10

MEASUREMENT AND GEOMETRY

SURFACE AREA AND

VOLUME
The Sydney Opera House was designed by Danish architect Jørn Utzon, with construction
taking place between 1959 to 1973 at a cost of $102 million. Its shell design represents a
ship under full sail, with each concrete shell being segments of a sphere of radius
75.2 metres. The shells are covered by 1 056 006 white and cream 120 mm square tiles.
The Sydney Opera House stands on 1.3 hectares of land, is 183 metres long and
120 metres wide at its widest point.

350 New Century Maths 9 Advanced 9780170453325

 Shutterstock.com/structuresxx
Chapter outline Wordbank
capacity The amount of fluid (liquid or gas) in a container
Working mathematically
cross-section A ‘slice’ of a solid, taken across the solid rather
10.01 Metric units U F PS R C than along it
10.02 Limits of accuracy of U R C limits of accuracy How accurate a measured value using a
measuring instruments# measuring instrument can be, equal to half of a unit on the
instrument’s scale
10.03 Perimeters and areas of U F PS R mega- (abbreviated M) A metric prefix meaning one million
composite shapes
micro- (abbreviated µ) A metric prefix meaning one-millionth
10.04 Areas of quadrilaterals U F PS R prism A solid shape with identical cross-sections and ends with
10.05 Circumferences and areas of U F PS R straight sides
circular shapes sector A region of a circle cut off by 2 radii, shaped like a slice
of pizza
10.06 Surface area of a prism U F PS R
surface area The total area of all the faces of a solid shape
10.07 Surface area of a cylinder U F PS R
10.08 Volumes of prisms and U F PS R C
cylinders
10.09 Volumes of pyramids and U F R
cones*
*STAGE 5.3
#
NSW ONLY, NOT AUSTRALIAN CURRICULUM

U = Understanding | F = Fluency | PS = Problem solving | R = Reasoning | C = Communication

9780170453325 Chapter 10 | Surface area and volume 351

 In this chapter you will:
• learn the metric prefixes for very small and very large units of measurement
• convert between units of digital memory, such as megabytes and gigabytes
• calculate the limits of accuracy of measuring instruments
• calculate the perimeters and areas of triangles, quadrilaterals, circles, sectors and composite
shapes
• calculate the surface areas, volumes and capacities of right prisms and cylinders
• (STAGE 5.3) calculate volumes of pyramids and cones

SkillCheck ANSWERS ON P.592

1 The measurements shown on the shapes below are in centimetres. For each shape, find
(correct to one decimal place if necessary):
i its perimeter ii its area

32
6
a b c
50
14
12

48

d e f
10 8 26 24 30 30

25 28

a 34 m to mm b 925 cm to m 1500 m to km
2 Convert:
c
d 3750 mL to L e 38 cm to mm f 7.5 L to mL

3 What fraction of each circle is shaded?

a b c d

120°
130°

4 Calculate the volume of each prism (all units are in cm).

a b c d

15 20
15 45
A = 12 cm2 10
12 8
13

352 New Century Maths 9 Advanced 9780170453325

 5 For each prism:
i count the number of faces
ii name the shapes of the different faces
iii name the prism.
a b c

10.01

6 Find the value of each variable.

b c
9 16.8
a h w
p

40 7.5
19.5 21

Metric units 10.01

Length Capacity
1 cm = 10 mm 1 L = 1000 mL
1 m = 100 cm = 1000 mm 1 kL = 1000 L
1 km = 1000 m 1 ML = 1000 kL = 1000 000 L

× 1000 × 1000 000

× 1000 × 100 × 10 × 1000 × 1000 × 1000

ML kL L mL
÷ 1000 ÷ 1000 ÷ 1000
km m cm mm
÷ 1000 ÷ 100 ÷ 10

÷ 1000 ÷ 1 000 000

Mass Time
1 g = 1000 mg 1 min = 60 s
1 kg = 1000 g 1 h = 60 min = 3600 s
1 t = 1000 kg 1 day = 24 h

