Mathematics-MYP1-1

12:10 | Timetables

Timetables are part of everyday life. We use them to predict tides and to catch trains. The skills
covered in the previous exercise should help you with the following questions.

Tide sure goes out fast around here!

Exercise 12:10

1 This timetable shows the TV programs on Star Sports Time
for one day.
a At what time do the following programs begin? 01:29 Sport News
i Golf? 01:30 Tennis
ii Soccer/Futbol? 02:30 Auto Racing
iii Rugby? 03:30 Golf
b How many hours are there of: 04:00 Rugby
i tennis? 06:00 Auto Racing
ii surfing? 07:00 Tennis
iii sport news? 08:00 Tennis
c I have just finished watching the first screening of 14:30 Surfing
Auto Racing. How long will it be before Auto Racing 15:30 Extreme Sports
is on again? 16:30 Tennis
d What was the most common sport on this day? 20:00 Tennis
Can you give a reason why this might be the case? 21:00 Wrestling
21:59 Sport News
2 united cinemas 22:00 Soccer/Futbol
22:30 Tennis
HARRY POTTER

& THE PHILOSOPHER’S STONE (PG)
FRI, SAT 10:10, 12:30, 2:40, 5:00,

7:15, 9:30, 11:45PM
SUN 10:10, 12:30, 2:40, 5:00,

7:15, 9:30PM

a How many times is Harry Potter & the Philosopher’s Stone shown on Fridays?
b If I arrive at the cinema at 5:25 pm, how long must I wait before the next session begins?
c Estimate the finishing time of the last session on Sunday.
d What is the average time from one session to the next on Sunday.

378 INTERNATIONAL MATHEMATICS 1

3 Here is a timetable for the Oughtred Rd – Tartaglia Station bus route.

BUS TIMETABLE

M O N DAY T O F R I DAY
• ROUTE 509 • OUGHTRED RD TO TARTAGLIA STATION
DEPARTS ARRIVES ARRIVES ARRIVES ARRIVES ARRIVES

Oughtred Rd
Terminus
Cnr Samos Rd
and
Pythagoras Lane
Agnesi Rd
Shops
Pascal High
School
Cnr Tycho Ave
and
Brae St
Tartaglia
Station

2:38 B 2:47 2:52 2:58 3:01 3:04

3:20 3:29 3:34 3:40 3:43 3:46

3:48 3:57 4:02 4:08 4:11 4:14

4:10 4:19 4:24 4:30 4:33 4:36

4:35 4:44 4:49 4:55 4:58 5:01

5:08 5:17 5:22 5:28 5:31 5:34

B − On school days, diverts to Oughtred Road School

a At what time does the bus that leaves Oughtred Road Terminus at 2:38 arrive at:
i the corner of Samos Road and Pythagoras Lane?
ii the Agnesi Road shops?
iii Tartaglia Station?

b How long is the bus ride from:
i Oughtred Rd Terminus to the corner of Tycho Avenue and Brae Street?
ii the Agnesi Road shops to Tartaglia Station?
iii the corner of Samos Road and Pythagoras Lane to Pascal High School?

c If the bus is 8 minutes late, at what time would the school bus reach the Agnesi shops?
d If an evening bus leaves Oughtred Road Terminus at 6:45 pm, at what time would it reach

Pascal High School?
e A woman leaves her home on Samos Road at 3:45 pm and takes 9 minutes to walk to the

bus stop. At what time could she expect to reach Tartaglia Station?

4 Study the train timetable on the next page and answer the following questions.

a How many trains stop at Lansdowne Road?
b What time would you catch a train from:

i Sutton to be in Lansdowne Road just before 9:30 am?
ii Dublin Pearse to be in Dun Laoghaire by noon?
iii Kilbarrack to be in Shankill by 10:30 am?
c Looking at the times in the third column, how long does it take to get from:
i Howth to Raheny?
ii Blackrock to Killiney?
iii Bayside to Bray?

379CHAPTER 12 MEASUREMENT: LENGTH AND TIME

HOWTH – DUBLIN – GREYSTONES
Monday to Friday

Howth .............................. 0700 0740 0850 0953 1030 1102 1115 1146
Sutton ............................... 0703 0744 0853 0956 1033 1105 1118 1149
Bayside.............................. 0705 0746 0856 0958 1035 1107 1120 1151
Howth Junction.............. 0709 0750 0859 1002 1039 1111 1124 1155
Kilbarrack ........................ 0710 0751 0901 1003 1040 1112 1125 1156

Raheny .............................. 0712 0753 0903 1005 1042 1114 1127 1158
Harmonstown................. 0714 0755 0905 1007 1044 1116 1129 1200
Killester ............................ 0717 0758 0907 1010 1047 1119 1132 1203
Clontarf Road................. 0720 0801 0911 1013 1050 1122 1135 1206
Dublin Connolly........ arr 0914 1125 1210
• • 0925 • • 1135 •
dep 0724 0805 1017 1054 1139 •
0928 1136
Tara Street....................... 0727 0808 0921 1020 1057 • 1142 1212
Dublin Pearse ............ arr 0728 0809 0931 • 1058 • 1214
0734 0811 0933 1106 1139
dep 0737 0813 0935 1022 1108 • 1144 •
Grand Canal Dock........ 0815 0937 1024 1110 • 1146 1216
Lansdowne Road ........... • 0817 1026 1112 • 1148 1218
Sandymount..................... • 0939 1028 1150 1220
0819 0941 1114 •
Sydney Parade ................ 0741 0822 0944 1030 1116 • 1152 1222
Booterstown ................... • 0824 0946 1033 1118 • 1155 1225
Blackrock ......................... 0826 0948 1035 1120 • 1157 1227
Seapoint............................ 0746 0828 1037 1122 • 1159 1229
Salthill ............................... • 0951 1039 1201 1231
• 0830 0954 1125 •
Dun Laoghaire ................ 0833 0956 1041 1128 • 1203 1233
Sandycove ........................ 0752 0835 0959 1044 1130 • 1206 1236
Glenageary....................... • 0838 1003 1046 1132 • 1208 1238
Dalkey............................... • 0842 1049 1136 • 1211 1241
Killiney .............................. 1006 1053 1215 1245
0759 0844 1010 1139 •
Shankill.............................. 0807 0849 1012 1055 • • 1217 1247
Bray ..............................arr 0909 1022 • 1214 • 1253
• 0918 1145 1224
dep • 1101 1154 1223 •
Greystones ...................... 0813 1110 1232 1302
0824

380 INTERNATIONAL MATHEMATICS 1

5 Vegetable Quantity Cooking Time This chart shows the time taken to cook
various vegetables in a microwave oven.
Beans 500 g 15 min Vegetables that have similar cooking times
can be cooked at the same time, but the
Carrots 6 12 min cooking time should be doubled.

Corn 1 cob 4 min

Peas 500 g 10 min

Potatoes 500 g 12 min

Spinach 500 g 8 min

a How long would it take to cook the following?

i 1 kg beans ii 3 cobs of corn

iii 250 g potatoes iv 4 carrots

b How long would it take to cook 6 carrots and -1- kg of potatoes?

