(d) Use algebra to generalise the pattern.
MEASUREMENT
101
DIY summary
Copy and complete the following using the words and phrases
from the list where appropriate to write a summary for this chapter.
A word or phrase may be used more than once.
1 The area of a _______or a _______is half the product of the
lengths of the two _______s.
2 The area of a _______ is half the sum of the parallel sides
multiplied by the perpendicular height.
3 The volume of a _______is found by multiplying the area of the
base by the height.
4 Despite meticulous care in measuring length, most answers
found contain _______.
5 The _______ of a container is usually measured in millilitres
or litres.
6 The word ______refers to the total distance around the edges of
a figure.
area
capacity
diagonal
error
kite
net
percentage error
perimeter
prism
quadrilateral
relative error
rhombus
trapezium
volume
7 A rhombus is a ______ but a kite is not necessarily a rhombus.

8 Any four sided figure can be called a ______.
9 The _________ of the length of a rectangle is found by dividing
the error by the length of the side and then converting the
answer to a percentage.
10 The ________of one litre is equivalent to a volume of 1000 cm3.
VELS personal learning activity

1 Write down all the merric units of measurement that you can think of and how to convert
between them.
2 Think of an example of several items that need to be measured. The items lengths then need to
be added. Explain clearly all the errors that might be involved in measuring and adding and how
the answers could be affected.
3 Write a summary with diagrams of the different shapes studied and how to calculate the area of
each.
4 Explain how to find the surface area and volume of a prism.
5 Write down in your own words the difference between capacity and volume.
102
Skills
1 (a) 25.7 cm is equivalent to:

A 25.70 mm
B 2570 m
D 257 mm
E 20.57
(b) 4.28 m is equivalent to:
A 402.8 cm
B 420.8 cm
D 0.0428 cm
E 42.8 cm
(c) 0.75 m is equivalent to:
A 7500 mm
B 750 mm
D 75 000 mm
E 1000.75 mm
(d) 0.045 km is equivalent to:
A 4.5 m
B 45 000 cm
D 450 cm
E 45 000 mm
2.1
C 2.57 mm
C 428.0 cm
C 75 mm
C 45 mm
2 (a) Calculate the percentage error, to one decimal place, for a

measurement of 6.3 0.4 m.
(b) Which has the larger relative error, a measurement of 5.0 0.3 m or
60 3.6 cm?
(c) Find the range of areas possible for a rectangle with length
102.5 0.2 cm and width 65 0.2 cm.
3 Calculate the areas of the following figures, correct to two decimal places.
(a)
(b)
2.2
2.3
70 cm
12 cm
30 cm
(c)
(d)
155 cm
45.4 m
34 cm
55 m
17 m
14 m
MEASUREMENT
103
4 Find the area of each of these shapes.

(a)
2.4
(b)
(c)
27 cm
26.5 cm
3.1 m
14 cm
18 cm
37.8 cm
4.2 m
(d)
(e)
26 cm
2.7 m
(f)
2.7 m
26 cm
5.4 m
18 cm
5 Find the total surface area of these solids.

(a)
(b)
2.5
5 cm
8 cm
4 cm
3.5 cm
9 cm
6 cm
8 cm
6 cm
6 Find the volume of the following figures.

(a)
(b)
2.6
(c)
8 cm
6 cm
7 cm
3 cm
10 cm
10 cm
12 cm
12 cm
15 cm
10 cm
(d)
(e)
(f)
4 mm
13 m
10 cm
20 cm
15 cm
2 mm
7 mm
104
10 mm
5m
15 m
12 m
Applications
7 A book cover is measured as 25.6 cm by 18.5 cm with a possible error of

0.2 cm in each measurement.
(a) What are the smallest possible side lengths?
(b) What are the largest possible side lengths?
(c) What are the smallest and largest possible perimeters?
(d) What are the smallest and largest possible areas?
2.2
8 A coffee cup 8.2 cm wide left a ring mark on a table that was 2 mm wide.
What is the area of the ring?
9 Find the area of the following shapes.
(a)
2.3
2.4
(b)
0.9 m
6.10 m
6.92 m
0.5 m
2.00 m
4.5 m
2.0 m
7.20 m
0.5 m
3.5 m
Analysis
10 (a) The inside of a rectangular swimming pool is to be tiled. If the pool is

12 m long, 5.5 m wide and 2.2 m deep, what is the tiled area?
(b) If the tiles are 12 cm long and 8 cm wide, how many will be needed
(to the nearest 100)?
(c) How many litres will the pools capacity be when it is full?
11 A cylindrical paint tin has a radius of 11 cm and volume of 12 164.3 cm3.
(a) What is the height of the tin?
(b) What is its capacity, in litres? Round your answer off to two decimal
places.
(c) If the paint just fills a rectangular paint tray that is 33 cm long and
22 cm wide, how deep is the tray?
MEASUREMENT
105
1 Draw a stem-and-leaf plot to represent the following data.