× 3600
× 1000 × 1000 × 1000
× 24 × 60 × 60
t kg g mg
÷ 1000 ÷ 1000 ÷ 1000
day h min s
÷ 24 ÷ 60 ÷ 60

÷ 3600

9780170453325 Chapter 10 | Surface area and volume 353

 Other than for time, units of the metric system have prefixes based on powers of 10.
• kilo- means 1000 (thousand) or 103 1 kilogram = 1000 grams
• mega- means 1 000 000 (million) or 10 6
1 megalitre = 1 000 000 litres
1 1
• centi- means (hundredth) or 10–2 1 centimetre = metre
100 100
1 1
• milli- means (thousandth) or 10–3 1 millilitre = litre
1000 1000

Example 1
Convert:
a 5.8 t to kg b 16 km to cm c 23 700 mL to L d 1500 min to h

Solution
a 5.8 t = 5.8 × 1000 kg To convert from large to small units, multiply (×).
= 5800 kg × 1000
t kg

16 km = 16 × 1000 × 100 cm
× 1000 × 100
b
= 1 600 000 cm km m cm

c 23 700 mL = 23 700 ÷ 1000 L To convert from small to large units, divide (÷).
= 23.7 L

÷ 1000
L mL

d 1500 min = 1500 ÷ 60 h

= 25 h
÷ 60
h min
Shutterstock.com/Shuang Li

354 New Century Maths 9 Advanced 9780170453325

Metric prefixes
This table shows a more detailed list of metric prefixes.
Prefix (abbreviation) Meaning Example
pico (p) 10–12

nano (n) 10–9 =

1 nanosecond (ns) = billionth of a second
1000000000
10.01
micro (μ) 10 =
–6
1 microgram (μg) = millionth of a gram
1000000

10–3 = 1000 millibar (mbar) = thousandth of a bar (air pressure)

1
milli (m)

centi (c) 10–2 =

1 centimetre (cm) = hundredth of a metre
100

deci (d) 10–1 =

1 decibel (dB) = tenth of a bel (sound level)

101 = 10 decametre (dam) = ten metres

10
deca (da)
hectopascal (hPa) = hundred pascals (air
hecto (h) 102 = 100
pressure)
kilo (k) 103 = 1000 kilojoule (kJ) = thousand joules (energy)
mega (M) 106 = 1 000 000 megahertz (MHz) = million hertz (frequency)
giga (G) 10 9
gigalitre (GL) = billion litres (water)
tera (T) 10 12
terawatt (TW) = trillion watts (power)

Digital memory
Metric prefixes are used in units of digital (computer) memory. A byte, abbreviated B, is the
amount of memory used to store one character, such as @, g, #, 7 or ?.
Unit Relationship Size
kilobyte (kB) 1 kB = 1000 B About half a page of text.
megabyte (MB) 1 MB = 1000 kB A million bytes, about the size of a novel.
gigabyte (GB) 1 GB = 1000 MB A billion bytes, about 500 photos or 7.5 hours of
video.
terabyte (TB) 1 TB = 1000 GB A trillion bytes, about the size of all of the books in a
large library.

× 1000 × 1000 × 1000 × 1000

TB GB MB kB B
÷ 1000 ÷ 1000 ÷ 1000 ÷ 1000

Some examples of digital memory sizes:

E-mail (without attachment): 75 kB Photo: 2 MB
Music file: 4 MB CD: 750 MB
Streaming a video: 790 MB per hour DVD: 4.7 GB
Blu-ray disc: 25 GB Online game: 40 to 300 MB per hour

9780170453325 Chapter 10 | Surface area and volume 355

 Example 2
Convert:
a 3.2 MB to kB b 147 000 MB to GB c 2 TB to GB

Solution
a 3.2 MB = 3.2 × 1000 kB To convert from large to small units, multiply (×)
= 3200 kB × 1000
MB kB

b 147 000 MB = 147 000 ÷ 1000 GB To convert from small to large units, divide (÷)
= 147 GB
GB MB
÷ 1000

2.6 TB = 2.6 × 1000 GB

× 1000
c
= 2600 GB
TB GB

Very small and very large units

Some very small and very large units of length, distance and time also use metric prefixes.