2

c What total time would it take to cook the following lists of vegetables?

i 6 carrots, 4 cobs of corn, -1- kg peas, -1- kg spinach

24

ii 250 g beans, 10 carrots, 750 kg potatoes, 1 kg spinach

6 The tides for a period of 3 days are given in this table. Day Time Tide (m)
The times are obviously in 24-hour time.
1 0414 0·3
a What is the lowest tide and at what time does
it occur? 1012 1·4

b How long is it between high tides on the first day? 1607 0·4
c How long is it between low tides on the second day?
d How long is it between the first low tide and the first 2227 1·7

high tide on day 3? 2 0458 0·4
e What is the difference in height between the lowest
1052 1·3
and highest tides on day 2?
f At what time during the three days did the highest 1637 0·5

tide occur? 2304 1·7
g Barney believes the best fishing occurs half an hour
3 0544 0·5
after high tide. If this is so, what would be the best
fishing times on day 3? 1135 1·2

1709 0·6

2343 1·6

Refer to ID Card 1 (Metric Units) on page xiv. Identify the units (1) to (24). id
Learn the units you do not know.
12:10

12:10A: Tide times
12:10B: Flying and arrival times
12:10C: Fencing Australia

381CHAPTER 12 MEASUREMENT: LENGTH AND TIME

fun spot Fun Spot 12:10 | And now for something light

12:10 A kilometre may seem long to us at times but, when we look beyond our Earth to the solar
system and the stars, a kilometre is very, very small.

Astronomers need to use much greater units of distance than a kilometre. The average

distance from the Earth to the sun is 150 million km, ie 150 000 000 km. For convenience

astronomers call this 1 astronomical unit. Another unit is the light year, which is the

distance light can travel in 1 year.

1 Considering that light travels about

300 000 km in 1 second, calculate how

Wow! far light would travel in:
Units like
light years a 1 minute
are pretty
heavy stuff! b 1 hour

c 1 day This answer is the number

d 1 year of kilometres in 1 light year.

You might need
pencil and paper.

2 Using your answer to 1d, calculate approximately
how many astronomical units equal 1 light year.
(Give your answer to the nearest thousand.)
Note: This answer means that travelling at the
speed of light, you could travel to the sun from
the Earth this many times in one year.

3 This table gives the distances from the sun of some of the planets, in astronomical units.
Calculate each distance in kilometres.

Planet Astron. units Kilometres Planet Astron. units Kilometres

Mercury 0·39 Mars 1·52

Venus 0·72 Saturn 9·54

Earth 1·00 150 000 000 Pluto 39·4

4 If you think Pluto is far away, the distance to the nearest star is 4·35 light years.

a How many km is this? b How many astronomical units is this?

5 A further unit astronomers use is the parsec.

1 parsec = 206 265 astronomical units.

a How many km is this? b How many light years is this?

Appendix G G:02 Distance, speed and time

382 INTERNATIONAL MATHEMATICS 1

Investigation 12:10 | Distance, speed and time investigation
12:10
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.

1 Use a tape measure or trundle wheel to measure:
a how far you can walk in 10 seconds
b how far you can run in 10 seconds

2 Use your measurements from question 1 to find:
a your average speed when walking, in m/s
b your average speed when running, in m/s
(Use S = D ÷ T)

3

Set up two sticks 20 m apart. Run 80 m around these sticks while a friend measures the time
taken in seconds. Calculate your speed using S = D ÷ T.

4 Measure out a 100 m section of a road. Find the speed of passing cars. How many cars
exceeded the speed limit? How many did not? Draw a column graph of your results using
such categories as: 50–55 km/h, 55–60 km/h, 60–65 km/h, 65–70 km/h etc.

• Estimate the height of the hurdles.
• Estimate the speed of the athletes in m/s.

383CHAPTER 12 MEASUREMENT: LENGTH AND TIME

Assessment Grid for Investigation 12:10 | Distance, speed and time

The following is a sample assessment grid for this investigation. You should carefully read
the criteria before beginning the investigation so that you know what is required.

Assessment Criteria (B, C, D) Achieved ✓

a No organised approach has been used to answer the 1
questions. 2
Criterion B 3
Application & Reasoning b An organised approach has been used and most of the 4
results are reasonable. 5
6
c An organised approach has been used and the results are 7
reasonable. 8
9
d Question 4 has been completed satisfactorily using a 10
reasonable number of vehicles. 1
2
e Question 4 has been completed accurately using a 3
reasonable number of vehicles. 4
5
Criterion C a Working out is not shown and answers are hard to find. 6
Communication 1
b Presentation is good and includes tables and graphs and 2
some correct mathematical terminology. 3
4
c Presentation is good and includes tables and graphs and
correct mathematical terminology throughout. 5

Criterion D a Some attempt has been made to explain the processes 6
Reflection & Evaluation involved in the investigation.
7
b The methods and processes used are justified and results 8
have been checked with some success.

Reasoned explanations are given to accompany the
c working out and there is comment on the reliability of the

results.

d There is comment on the accuracy of the results and some
discussion on how to improve the accuracy of the results.

384 INTERNATIONAL MATHEMATICS 1

Mathematical terms 12 dimension mathemrmsatical te
• A measurement of the size of an object in a
measuring (to measure) 12
• The process of recording the size of a particular direction.

quantity by comparing it to a known second, minute, hour, day, week, fortnight,
amount of the quantity (called a unit). month, year
• Commonly used units of time.
measurement • See sections 9:06 and 9:07 for the
• The result of measuring.
• Contains a numeral and a unit. relationship between the units.

eg 65 mm leap year
• A year containing 366 days.
unit • Occurs every fourth year (except for
• A known amount of a quantity used to
century years whose number is not
record measurements. divisible by 400).

scale century
• A set of markings on a measuring • A period of 100 years.

instrument that allow a measurement AD
to be read using the instrument. • Anno Domini: ‘In the year of our Lord’.

analog display BC
• Uses a scale and a pointer to give the size • Before Christ.

of a measurement. midnight
• 12 pm, the ‘middle’ of the night.
digital display
• Gives the size of a measurement directly noon
• 12 am, the ‘middle’ of the day.
as a number.
am
metre, kilometre, centimetre, millimetre • Means between 12 midnight and 12 noon.
• Metric units of length.
• For relationships between the units see pm
• Means between 12 noon and 12 midnight.
section 9:02.
average speed
odometer • Obtained by dividing the distance travelled
• An instrument that records the distance
by the time taken.
travelled by a vehicle.
24-hour time
trundle wheel • Time given as a 4-digit number, the first
• An instrument comprised of a wheel and
two digits indicating the hour, the second
a counter that allows distances to be two indicating minutes.
measured in metres. eg 13:20 (or 1320) means 20 minutes
past 1 in the afternoon.
perimeter
• The sum of the lengths of the sides of a

figure.
• The distance around the figure.

385CHAPTER 12 MEASUREMENT: LENGTH AND TIME

diagnostic test Diagnostic Test 12: | Measurement: Length and Time

12 • Each section of the test has similar items that test a certain type of example.
• Failure in more than one item will identify an area of weakness.
• Each weakness should be treated by going back to the section listed.

  Section
12:02
1 Write down each measurement to the nearest mm. Answer in
centimetres. 12:02
abcd 12:02
12:03
cm 1 2 3 4 5 6 7 8 9

2 a 3 cm = . . . mm b 7 km = . . . m c 2·5 m = . . . cm
3 a 50 mm = . . . cm b 650 cm = . . . m c 7150 m = . . . km
4 Measure these lines to the nearest mm.
d

b
a

c

5 Calculate the perimeter of each figure (all measurements are in cm). 12:05

a9 b 1·2 12:07
12:08
47 1·6 1·2 12:08
12:08
2·4 12:08

cd

2·1

2·5 3·6

6 Write these times in 24-hour time. d 12:20 am
a 5:30 am b 1:20 pm c 7:57 pm

7 Find the difference from the first time to the next.

a 8:30 am to 10:20 pm b 2:45 pm to 7:57 pm

c 10:12 am to 1:05 pm d 6:47 pm to 12:05 am

8 If it is 1:30 pm (EST) in NSW, what time is it in:

a WA? b SA? c Qld? d Vic?