17, 34, 51, 33, 39, 26, 18, 42, 31, 25, 40
2 Convert the following decimal hours to hours and minutes.
(a) 3.25 hours
(b) 10.5 hours
(c) 0.1 hours
3 Evaluate:
(a) 11 3
(b)
84
(7)
(c) 150 (4)
4 Calculate:
(a) 85% of 250
(b) 20% of $0.40
(c) 90% of 80
5 Simplify:
64
(a)
(b) 92
(c)
6 Substitute x = 2, y = 3 and z = 5 into the following expressions and then

simplify.
4z
(a) 3y + 4x
(b) ----(c) x + y3
xy
7 Simplify these expressions.
(a) 16a 12ab a
(b) 12xy + 17x + 48yx y
(c) 7j k + j k
8 Simplify:
(a) 10g 4h 2
(b)
12k
3k
(c) 9ef
6e2
Worksheet R2.6
Worksheet R2.7
Worksheet R2.8
Worksheet R2.9
Worksheet R2.10
Worksheet R2.11
Worksheet R2.12
Worksheet R2.13
9 Expand these expressions and simplify where possible.

e
(b) 5y(6 11y)

(c) 9(p + 7) + p(3p 5)
(a) 8(x + 2)
10 Factorise:
(a) 4m 28
(b)
7ed2
+ 35d
(c) f(g 3) + 5(g 3)
11 Convert each of these fractions to (i) a decimal and (ii) a percentage. If the
answer is not exact, give it correct to two decimal places.
2
3
5
(a) --(b) --(c) -----8
12
5
12 Simplify the following expressions.
x3 x6
(a) x3 x5
(b) x8 x4
(c) --------------x5
106
Worksheet R2.14
Worksheet R2.15
1.1
1.3
e
e
e
Worksheet C2.7
Worksheet C2.8
Assignment 2

Heinemann Maths Zone 9 - Chapter 2

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F

or centuries ships navigators had used the

1 Solve the following.

(b) 18.21 1000

2 If x = 2, y = 4 and z = 5, find the value of the following.

3 (a) What is the length of AB in each of the following diagrams?

(d) z(4x + 5y)

(b) Find the perimeter of each shape in part (a).

4 Match the name to its shape.

5 Calculate the area of each of these shapes:

02HMZVELS9EN_text Page 67 Monday, June 30, 2008 10:57 AM

Hardly a day goes by when we are not required to express

Metric units of length

very small measurements, such as the length of an ant.

02HMZVELS9EN_text Page 68 Monday, June 30, 2008 10:57 AM

(b) 2.73 m = ____ mm

(a) 1. Decide on the type of conversion.

2. Multiply by 1000 and state the answer. Note

2. Divide by 10 and state the answer. Note that

1 Fill in the gaps:

(c) 7.5 km = ____ m

HEINEMANN MATHS ZONE

02HMZVELS9EN_text Page 69 Monday, June 30, 2008 10:57 AM

(d) distance to Sydney

(e) width of a fingernail

(f) distance you can kick a

02HMZVELS9EN_text Page 70 Monday, June 30, 2008 10:57 AM

13 An electric light pole is to be

HEINEMANN MATHS ZONE

02HMZVELS9EN_text Page 71 Monday, June 30, 2008 10:57 AM

From fingers to feet to metres

02HMZVELS9EN_text Page 72 Monday, June 30, 2008 10:57 AM

a = 1 cubit = 1 1--2- feet

HEINEMANN MATHS ZONE

02HMZVELS9EN_text Page 73 Monday, June 30, 2008 10:57 AM

02HMZVELS9EN_text Page 74 Monday, June 30, 2008 10:57 AM

The accuracy of any measurement that is made depends on a number of

Approximation and rounding

HEINEMANN MATHS ZONE

02HMZVELS9EN_text Page 75 Monday, June 30, 2008 10:57 AM

Sometimes it is important to specify the size of the uncertainty in the last

(a) (i) 1. State the formula.

2. Substitute the error and measurement

3. Convert to percentage error.

(ii) 1. State the formula.

(b) Compare the percentage errors.

02HMZVELS9EN_text Page 76 Monday, June 30, 2008 10:57 AM

Addition of quantities involving

2. Rearrange so the errors are grouped together.

= (11 + 15) (0.2 + 0.4)

(11 0.2) + (15 0.4)

When measurements are added the errors are added together.

Multiplication of quantities with

HEINEMANN MATHS ZONE

02HMZVELS9EN_text Page 77 Monday, June 30, 2008 10:57 AM

02HMZVELS9EN_text Page 78 Monday, June 30, 2008 10:57 AM

Errors and approximation

1 State the range of values possible for the following measurements.