1
Length and distance
• One nanometre (nm) = m: your fingernails grown 1 nm every second, the
1 000000000
diameter of a helium atom is 0.1 nm
1
• One micrometre (μm) = m: used to measure diameters of bacteria
1 000000
• One astronomical unit (AU) = 149 597 871 km: the average distance between the Earth
and the Sun, used to measure distances between planets in our solar system
• One light year (ly) = 9.46 × 1012 km: the distance light travels in a year, for measuring
distances between stars
• One parsec (pc) = 3.0857 × 1013 km or 3.26 ly: for measuring distances between stars, the
nearest star Proxima Centauri is 1.30 parsecs away

1
Time
• One nanosecond (ns) = s: the time it takes electricity to travel along a 30 cm
1 000000000
wire
1
• One microsecond (μs) = s
1 000000
1
• One millisecond (ms) = s: the duration of the flash on a camera
1 000
• One decade = 10 years
• One century = 100 years
• One millennium = 1000 years
• One mega-annum (Ma) = 1 000 000 years: used in geology and paleontology, the
Tyrannosaurus Rex dinosaur lived 65 Ma ago
• One giga-annum (Ga) = 1 000 000 000 years: the Earth formed 4.57 Ga ago

356 New Century Maths 9 Advanced 9780170453325

 Example 3
Convert:
a 0.73 s to μs b 25 610 000 km to AU
c 4.57 Ma to years d 1 920 000 nm to m

Solution
a 0.73 s = 0.73 × 1 000 000 μs To convert from large to small
units, multiply (×)
10.01
    = 730 000 μs
25 610 000 km = 25 610 000 ÷ 149 597 871 AU To convert from small to large
units, divide (÷)
b
= 0.17119…
≈ 0.17 AU

c 4.57 Ma = 4.57 × 1 000 000 years

= 4 570 000 years

d 1 920 000 nm = 1 920 000 ÷ 1 000 000 000 m

= 1.92 × 10–3 m
= 0.001 92 m

EXERCISE 10.01 ANSWERS ON P. 592

Metric units U F PS R C
1 What metric unit would you use for each measurement?
Width of a computer screen b Mass of a plane
R C

a
c Distance between 2 towns d Capacity of a car’s fuel tank
e Amount of water in a pool f Time taken to fly overseas

2 Convert:
750 g to mg b 36 kL to L 67 kg to g
R
EXAMPLE
a c
d 1.5 L to mL e 6.7 cm to mm 3.5 km to m
1
f
g 9.4 t to kg h 3.75 g to mg i 4.25 m to cm
j 0.2 kg to g k 26.2 L to mL l 0.3 km to m
m 3 kL to mL n 4 days to min o 15 hr to s
p 2.7 km to cm q 5 kg to mg r
0.7 m to mm
s 29.375 t to g t 18.14 km to mm u 3.27 g to mg

3 How many mL in 18.9 kL? Select the correct answer A, B, C or D.

A 1 890 000 B 18 900 000 C 189 000 D
18 900

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9780170453325 Chapter 10 | Surface area and volume 357

 4 Select the most appropriate answer A, B or C for each measurement.
a height of a door
A 20 cm B 200 cm C 20 m
b capacity of a soft drink can
A 37.5 mL B 375 mL C 3.5 L
c time to run 100 metres
A 12.3 s B 34.3 s C 50.3 s
d height of Sydney Tower
A 30.5 m B 3050 m C 305 m
e mass of a family car
A 100 kg B 1000 kg C 10 000 kg