9 a 3 h 45 min + 2 h 25 min b 2 h 40 min + 1 h 55 min

10 a 6 h 30 min − 1 h 35 min b 3 h 40 min − 2 h 55 min

386 INTERNATIONAL MATHEMATICS 1

Chapter 12 | Revision Assignment assignment

1 Give the type of angle in each case. 4 Find the value of each pronumeral. 12A
ab a a° 135°

c

2 Find the value of the pronumeral in each. b
a
b° 88°

a° 50° c

b 75°
c° 150°
b° 70°
d
c
a°
102°
c° 50° 78° x°

3 Find the value of each pronumeral. e
a
77° y°
110°
100° m° b°

b

55° x° f
125°
c° 87°
c d°
110°
y°

118°

387CHAPTER 12 MEASUREMENT: LENGTH AND TIME

5 Complete the table of values using the f P=2×M+1
rules shown at the top of each table. M2 3 4 5
a L=M+4 P
M 1 2 5 10
L 6 Simplify: b -1- of 72
b P=2×B+4
B0124 a -1- of 42 4
P
c n = (m + 3) × (m − 2) 2 d 2-- of 48
m 2 3 4 10
n c -1- of 60 3
d y=x+2
x1234 5
y
e m+n=6 7 If x = 7, find the value of:
m0 1 2 3
n a 5×x b 5×x+3

c 5×x−4 d 5×x−x

8 How many vertices has each solid?
ab

c

• If the height is 10 cm, estimate the volume of this rectangular prism.

1 Units of length
2 Clocks
3 Perimeter of shapes

388 INTERNATIONAL MATHEMATICS 1

Chapter 12| Working Mathematically assignment

1 Use ID Card 7 on page xix to identify: 4 A block of land is rectangular in shape. Its 12B
a1 b2 c3 d4 e5 perimeter is 360 m and it is known to be
f 6 g 7 h 8 i 17 j 18 8 times longer than it is wide. Find its id
length and width.
2 Two fruit growers send their fruit to 12
market. One grower sends 70 cases and the 5 While waiting to go on a long train trip I
other sends 96 cases. They receive $1660 decide to buy some magazines for the trip.
altogether. How much should each On the news-stand I see five magazines
person get? priced at $2.20, $3.40, $1.20, $1.60 and
$2.95. What possible combinations of
3 My car averages 9 km per 1 litre of petrol. magazines could I buy for exactly $5.
How much will it cost for petrol if I drive
from Bega to Bombala, a distance of 6 When a truck is loaded with 500 boxes,
126 km, if petrol costs 89 cents per litre? each of mass 5·8 kg, the total mass shown
on a weighbridge is 8·1 tonnes. What is the
mass of the truck when it is unloaded?

7 The speedometer below shows an estimate of the braking distance of a car for speeds up to
160 kilometres per hour. The innermost scale shows the speed in metres per second.

70 80 90 Whoah...Those
254.80.127.859.4100 are good brakes!
60 29.1 38.0
21.4 22.2
5.3 9.5 50 16.7 19.4 110

10 20 30 40 14.8 71.8 85.5 100.3120 130 140 150
13.9 30.6

11.1 Speedometer 33.3
and braking
8.3 36.1
distance
2.4 3181.69.3141.3734.54.4151.9160
0.6

2.8 5.6
0

0 Speed (m/s)
Average Braking Distance (m)
0
Speed (km/h)

a What is the average braking distance if the speed is:

i 30 km/h? ii 100 km/h? iii 25 m/s? iv 2·8 m/s?
iv 38·0 m?
b At what speed (in km/h) is the braking distance: iv 151·9 m?

i 21·4 m? ii 48·1 m? iii 85·5 m?

c At what speed (in m/s) is the braking distance:

i 2·4 m? ii 14·8 m? iii 9·5 m?

d Use the speedometer to change 60 km/h into m/s.

e Use the speedometer to change 22·2 m/s into km/h.

389CHAPTER 12 MEASUREMENT: LENGTH AND TIME

I hope I’ve got enough paint. 13

Area and
Volume

Measuring this
window is a real pain!

Chapter Contents 13:06 Volume of a rectangular prism
13:07 Capacity
13:01 The definition of area
Reading Mathematics: Capacity
Investigation: Finding area Practical Activity: Estimating capacity
13:02 Area of a rectangle Fun Spot: What makes money after 8 pm?
Challenge: The fantastic Soma cube
Investigation: Area of a rectangle Mathematical Terms, Diagnostic Test, Revision
13:03 Area of a triangle Assignment, Working Mathematically
13:04 Area problems

ID Card
13:05 Measuring 3D space (Volume)

Investigation: Measuring 3D space

Learning Outcomes

Students will:
• Use formulae to calculate the perimeter and area of circles and figures composed of

rectangles and triangles.
• Calculate surface area of rectangular and triangular prisms and volume of right prisms.

Areas of Interaction

Approaches to Learning, Homo Faber, Environment

390

13:01 | The Definition of Area

When we measure the area of a shape, we are
measuring the amount of space inside that shape.

The living area in a
home is measured
in square metres.

Do not confuse area with
perimeter. Remember that
the perimeter of a figure
is the length of its boundary.

Floorplan courtesy
of Pioneer Homes

We measure area by dividing it into square units,
and counting how many there are inside the figure.

AB That means B
has a bigger
area than A.

Number of squares = 16 Number of squares = 18

We say that the area of figure A is 16 square units, and the area of figure B is 18 square units.

Obviously, if we are going to be consistent, we need a standard unit of area. A convenient unit is
the square centimetre.

The area of a square 1 cm This can be written as
with a side of 1 cm is 1 cm
1 square centimetre. 1 cm2.

391CHAPTER 13 AREA AND VOLUME

The areas of these figures would be: Area = 14 units2 There’s got to be
a catch. This is
123
456 too easy!
7 8 9 10 11
12 13 14 15 16 Foundation Worksheet 13:01
The definition of area
Area = 16 units2 1 These shapes are drawn on

Exercise 13:01 centimetre grid paper.
Find the area of each shape.
1 Find the area, in square units, of each figure. a
(‘Square units’ can be written as ‘units2’.)
ab

cd

ef

gh i

392 INTERNATIONAL MATHEMATICS 1

2 Each of these shapes is drawn on centimetre grid paper. Find the area of each shape in cm2.
a bc

d
e

f

3 a What is the area of each shape?

b What is the perimeter of each shape?
c Do you think shapes with the same area will always have the same perimeter?
4 Use a plastic centimetre grid sheet, or a square with a side of 1 cm cut from a grid sheet, to find
the areas of these shapes in square centimetres.
ab

c

393CHAPTER 13 AREA AND VOLUME

de

5 Not all shapes are made up of whole unit squares. What fraction of a square centimetre would
these shaded areas be?
a bc d

6 Use your answers to question 5 to find the area, in cm2, of these shapes.
a bc

e
d

f h
g

7 Square centimetres are not the only units of area used. How
Smaller areas might be measured in square millimetres (mm2). about
that!
Larger areas would be measured in square metres (m2).
100 cm
a 10 mm Each side of this 1 square centimetre is,
1 m2 100 cm
10 mm of course, 10 mm. How many square mm
are in this square cm?

b Of course a square that has sides of 1 m would measure
100 cm on each side. How many square cm would fit
into 1 square m?