5 Convert:
a 8000 L to kL b 90 mm to cm c 180 s to min
d 240 min to h e 375 mL to L f 14 700 kg to t
g 8600 mg to g h 750 cm to m i 223 mm to cm
j 800 g to kg k 125 L to kL l 3.5 kg to t
m 17 000 000 mg to kg n 7200 min to days o 45 000 000 g to t
p 144 000 s to h q 86 400 s to days r 2500 cm to km
s 75 000 mL to kL t 34 700 mm to m u 628 L to ML

6 How many km in 408 m? Select the correct answer A, B, C or D.

A 4.08 B 0.408 C 0.0408 D
0.004 08

7 The capacity of Sydney Harbour is approximately a quarter of Warragamba Dam’s

capacity. If the capacity of Warragamba dam is 2013 GL, what is the capacity of
Sydney Harbour in litres?

8 Convert:
4.8 MB to kB b 2.1 TB to GB 8.5 GB to MB
R C
EXAMPLE
a c
d 10.8 MB to B e 2910 B to kB 740 MB to GB
2
f
g 1050 kB to MB h 5900 GB to TB i 0.94 kB to MB
j 57 GB to MB k 0.43 GB to kB l 21 TB to GB
m 7.8 kB to B n 2180 MB to GB o 42 000 B to MB
p 940 MB to kB q 0.64 TB to kB r 278 MB to GB

9 Mala made a 3-minute phone call to her friend in Rockhampton using the Internet
at 3.3 MB per minute. How many kB did she use?

10 Ritu made a video call lasting 1 hours. If it used 3MB/min, how much data did the
call take?
1
4

Foundation Standard Complex

358 New Century Maths 9 Advanced 9780170453325

 11 Calculate the number of MP3 files of average size 5 MB that can be stored on a 64 GB
MP3 player. PS R

12 Convert, correct to 2 significant figures:

3.5 AU to km b 6000 ms to s 7.5 centuries to years
R C
EXAMPLE
a c
d 12 000 000 nm to m e 0.64 Gm to m 2.1 ly to km
3
f
g 0.07 s to ms h 0.25 Ma to years i 4 800 000 μs to s
5 290 000 km to Mm k 0.2 pc to km 6 mm to μm
10.01
j l
m 0.94 Ma to millennia n 849 years to centuries o 8 × 1014 km to ly
p 0.000 008 cm to nm q 20 ly to pc r 156 millennia to Ma

Did you know?

Is 1 MB = 1000 or 1024 kB?
Computers represent all information as a series of 0s and 1s. These binary digits, called
bits (b), give their name to the smallest unit of computer memory. Computer memory is
measured in bytes, where one byte is equal to 8 bits. A byte (B) is the amount of memory
needed to store one character such as #, d, D, 8 or $. For example, the letter ‘d’ is represented
by the byte 01100100, while ‘D’ is the byte 01000100.
On most storage devices such as USB drives, hard drives and flash drives, 1 MB = 1000 kB.
Internet and phone companies use the same. Originally, however, computer programmers
used 1 MB = 210 kB = 1024 kB because that was actually how computer hardware stored data.
This table shows the original conversions.
Unit Meaning Number of bytes
byte (B) 1B=8b 1B
kilobyte (kB) 1 kB = 1024 B 1024 B or 210 B (approx. 1000)
megabyte (MB) 1 MB = 1024 kB 1024 × 1024 = 1 048 576 B or 220 B (approx. one million)
gigabyte (GB) 1 GB = 1024 MB 10243 = 1 073 741 824 B or 230 B (approx. one billion)

terabyte (TB) 1 TB = 1024 GB 10244 = 1 099 511 627 776 B or 240 B (approx. one trillion)

Find the byte (code) for ‘8’ and ‘$’.

Shutterstock.com/Anton Marchenkov

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 359

 Limits of accuracy of measuring
10.02
instruments# NSW ONLY, NOT AUSTRALIAN CURRICULUM
#

All measurements are only approximations.

No measurement is ever exact.
WS
Eraser
For example, we might measure this eraser to be 5.5 cm
2 3 4 5 6
long, because the scale on the ruler measures to the
Worksheet
Homeworkin
Accuracy
measurement
cm1

nearest 0.5 cm.

If we used a ruler with a more precise scale, such as 0.1 cm
markings, then we may find the length to be 5.4 cm. Eraser
If we used an instrument such as a micrometer that
cm 1 2 3 4 5 6
measured to the nearest 0.01 cm, we may find that the
length is 5.41 cm.
Notice that we can always find a more accurate measurement using a more precise measuring
instrument. So all measurements are approximations.
A measuring instrument such as a ruler has limits of accuracy due to its level of precision.
‘Accuracy’ means how close a measured value is to the true value, while ‘precision’ means how
fine the scale is on the measuring instrument.
For example, this ruler is marked in centimetres, so any
length measured with it can only be given to the nearest cm 10 11 12 13 14
centimetre.

{
Any measurement in the shaded region should be recorded as 12 cm. The measured length
is 12 cm, but the actual length is 12 ± 0.5 cm, meaning ‘12 centimetres, give or take 0.5
centimetres’ or ‘11.5 to 12.5 cm’.
The limits of accuracy of this ruler are ±0.5 cm or ‘plus or minus half a centimetre’.

Limits of accuracy of measuring instruments

The limits of accuracy of a measuring instrument are ±0.5 of the unit shown on the
instrument’s scale.

Example 4
Find the limits of accuracy for each measuring scale.

a
25 mL
b
80 90 20
15
10
kg
5

Solution
a The size of one unit on the scale is 1 kg.
The limits of accuracy are ±0.5 × 1 kg = ±0.5 kg
b The size of one unit on the scale is 5 mL.
The limits of accuracy are ±0.5 × 5 mL = ±2.5 mL.

360 New Century Maths 9 Advanced 9780170453325

 EXERCISE 10.02 ANSWERS ON P. 593
Limits of accuracy for measuring instruments U R C
1 For each measuring instrument, state:
i the size of one unit on the scale ii its limits of accuracy.
R C
EXAMPLE

b
4
a
60 70 80 90 0 10 20 30 40 50 60 70 80 90 100 °C
mm

10.02
c d e
12
11 1
10 2

9 3
500
8 4

4
3
7 5

2
1000
6
0

1
cm

0
GRAMS

f g h
60
70
80 90 100 110
120
50 60 70
2000 gv
40 80
100 90 80 13
50 110 70 0
30 90
60

1500 g
120 50
0
13
14

20 100
40

100
0

150
40
30

14

1000 g
150

30

10 110
160
0
10 2

20
180 170 16

170 180

0 120
10

088724

km/h 500 g
0

0g

i j k F

mL 200 kL 7
6
150 5
4
100 3
2
50 1
E

2 What are the limits of accuracy for this measuring cylinder? 300 mL
Select the correct answer A, B, C or D.
A ± 0.5 mL 200 mL
B ± 10 mL
C ± 20 mL 100 mL

D ± 5 mL

3 Measuring with a ruler, Gemma correctly gives a measurement as 26.5 to 27.5 cm.
What is the size of a unit on the ruler?
R C

a
b What are the limits of accuracy of this measurement?
Using a ruler marked in millimetres, Emily gives the same measurement as 26.8 cm.
What are the limits of accuracy of this measurement?
c

d What is the range within which this measurement must lie?

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 361

 Perimeters and areas of composite
10.03
shapes
The perimeter of a shape is the distance around the shape, the sum of the lengths of the sides
of the shape.
What is
area?
The area of a shape is the amount of surface covered by the shape, measured in square units.

Areas of rectangles and triangles

Rectangle A = length × width 1
A = × base
2
A = lw
Triangle

h × perpendicular height
1
A = bh
w

2
l b

Example 5
For each composite shape, find:
i its perimeter ii its area.