394 INTERNATIONAL MATHEMATICS 1

Investigation 13:01 | Finding area investigation
13:01
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.

Discuss how you could find the area of a figure with curved sides.
1 An approximation for the area inside a curved boundary can be found by placing a grid over

it and counting squares. If more than 1-- of a square is included in the figure, count it.
If less than 1-- of a square is included in2the figure, don’t count it. Find the approximate areas
of these sha2pes.
a

Count this square.

b

But not
this one.

c

2 Try to estimate the areas of various shapes and then trace the shapes on grid sheets.
Calculate their areas by counting squares.

3 Use the method above to find the area of your hand or your shoe.
4 Tape or glue sheets of paper together to make a square with sides of 1 metre. The area of this

square will, of course, be 1 m2. Use this square metre to find the area of some large shapes.
5 Draw several rectangles on grid paper, each with a perimeter of 24 cm. Then find the area of

each. Do you notice any pattern?

395CHAPTER 13 AREA AND VOLUME

Assessment Grid for Investigation 13:01 | Finding area

The following is a sample assessment grid for this investigation. You should carefully read
the criteria before beginning the investigation so that you know what is required.

Assessment Criteria (B, C, D) Achieved ✓

a No systematic or organised approaches have been used to 1
find the areas. 2
3
Criterion B b An organised approach has been used with some success. 4
Application & Reasoning 5
c An organised approach has been used successfully. 6

Organised approaches have been successfully used, and 7
d an attempt has been made to describe the patterns in
8
question 5.
9
e Organised approaches have been used successfully. There 10
is a successful discussion of question 5.
1
Criterion C a Very little working out is shown. 2
Communication 3
b Working out is shown; some diagrams and mathematical 4
terminology have been used. 5
6
c Presentation is good and working out is easy to follow. 1
Diagrams and mathematical terminology are well used. 2
3
Criterion D a Some attempt has been made to explain the method used 4
Reflection & Evaluation and to check the results. 5
6
b The method used is justified and the results have been 7
checked with some success. 8

c The method is justified and some comment is made on
the reliability of the results.

d There is some discussion on the method used and its
reliability.

396 INTERNATIONAL MATHEMATICS 1

13:02 | Area of a Rectangle prep quiz

Write the area and perimeter of each rectangle. 13:02

1 A = . . . units2 3 A = . . . units2 5 A = . . . units2
2 P = . . . units 4 P = . . . units 6 P = . . . units

7 A = . . . units2 9 A = . . . units2
8 P = . . . units 10 P = . . . units

If you didn’t know it already, you may have seen in the Prep Quiz a quicker way of finding the area
of a rectangle than counting the squares.

If we know the number of units in both the length I knew that!
and breadth of the rectangle, we can simply use the
following rule or formula.

l
A b A=l×b

This can be seen from the following diagrams. 5 cm

5 cm

3 cm 3 cm

Number of squares = 15 A=l×b
∴ Area = 15 cm2 =5×3
= 15

∴ Area = 15 cm2

Important note!
To use this rule we must be careful. The length and breadth must be measured in the same units.
If not, then one dimension must be converted.

397CHAPTER 13 AREA AND VOLUME

worked examples

1 Calculate the areas of these rectangles. Check out
these facts.
a 12 cm b 2·2 cm

6 mm

7 cm 1 cm2 = 100 mm2
1 m2 = 10 000 cm2
The units are different

2·2 cm could be ⎧ OR ⎧ 6 mm could be
written as 22 mm. ⎨ ⎨
Here the units are ⎩ ⎩ written as 0·6 cm.

both cm, Then A = 22 × 6 Then A = 2·2 × 0·6
so: A = 12 × 7
= 132 = 1·32
= 84
∴ Area = 84 cm2 ∴ Area = 132 mm2 ∴ Area = 1·32 cm2

(Note: 132 mm2 = 1·32 cm2)

2 A rectangular paddock measured 250 m by 100 m. ■ LAND AREAS
A square with sides
Its area would therefore be given by: 100 m long has an area
A = 250 × 100 of 1 hectare (ha).
= 25 000
100 m
∴ Area of land = 25 000 m2
But land is usually measured in units called 1 ha 100 m

hectares (see box). 1 ha = 10 000 m2
∴ Area of this paddock in hectares would be 2·5 ha.

3 The areas of some shapes can often be found by dividing them into smaller rectangles.

Before finding the area of each smaller rectangle, you may need to find some

unknown lengths.

a 7 cm Area of A Area of B

= (7 × 6) cm2 = (3 × 2) cm2

This side 4 cm = 42 cm2 = 6 cm2
must be
6 cm, AI 3 cm ∴ Total area of figure
(4 + 2) cm.
BII 2 cm = 42 cm2 + 6 cm2

= 48 cm2

b Area of C Area of D 10 cm
= (10 × 3) cm2 = (5 × 6) cm2
= 30 cm2 = 30 cm2 C 3 cm

∴ Total area of figure = 30 cm2 + 30 cm2 2 cm 2 cm
= 60 cm2 D 5 cm

(Note: The figures are not drawn to scale.) ⎧
⎨
⎩

This side must be 6 cm,
[10 − (2 + 2)] cm.

398 INTERNATIONAL MATHEMATICS 1

Exercise 13:02 Foundation Worksheet 13:02
Area of a rectangle
1 Find the area of each rectangle in square units. Find the area in square units:
ab 1a

2 3

8 2a
7

2 Determine the area of each rectangle using the dimensions given.

a b 9 cm c 8 cm

2 cm

5 cm

7 cm

10 cm

d 12 mm e 25 mm f 17 m

8 mm 9m

10 mm

3 A square is, of course, a rectangle whose length is equal to its breadth. Hence if a square has
sides of s cm, then its area = s × s cm2. Find the areas of these squares.

ab c
■ Area of a square

7 mm s

8 cm s
A = s2

20 cm

4 Measure these squares and calculate their areas in cm2.
ab

399CHAPTER 13 AREA AND VOLUME

5 Calculate the area of each rectangle in the units indicated. c 70 cm
ab

Area = . . . . mm2 9 mm

2 cm Area = . . . . m2 3 m 1·5 m Area = . . . . cm2

■ To calculate the area, 50 cm
both sides must be given
the same units.

d

Area = . . . . cm2 25 mm

4 cm 10 mm = 1 cm
100 cm = 1 m
e 7·3 m 1000 mm = 1 m

Area = . . . . m2 400 mm

6 Calculate the areas of these figures by dividing them into rectangles.

ab

5 cm

8 cm 5 cm

10 cm

5 cm 6 cm I can divide these
shapes in two ways.

6 cm 9 cm

c 12 cm d 4 cm 4 cm

7 cm 2 cm 5 cm
3 cm
8 cm

4 cm

400 INTERNATIONAL MATHEMATICS 1

e 10 cm f 4 cm

2 cm 3 cm
5 cm

3 cm 3 cm 3 cm
6 cm
4 cm

2 cm

7 Determine the coloured area in each figure. c
ab
6 cm
2 cm 20 cm

8 cm 7 cm
12 cm
4 cm 3 cm
10 cm 5 cm 15 cm
3 cm
■ Note: We can take the smaller
yellow areas away from the large
area to get the coloured area.