7m
a b

38 cm
13 m 18 cm
11 m
6m
15 cm

Solution
Find any unknown lengths on the shape.
a i  11 – 6 = 5 7
Perimeter = 6 + 13 + 5 + 7 + 11 + 20
13
= 62 m 11
6

13 + 7 = 20

ii Area = (7 × 5) + (20 × 6)   Sum of areas of 2 rectangles

= 155 m2
Divide the shape into a rectangle and a
triangle, and find the unknown length, h,
b i 

using Pythagoras’ theorem. h 38 – 18 = 20

h = 15 + 20
2 2 2

= 625
h = 625 18 18
= 25 cm
Perimeter = 18 + 25 + 38 + 15
= 96 cm 15

362 New Century Maths 9 Advanced 9780170453325

 Area =  × 15 × 20 + (15 × 18)   Area of triangle + Area of rectangle
1 
ii

= 420 cm2
2 

Example 6
This is the plan of a farm up for sale.
All measurements are in metres.

900
1500
10.03
a Find the area of the farm.

1900
b Express the area in square kilometres.
1800
c Find the area of the property in hectares, where 1 ha = 10 000 m2.

Solution
a Area = 900 × 1500 + 1900 × 1800
= 4 770 000 m2

b Since 1 km = 1000 m, 1 km2 = 10002 m2 = 1 000 000 m2

∴ Area = 4 770 000 ÷ 1 000 000
= 4.77 km2

c 1 ha = 10 000 m2
∴ Area = 4 770 000 ÷ 10 000

= 477 ha

EXERCISE 10.03 ANSWERS ON P. 593

Perimeters and areas of composite shapes U F PS R
1 For each composite shape, find:
i  its perimeter its area.
PS R
EXAMPLE
5
ii
7m c 25 mm
33 cm
a b
15 cm

11 m
57 mm
28 cm

6m
16 m

48 mm
5m

15 cm 22 m
32 mm

8 cm f
24
d e
c m
20 cm
9 cm

20 cm
52 cm
10 cm

30 cm
40
cm

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9780170453325 Chapter 10 | Surface area and volume 363

 2 The length of a rectangular block of land is 112 m and its perimeter is 334 m.
How wide is the block? Select the answer A, B, C or D.
A 222 m B 110 m C 55 m D 224 m
R

3 Find the area of each composite shape.

9 cm c
PS R

11 cm
a b

4
4 cm cm 4 cm
15 cm

17 cm
12 cm
8 cm

11 cm

11 cm
24 cm 15 cm 4 cm 4 cm
11 cm

d e f 17 cm
10 cm
11 cm
9 cm

9 cm
5 cm

15 cm
8 cm

11 cm
16 cm

9 cm
20 cm

g 10 cm h i 34 cm

11 cm
2 cm

4 cm 12 cm
19 cm

20 cm
8 cm

17 cm
8 cm

j 12 cm 12 cm k l 20 cm
15 cm

22 cm
cm

12
24 cm
25 cm

.6
.6

12 cm
22

30 cm
50 cm

22 cm
cm

25 cm
15 cm

12 cm 12 cm

4 What is the area of the shaded arrow? 3 cm 3 cm

Select the correct answer A, B, C or D.
A 14 cm2 B 8 cm2
2 cm

C 9 cm2 D 12 cm2
1 cm

Foundation Standard Complex

364 New Century Maths 9 Advanced 9780170453325

 5 Find each shaded area. PS R

a 0.5 cm b 

3 cm 3.8 cm 3 cm

4 cm
6.4 cm 3 cm
19.6 cm

6 a Find the area of the large block of land shown in 800 m 10.03
the diagram (in m2).
b Express the area of the land in:

500 m
EXAMPLE

1300 m
6

i km 2
ii hectares

7 The area of a large rectangular property is 8.6 ha. Calculate the width if its length is
400 m. Select the correct answer A, B, C or D.
A 21.5 m B 215 m C 2150 m D 0.0215 km