8 Find the area, in hectares, of these A common unit for I used to
fields. (Remember: 1 h = 10 000 m2) land areas was the have a tooth
a ACRE. Find out
how big this was. that was
200 m an acre!
c
500 m 4 km

b 600 m

200 m 350 m 2 km
150 m

9 Select an area on the right to match 6 cm2
the area of each part.
2 ha
a A ruler 800 m2
b A school
c A table 8 ha
d A house block
e A postage stamp 10 ha
f An Australian rules football field 90 cm2
g A road 10 km long 1 m2

401CHAPTER 13 AREA AND VOLUME

10 3 m 2·5 m 3m This basic holiday house has the
dimensions shown. Find the floor
BED BATH BED 3m area, in m2, of the:
1 2 1·5 m
a kitchen
b bathroom
c lounge
d two bedrooms
e whole house

LOUNGE KITCHEN 3 m

11 The plans shown are for an 3200 3200
extension to a house.
Measurements on plans are BED BED 4100
normally given in millimetres. 3 4
Find the area, in square metres, of:

a bedroom 4
b the family room
c the whole extension

2800 FAMILY 1100
ROOM 2000

3600

inve stigation Investigation 13:02 | Area of a rectangle
13:02
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.

1 Advertisements in newspapers cost more if they take up more area on the page.
Take a page from a newspaper and find the areas in cm2, taken up by advertisements,
articles and photographs.

2 Calculate the area of the face of a brick. Use this to determine the number of bricks
needed to build a wall 10 m long and 2 m high.

3 Measure the area of a paving brick and calculate the number needed to pave a rectangular
driveway 3 m by 6 m.

4 Mark out an area 100 m by 100 m. This area is 1 hectare.

5 Determine the area of the school grounds in hectares. How many housing blocks would
fit into it?

402 INTERNATIONAL MATHEMATICS 1

Assessment Grid for Investigation 13:02 | Area of a rectangle

The following is a sample assessment grid for this investigation. You should carefully read
the criteria before beginning the investigation so that you know what is required.

Assessment Criteria (B, C, D) Achieved ✓
1
a No systematic approach has been used to find the areas. 2
3
Criterion B b A systematic approach has been used and some questions 4
Application & Reasoning are answered correctly. 5
6
c A systematic approach has been used and most of the 7
questions are answered correctly. 8
9
d The exercises are answered correctly and most have been 10
worked through in an organised way. 1
2
e All exercises have been completed correctly and 3
instructions have been carried out efficiently. 4

Criterion C a No working out is shown and no explanations are given. 5
Communication
b Working out is shown and is easy to follow. Diagrams and 6
some correct mathematical terminology have been used.
1
Presentation is easy to follow. Diagrams are used 2
c effectively and correct mathematical terminology is used 3
4
throughout. 5
6
Criterion D a Some attempt has been made to explain the methods used
Reflection & Evaluation in the questions. 7

b The methods used are justified and the answers have been 8
checked for reasonableness with some success.

c Reasoned explanations are given for the methods used
and the answers have been checked successfully.

Methods are justified and answers have been checked
d successfully. There is some discussion on the reliability of

the methods used.

403CHAPTER 13 AREA AND VOLUME

13:03 | Area of a Triangle

prep quiz Calculate the areas of these rectangles. 2 10 cm
1 5 cm
13:03 112 cm
2 cm
4 6 cm
3 12 cm
2·5 cm
9 cm

Find the area of: 5 a rectangle with dimensions of 8 mm and 5 mm
6 a rectangle with a length of 20 cm and a breadth of 15 cm
7 a square with a side length of 7 cm

Complete: 8 1 cm2 = . . . mm2 9 1 m2 = . . . cm2 10 1 ha = . . . m2

A B
A B

These two These two areas
areas are are equal.
equal.

Looking at either of these two figures, you should be able to see that the area of the green triangle
is half the area of the rectangle around it.

Knowing that the area of a rectangle is given by l × b,
we could say:
b area of a triangle = -1- l × b units2.

2

l height (h)
base (b)
But we usually call the dimensions of a triangle its
base and height. (Note: these two measurements
must be perpendicular to one another.)

Area of a triangle is given by: h h h
A = -b----×-----h- or -1- bh b b b

22

404 INTERNATIONAL MATHEMATICS 1

worked examples

1 Find the area of each triangle. b
a

5 cm 6 cm

7 cm

8 cm

A = -b----×-----h- A = -b----×-----h-
2 2

= 8-----×-----5- = 7-----×-----6-
2 2

= 20 = 21
∴ Area = 20 cm2 ∴ Area = 21 cm2

c

5 cm ■ Note: Although this
dimension is ‘outside’ the
7 cm triangle, it is still the
height of the triangle
A = -b----×-----h- above the given base line.
2

= -7----×-----5-
2

= 17 -1- or 17·5

2

∴ Area = 17·5 cm2

2 These shapes may be divided into triangles and rectangles to find their areas.
ab

A 6 cm
10 cm I
6 cm
15 cm 6 cm
10 cm
6 cm 12 cm

B II 8 cm 8 cm

AB = 15 cm 12 cm 12 cm

Area of triangle I Area of triangle II Area of triangle Area of rectangle

A = 1----5----×-----6-- A = 1----5----×-----1---0-- A = 1----2----×-----6-- A = 12 × 8
2 2 2 = 96

= 45 = 75 = 36

∴ Total area of figure = 45 cm2 + 75 cm2 ∴ Total area of figure = 36 cm2 + 96 cm2
= 120 cm2 = 132 cm2

405CHAPTER 13 AREA AND VOLUME

3 Areas like the green one shown can be calculated by completing

I a rectangle around the figure and subtracting the area of the
II
resulting triangles from the area of the rectangle. The remainder

must be the area of the shaded figure. Thats pretty
Area of triangle I = -5----×-----4- cm2 handy!

2

= 10 cm2

Area of triangle II = -3----×-----2- cm2
2

= 3 cm2
Area of rectangle = 5 × 4 cm2

= 20 cm2
∴ Area of green figure = 20 − (10 + 3) cm2

= 7 cm2

Exercise 13:03 Foundation Worksheet 13:03
Area of a triangle
1 Calculate the area of the green triangle inside each rectangle. 1 Use the cm grid to find the

ab c area of each triangle.
a
6 cm
2 Other basic cases.
6 cm

10 cm 14 cm

9 cm

2 Calculate the area of each triangle. 8 cm 18 cm
ab
c
5 cm 7 cm 12 cm
f
10 cm

10 cm

d 8 cm e 3 cm
14 cm
7 cm 9 cm
11 cm

3 Calculate the areas of these triangles. 8 cm c
ab
7 cm
4 cm

8 cm 5 cm

406 INTERNATIONAL MATHEMATICS 1 10 cm

4 a A triangle has a base of 20 cm and a height of 12 cm. What is its area?
b A triangle has an area of 48 mm2 and a height of 12 mm. What is the length of its base?
c A triangle has an area of 72 cm2 and a base length of 6 cm. What is its height?

5 Determine the areas of these figures. 6 cm c 6 cm

ab

3 cm

5 cm 4 cm 5 cm

10 cm 8 cm 6 cm

dC e f 14 cm

3 cm 20 cm
5 cm6 cm
4 cm 8 cm
A B 2 cm
D

AB = 10 cm E
CD = 4 cm
DE = 9 cm

6 Determine the area of each shaded figure from the centimetre grid it is drawn on.
ab

cd

407CHAPTER 13 AREA AND VOLUME

13:04 | Area Problems

prep quiz Calculate the area of each figure. 3
12
13:04 4 cm 2 cm
3 cm 10 cm

8 cm 6 cm

4 5

4 cm

6 cm 7 cm

How many square tiles 10 cm by 10 cm will it take to cover a rectangular area:

6 10 cm × 80 cm? 7 10 cm × 2 m? 8 50 cm × 50 cm?