Investigation
Area of a trapezium, kite and rhombus
1 Draw 2 copies of this trapezium onto a sheet
Side 1
a
of paper. Draw a dotted line, parallel to the
2
h

sides labelled a and b, halfway down the

height of each trapezium and label the sides 2 Side 2
h

as shown. b
2 Cut out one trapezium and cut along the
dotted line to make 2 smaller trapeziums. Join the pieces to make a long parallelogram.
3 By measuring the height and base of the parallelogram to the nearest millimetre, find its area.
4 By comparing the measurements of the parallelogram to the measurements of the
original trapezium, suggest a general formula for finding the area of any trapezium.
Check your answer with your teacher.
5 Cut out the original trapezium and paste it and the parallelogram into your book.
6 Draw 2 copies of the kite and rhombus on a sheet of paper. Draw diagonals of length x and
y and label them as shown. Notice that they cross at right angles. For the kite, one diagonal
(x) is bisected by the other (y), while for the rhombus, both diagonals bisect each other.

1x 1x
2 2
1x 1y
y 2 2
1y
2 1x
2 x and y are the lengths
of the diagonals

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9780170453325 Chapter 10 | Surface area and volume 365

  7 Cut out one of your kites and rhombuses and cut along their diagonals to make
4 triangles each. Join each set of triangles to make a rectangle.
 8 By measuring the sides of each rectangle to the nearest millimetre, find the area of each one.
 9 By comparing the measurements of the kite to the measurements of the rectangle
formed from it, suggest a general formula for finding the area of any kite. Check your
answer with your teacher.
10 Is a rhombus a special type of kite?
11 Suggest a general formula for finding the area of any rhombus. Check your answer with
your teacher.
12 Cut out the original kite and rhombus and paste all shapes into your book.

10.04 Areas of quadrilaterals

Areas of special quadrilaterals

Parallelogram Trapezium
a

h h

b
b
1
Area = base × perpendicular height Area = × (sum of parallel sides)
2
A = bh × perpendicular height
1
A = (a + b)h
2
Rhombus Kite

x y
y x

1
Area = × (product of diagonals)
2
1
A = xy
2

366 New Century Maths 9 Advanced 9780170453325

 Example 7
Find the area of each quadrilateral.
a b c 16 mm
9m
20 cm 12 mm

38 cm 14 m 30 mm
10.04
d 9m e

15 cm
10 cm
14 m
11 cm

Solution
1 1
Area of rhombus = × 38 × 20
2 2
a A = xy
= 380 cm2
b Area of parallelogram = 14 × 9 A = bh
= 126 m 2

1 1
Area of trapezium = × (16 + 30) × 12 A = (a + b)h
2 2
c
= 276 mm2
1 1
Area of kite = × 9 × 14
2 2
d A = xy
= 63 m2
1 1
Area of trapezium = × (10 + 15) × 11 A = (a + b)h
2 2
e
= 137.5 cm2

EXERCISE 10.04 ANSWERS ON P. 593

Areas of quadrilaterals U F PS R
1 Find the shaded area of each shape.
b
EXAMPLE
a c 7
17 mm

4m
6 cm

20 mm
8 cm
13.6 m

f
12 cm
d e
10 mm
1.5 cm

30 mm
25 mm

2 cm
7 cm

50 mm

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9780170453325 Chapter 10 | Surface area and volume 367

 h i
11 m
30 mm
g

4m

12 mm
22 mm

7 cm
m
64 mm

4.4
10.4 5m

4.4 c

7m
10.4

c m
2m 3m
cm cm
2 Find the area of each shape.
b
12 cm 62 mm
a c

49 mm

28 cm
17 cm
15 cm

54 m e f
20 mm
d
25 mm 23 cm
m

31 mm 25 mm
20 mm 31 cm

18
g h i
10 cm 14 m
14 cm 14 cm 9m
m

19 cm

m
32
3 Find the area of each shape.
a 9m b 5.5 m c 48 mm

7m 8m 20 mm

15 m 11 m 32 mm

d 16 m e f 2.1 m

10 m 10.5 cm 1.8 m
7.5 cm

22 m 1.4 m
6 cm
h i 20 mm
12 m
g
7 cm 14.5 m

35 mm
9 cm 6 cm 9.6 m

28 mm

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368 New Century Maths 9 Advanced 9780170453325