9 50 cm × 5 m?

10 What is the perimeter of the rectangle in question 1?

Obviously the measurement of area can occur in a variety of situations. Some everyday examples
would be tiling, wallpapering, making curtains, laying turf and painting.

The problems in this section ask you to apply your ability to find the areas of rectangles and
triangles in various situations. It is also possible to find the areas of different common shapes from
the area of a rectangle.

worked example 1

By cutting and pasting this parallelogram, • Square tiles are often used to cover walls.
form a rectangle and hence find its area.
3 cm
Solution
4 cm
If we cut the parallelogram as shown,
and slide the left triangle across to the
right, we get a rectangle 4 cm by 3 cm.
A=4×3

= 12
∴ Area of rectangle = 12 cm2
∴ Area of the parallelogram is 12 cm2.

408 INTERNATIONAL MATHEMATICS 1

worked example 2

A rectangular garden bed is 5 m by 4 m. A path 1 m wide is to be put around the garden.
What will be the area that needs to be paved?

Solution ■ Important Notice!
Always draw a diagram
The path and garden would look like the diagram below. of the problem if there
is not one with the
7m Because the path is 1 m wide, question.
1m the outer dimensions would
be 7 m by 6 m.
5m

Garden 4 m 6 m

Area of path = area of outer rectangle − area of garden
= (7 × 6) m2 − (5 × 4) m2
= 42 m2 − 20 m2
= 22 m2

∴ Area of path will be 22 m2.

Exercise 13:04

1 Draw a diagram to show how each figure could be cut along the dotted lines and pieced
together to form a rectangle. Thus find the area of each figure.
a b 5 cm

4 cm 3 cm Draw and
cut them
4 cm 5 cm out if you

c d like.

3 cm 2 cm e
2 cm 2 cm 2 cm 2 cm
3·5 cm
3 cm 4 cm

7 cm 2 cm

f 6 cm g 8 cm
11 cm 9 cm
2 cm 3 cm

409CHAPTER 13 AREA AND VOLUME

2 Tanith’s parents are going to carpet her bedroom. 4m
Her bedroom is rectangular, measuring 4 m × 3 m. 3m
If the carpet costs $67 per square metre, what will
be the cost of carpeting the room?

3 Jason’s fence needs painting with paint that covers 16 square metres for each litre of paint.
If the fence is rectangular in shape and is 80 m long by 2 m high, how many litres of paint
will be needed?

4 An athletics club gave pennants to the winners of each final race. 15 cm
The pennants had the dimensions shown in the diagram. 4m
Find the area of each pennant.

5 A garden 6 m by 2·5 m is to have a path 0·5 m wide around its border. 30 cm
If the garden is rectangular in shape, find:

a the area of the garden
b the total area of the garden and path
c the area of the path

6 Brent needed to cover his Maths textbook because he wanted to take care of it. If the book
measured 40 cm by 24·5 cm when opened out flat, what area of plastic would he need if he
wanted an extra 2 cm around each edge to tuck in?

7 5m How much fertiliser would be needed to cover a lawn
2m with the measurements shown if 1 kg will cover 10 m2?
6m

4m

8 a Fred the farmer has a rectangular field measuring 100 m 400 m
80 m by 400 m. How many hectares of potatoes 340 m
can he grow in this field?

b Fred’s neighbour, Barney, has a field like the one
shown in the diagram. Can he plant more potatoes
than Fred?

9 A square piece of land is one kilometre on each side. How many hectares would this be?

10 A curtain needs to be made for a window that is 2·5 metres wide. If you need a length of
material that is three times the width of the window to allow for pleats and folds, and the
curtain is to have a drop (length) of 2 m, what area of material is needed?

11 Wallpaper comes in rolls that are 50 cm wide. What length of wallpaper would be needed to
cover two walls 4 m by 3 m high and one wall 2·5 m by 3 m high?

12 Floor tiles measure 300 mm by 300 mm. How many would be needed to cover a kitchen floor
measuring 2·4 m by 3·6 m?

410 INTERNATIONAL MATHEMATICS 1

13 A wall is in the shape of a triangle on top of a rectangle
with the measurements shown. Find the area in m2 of

the wall.

Go to a brick wall, mark an area of 1 square metre and 9m
count the number of bricks in that square metre. 6m
Allowing for some breakages, calculate the number of
bricks required to build the wall. 10 m

14 a A company supplies turf, which comes in rolls 40 cm wide and 2 m long. How many rolls
must be ordered to cover an area 28 m by 24 m?

b If each roll costs $4.20, what is the cost of the turf?
c The same area could have been seeded. If a box of seed will cover 16 m2 and costs $18.50,

how much could have been saved on the cost of turf?

15 When laying tiles, an exact number may not cover an area, or a whole number may not lie

along each edge. Look at this diagram. ■ This is true even

If the tiles are 10 cm × 10 cm, though the area is
28 cm × 45 cm,
we can see that 15 tiles are ie 1260 cm2.

needed, presuming that the Dividing this by
100 cm2 (the tile area)
28 cm pieces of tiles cut off are not
would suggest that
good enough to be used

elsewhere.

only 12·6 or 13 tiles

45 cm might be needed.

a How many 10 cm × 10 cm tiles would be I can see
needed to cover an area 3·25 m by 2·17 m? an easy way to do
these. Can you?
b How many 300 mm by 300 mm tiles would
be needed to cover an area 2·5 m by 3·8 m?

Refer to ID Card 4 on page xvi. id
Identify figures (1) to (24).
Learn the terms you do not know. 13:04

13:04A Areas of plane figures
13:04B Carla’s chicken run

• Estimate the area
of the sail.

411CHAPTER 13 AREA AND VOLUME

13:05 | Measuring 3D Space (Volume)

When we measure the space an object occupies, we are measuring its volume.

There must be an easier
way to measure space!

When we say that one object is bigger or smaller than another, we are usually comparing
their volumes.

If we are going to measure the volume of an object, we need appropriate units. To measure area we
used square units. To measure volume we use cubic units.

1 If this is one worked examples
cubic unit,
then the volume
of this solid is
8 cubic units.

2 The volumes of these solids are found by counting the cubes.
Check that you can get the answers shown below.

Volume = 6 cubic units

Volume = 10 cubic units

412 INTERNATIONAL MATHEMATICS 1

Exercise 13:05 c

1 Count the number of cubic units in each solid.
ab

de

f gh

2 What is the volume, in cubic units, of each solid? c
ab

d
ef

413CHAPTER 13 AREA AND VOLUME

3 1 cm A cube that has edges 1 centimetre long has a volume of
1 cm 1 cubic centimetre (1 cm3). What is the volume, in cubic
1 cm centimetres, of the following solids? (Each block represents
1 cubic centimetre.)
a
c

b

d
e

4 This large cube is made of smaller
cubic units.
a How many smaller cubes are
there along each edge?
b How many smaller cubes are
there in the top layer?
c How many smaller cubes are
needed to make the large cube?

For the following
investigation, you will
need some centicubes

or similar cubes.

414 INTERNATIONAL MATHEMATICS 1

Investigation 13:05 | Measuring 3D space investigation iz
13:05
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.

1 There are only 2 differently shaped solids that can be formed by joining three cubes together
at their faces, as shown by these diagrams.

There are eight different solids that can be formed using 4 cubes. After stacking cubes
together, draw all eight.

2 By stacking cubes together, find out how many rectangular prisms can be made that have
different dimensions and a volume of:
a 8 cubic units
b 20 cubic units
c 24 cubic units

13:06 | Volume of a Rectangular Prism

What is the volume, in cubic units, of each solid? 4 5 prep qu
1 23
13:06

What is the area of each rectangle?

6 7 8 3 mm

5 cm

8 cm 9 cm 12 mm

9 6 mm 4 cm

10

10 cm

2·5 cm

415CHAPTER 13 AREA AND VOLUME

Assessment Grid for Investigation 13:05 | Measuring 3D space

The following is a sample assessment grid for this investigation. You should carefully read
the criteria before beginning the investigation so that you know what is required.

Assessment Criteria (B, C, D) Achieved ✓
1
a No organised approach has been used to make the solids. 2
3
Criterion B b An organised approach is used and some attempt has 4
Application & Reasoning been made to describe or draw the shapes. 5
6
c An organised approach has been used and the shapes are 7
described or drawn in some way. 8
9
d Most of the solids have been found in an organised way 10
and have been described or drawn successfully. 1
2
e All the solids are found and have been described 3
successfully. 4

Criterion C a There is no working out, and there is little description of 5
Communication the solids.
6
b Working out is shown with the aid of some diagrams or a
description of the solids. 1
2
Presentation and working is easy to follow and the solids 3
c have been successfully described with the use of correct 4
5
technology. 6

Criterion D a Some attempt has been made to explain the method used 7
Reflection & Evaluation to find the solids.
8
b The method used is justified and the reasonableness of
the findings checked.

c A reasoned explanation of the method used is given and
the findings are reasonable.

The method is justified and the findings are discussed.
d There is some discussion about the shapes formed and

about the areas of these shapes.

416 INTERNATIONAL MATHEMATICS 1

To calculate the volume of a rectangular prism,
we work out how many cubes are in one layer,
and multiply by the number of layers.

You may
have noticed

this.

For this prism we can see there are
2 layers of 5 × 2 cubes.

worked examples

1 To determine the volume of this prism,
we can see that there would be 4 × 3 cubes
in the bottom layer, ie 12 cubic units.

Since there are two layers, the volume
must be:
2 V = (4 × 3) × 2 cubic units

= 24 cubic units

4 3 ■ Notice
The number of cubes
2a The volume of this solid in each layer is the
would be given by: same as the area of
V = (4 × 2) × 4 the base rectangle
4 = 32 cubic units in square units.

2 4

b The volume of this rectangular prism would be
given by:
3 V = (6 × 4) × 3

= 72 cubic units

4

6

417CHAPTER 13 AREA AND VOLUME

To find the volume of a rectangular prism, use the rule:

V=l×b×h h
b
or simply l

V = lbh

Exercise 13:06 Foundation Worksheet 13:06

1 Find the volume of each rectangular prism in cubic centimetres. Volume of a rectangular prism
1 Find each volume in cm3.

a

ab

2 cm

3 cm

3 cm 2 Examples like Question 1 of text.

7 cm 2 cm

5 cm

c 2 cm d e 4 cm

3 cm 1 cm 7 cm
3 cm
7 cm

f 4 cm 4 cm

3 cm h

6 cm 2 cm 3 cm

g

8 cm

6 cm 2 cm 3 cm

6 cm

i 6 cm j

6 cm

3 cm 4 cm
5 cm 4 cm

418 INTERNATIONAL MATHEMATICS 1

2 By using the formula V = l × b × h, calculate the h
volume of each rectangular prism described below.
l
a length = 3 cm, breadth = 2 cm, height = 1 cm b
b length = 5 cm, breadth = 3 cm, height = 4 cm
c length = 2 cm, breadth = 1 cm, height = 6 cm
d l = 9 cm b = 3 cm h = 2 cm
e l = 5 cm b = 5 cm h = 5 cm
f l = 10 cm b = 8 cm h = 4 cm

3 a A cube with a side length of 1 metre would
have a volume of 1 cubic metre (1 m3).

What would the volume of this cube be

in cubic centimetres (cm3)?

b Similarly, a centimetre cube would have side

lengths of 10 mm. What would the volume

100 cm of this cube be in cubic millimetres (mm3)?

100 cm 100 cm 10 mm 10 mm
10 mm

4 Calculate the volume of a cube that has a side length of:

a 3 cm b 5 cm c 9 cm d 4 mm

5 Calculate the volume of each prism, in m3. c
ab
2m
3m
7m

7m 6m

10 m 10 m

d e 15 m

14 m 3m

112 m

20 m 12 m

9m
11 m

419CHAPTER 13 AREA AND VOLUME

6 If open boxes were made by folding along the dotted lines, what would be the volume of each?
ab

4 cm 3 cm

2 cm

3 cm

6 cm

4 cm

7 Look at the dimensions of these prisms.

The dimensions of prism A have been B

doubled to give prism B.

a What is the volume of each prism?

b How many of prism A would be needed

to make prism B? A

4

2

2 4
1

2

8 The dimensions of a rectangular prism are 5 cm × 4 cm × 2 cm.
a If these dimensions are doubled, what would the volume become?
b How many of the original prisms would fit into the larger prism?

9 Find the volume of these solids formed by joining rectangular prisms together.

a 2 cm b 2 cm

2 cm 3 cm
2 cm 2 cm

2 cm 3 cm
6 cm
10 cm

6 cm

420 INTERNATIONAL MATHEMATICS 1

13:07 | Capacity

When we measure the amount of a liquid that a container can hold, we are measuring its capacity.

We might say that an amount ■ Capacity is the
of a liquid is so many cupfuls volume of liquids.
or spoonfuls, but as with the
volume of solids, we need Note that you should
some standard units. always use a capital

The basic unit in the metric system for capacity is the litre (L). L for litre.

This means that:

1000 millilitres (mL) = 1 litre (L)

1000 litres (L) = 1 kilolitre (kL)

Medicine is Milk can be bought
usually measured in a 1 litre carton.

in millilitres.

Exercise 13:07 Foundation Worksheet 13:07

1 How many millilitres would there be in: Capacity
Complete:
a 2 L? b 5 L? c -1- L? d 0·4 L? 1 a 2 L = . . . mL b 7 L = . . . mL
h 1 kL?
2 c 4000 mL = . . . L
2 a 3 kL = . . . L b 2000 L = . . . kL
e 3·5 L? f 100 L? g 0·05 L? 3 a 3 cm3 = . . . mL

2 How many litres would there be in:

a 3000 mL? b 9000 mL? c 7500 mL? d 8300 mL?
g 20 000 mL? h 50 mL?
e 500 mL? f 300 mL?

3 Which measure of capacity, 50 mL, 500 mL or 50 L, would be most likely for:

a a bottle of drink? b the petrol tank of a car? c a medicine glass?

4 The capacity of a cup would be closest to 25 mL, 250 mL or 2500 mL?

5 How many 300 mL mugs could be filled completely from a 2 L kettle?

6 How many 375 mL cans of drink do I need to buy to have at least 5 L of drink?

7 Elizabeth’s backyard pool contains 20 000 L of water. How many kL is this?

8 How many times would a 1·5 L jug need to be filled to pour out 50 drinks at a party, if the
glasses hold 300 mL?

9 A 600 mL bottle of drink costs $1.20 and a 2 L bottle costs $4.20. Which is the better buy, and
by how much per litre?

421CHAPTER 13 AREA AND VOLUME

10 A leaking tap loses 1 mL of water every 10 seconds.

How much water will be lost in: Only a real drip
wouldn’t fix
a 10 minutes? b 1 hour? his tap!

c 1 day? d 1 week?

e 1 year?

There is a link between the capacity of liquids and the volume of solids.

A container with a volume of 1 cubic centimetre would hold 1 mL of liquid.
Also 1 m3 = 1 kL.

1 mL = 1 cm3 This also means that: 1 L = 1000 cm3 and 1 m3 = 1 kL .

11 Complete: b 8 cm3 = . . . mL c 183 mL = . . . cm3
e 35 L = . . . cm3 f 2·5 L = . . . cm3
a 15 mL = . . . cm3 h 4·5 kL = . . . m3 i 8·2 m3 = . . . kL
d 5000 cm3 = . . . L
g 15 m3 = . . . kL c

12 How many mL of liquid would each of these prisms hold? 2 cm
ab

4 cm

3 cm 5 cm 50 cm

5 cm 20 cm

13 If this measuring cylinder has a How can you 10 cm
capacity of 500 mL, what must its divide a cylinder 10 cm
volume be in cubic centimetres?
up into cubic
centimetres?

14 a What is the volume, ■ 10 cm = 1 decimetre (dm)
in cm3, of this cubic 1000 cm3 = 1 dm3 = 1 L
container?
100 cm
100 cm b How many litres of
water would fill this 100 cm
container? 100 cm
1 cubic metre = 1 kilolitre
100 cm

100 cm

15 a What is the volume, in cm3, of a cubic metre?
b What is the capacity of a 1 m3 container in:
i millilitres?
ii litres?
iii kilolitres?

422 INTERNATIONAL MATHEMATICS 1

16 A petrol can is in the shape of a rectangular prism with dimensions of 15 cm × 20 cm × 25 cm.

How much petrol will it hold?

17 The capacities of car engines are sometimes quoted in litres and ■ ‘cc’ stands for
sometimes in cubic centimetres (often using the abbreviation cc). ‘cubic centimetre’.

a A car’s engine is said to be 2·4 L. How many cubic centimetres is this?

b Another engine has a capacity of 1600 cc. What is this in litres?

Reading mathematics 13:07 | Capacity readingaticsmathem

Collect various drink containers and estimate their capacities. Check your accuracy by 13:07
reading the labels.

If a container is not marked, the capacity can be measured by filling it with water and then
pouring the water into a container whose capacity is known.

• Estimate the volume • Estimate the flow of water each hour.
of each item.
423CHAPTER 13 AREA AND VOLUME

Practical Activity 13:07 | Estimating capacity

1 Work with a friend. You will need a large measuring cylinder (big enough to take your
clenched fist) and a tray in which to stand the cylinder.
Begin by estimating the volume of your fist in cubic centimetres, then fill the measuring
cylinder to the brim with water and gently plunge your clenched fist into the water. Measure
how many millilitres of water have overflowed into the tray. Decide who has the bigger fist.

2 On average, each person in a town uses 30 litres of water each day. If there are 2500 people
in the town and the town needs a water tank capable of holding 14 days’ supply, find how
many litres of water the tank should hold. Design a possible tank in the shape of a
rectangular prism.

Appendix I Mass

un spotf
10
13:07 40 000
100
300

4
180
400
5000
4000

24
6000
1000
60 000
2000
3000

500
1800
Fun Spot 13:07 | What makes money after 8 pm?

Work out the answer to each part and put the letter for that part in the box that is above the
correct answer.

A 1 m = . . . cm A 1 cm = . . . mm D 5 kg = . . . g E 1 L = . . . mL

E 3 h = . . . min F 5 min = . . . s I 3 km = . . . m I 4 t = . . . kg
N 6 kg = . . . g N 5 cm2 = . . . mm2
M 2 kL = . . . L N 1 day = . . . h R 6 m2 = . . . cm2
T 1·8 m = . . . mm
N 4 ha = . . . m2 R 4 km2 = . . . ha

T 4 000 000 cm3 = . . . m3

• Estimate the volume of the
Parthenon in Athens.

424 INTERNATIONAL MATHEMATICS 1

Challenge 13:07 | The fantastic Soma cube challen ge

1 4 13:07
2 5

3

67

The Soma cube is formed by joining these 7 pieces together.
The first piece is formed by gluing 3 cubes together;
the other six pieces are formed from 4 cubes.
They represent all of the ways of joining 3 or 4 cubes
together, having at least one corner in the construction.
There are many, many ways of putting them together to
make the Soma cube. See how many you can find.
The pieces can also be put together to form other
interesting shapes. Some are shown below. See if you can
form them. Also try making some shapes of your own.

Cross Crystal
Wall
Tower

Serpent

425CHAPTER 13 AREA AND VOLUME

hematical terms Mathematical terms 13 volume
mat • The amount of space inside a three-
13 area
• The amount of space inside a dimensional shape.
• Units of volume
two-dimensional shape.
• Units of area cubic millimetre, cubic centimetre,
cubic metre
square millimetre, square centimetre,
square metre, hectare, square kilometre capacity
• The amount of fluid that can be held by a
hectare
• An area of 10 000 m2. container.
• 100 m × 100 m. • Units of capacity

litre, millilitre, kilolitre

diagnostic test Diagnostic Test 13: | Area and Volume

13 • Each section of the test has similar items that test a certain type of example.
• Failure in more than one item will identify an area of weakness.
• Each weakness should be treated by going back to the section listed.

  Section
1 What is the area of each figure, in square units? 13:01

a bc d

2 What is the area of each figure, in square centimetres? 13:01
a bc

3 Complete: b 1 m2 = . . . cm2 13:02
a 1 cm2 = . . . mm2 d 50 000 cm2 = . . . m2 13:02
c 1 ha = . . . m2

4 Find the areas of these rectangles. c 7m
ab 3m
15 mm
4 cm

9 cm

10 mm

426 INTERNATIONAL MATHEMATICS 1

  Section
13:02
5 Calculate the areas of these figures by dividing them into rectangles.
13:03
a 3 cm b 10 cm c 10 cm
13:05
2 cm 8 cm 2 cm 4 cm 7 cm
5 cm 7 cm

2 cm

6 Find the areas of these triangles.

a b 5 cm c

5 cm 4 cm 7 cm

8 cm 5 cm

7 What is the volume of each figure, in cubic units?
a bc

8 What are the volumes of these rectangular prisms, in cm3? 13:06
ab
13:06
2 cm 4 cm 5 cm 13:07
13:07
6 cm 13:07

3 cm

7 cm

9 What is the volume of:
a a cube with an edge length of 4 cm?
b a rectangular prism that has length 7 cm, breadth 3 cm and
height 4 cm?
c a rectangular prism with dimensions of 2 m × 3 m × 5 m?

10 Complete the following. b 2000 L = . . . kL
a 1 L = . . . mL d 1·5 kL = . . . L
c 3500 mL = . . . L

11 Complete the following. b 50 cm3 = . . . mL
a 3 mL = . . . cm3 d 1 m3 = . . . kL
c 1 L = . . . cm3

12 What would be the capacity of each prism in question 8, in millilitres?

1 Units of capacity and volume
2 Area puzzle

427CHAPTER 13 AREA AND VOLUME