 4 The area of the trapezium is 96 cm2. What is the length
of the missing side? Select the correct answer A, B, C or D.
k cm

A 10 cm B 15 cm 8 cm
C 17 cm D 12 cm
9 cm
5 Find the area of each shape.
13 cm c 5m
9m
4m
a b

5m 4m 7m 10 cm 8 cm 17 cm 5m 3m
10.04

4m 5m
3m
14 m 22 cm
5m

f 5.5 m
10 cm 10 cm
d e
6 cm 9.6 cm 6.6 m
8 cm 8 cm
8.8 m
10.5 cm
17 cm 17 cm
15 cm

6 Calculate each shaded area (all measurements are in metres). PS

15 c 1.2
R

5
a b
12
10 10 20 15 15
18.6 2
12 4.5
1.2
20

Mental skills 10A: Maths without calculators ANSWERS ON P. 593

Finding 10%, 20% and 5%
1
10
To find 10% or of a number, simply divide the number by 10 by moving the decimal point
one place to the left.
1 Study each example.
a 10% × 150 = 15 0. = 15
b 10% × $1256.80 = $125 6. 8 = $125.68
c 10% × 4917 = 491 7. = 491.7
d 10% × $48.55 = $4 8. 55 = $4.885

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 369

 2 Now find 10% of each amount.
a 190 b $75 c 875 d $202
e $37.60 f 400 g $9.25 h 896
i $2700 j $3.80 k $1527.60 l $72.50
m 3154 n $10.70 o 426 p $24 317.60

3 Study each example.

20% is 10% doubled, so to find 20% of a number, first find 10% then double it.

a 20% × 700 b 20% × $876

10% × 700 = 70 10% × $876 = $87.60
∴ 20% × 700 = 70 × 2 ∴ 20% × $876 = $87.60 × 2
= 140 = $175.20
c 20% × 325 d 20% × $38.50
10% × 325 = 32.5 10% × $38.50 = $3.85
∴ 20% × 325 = 32.5 × 2 ∴ 20% × $38.50 = $3.85 × 2
= 65 = $7.70

4 Now find 20% of each amount.

a 50 b 620 c $2450 d $8.60
e 72 f $12 700 g 390 h $5.80
i $45 j $84 k $4600 l 320
5% is half of 10%, so to find 5% of a number, first find 10% then divide it by 2.
5 Study each example.
a 5% × 180 b 5% × $76
10% × 180 = 18 10% × $76 = $7.60
∴ 5% × 180 = 18 ÷ 2 ∴ 5% × $76 = $7.60 ÷ 2
=9 = $3.80
c 5% × 120 d 5% × $142.20
10% × 120 = 12 10% × $142.20 = $14.22
∴ 5% × 120 = 12 ÷ 2 ∴ 5% × $142.20 = $14.22 ÷ 2
=6 = $7.11

6 Now find 5% of each amount.

a 2000 b $12 c 50 d $27
e $36.80 f $84 g 800 h 130
i $9.60 j $138 k $72 l 840

370 New Century Maths 9 Advanced 9780170453325

 Circumferences and areas of circular
10.05
shapes
The circumference of a circle is found by multiplying its diameter by a special number called
pi (pronounced ‘pie’), represented by the Greek letter π. WS

The area of a circle is found by multiplying π by the radius squared. Worksheet

Homework
A page of
circular shapes

10.05
Circumference of a circle WS

The circumference (perimeter) of a circle is:

Worksheet
Homework
Applications
of area

C = π × diameter
C = 2 × π × radius
or
d
C = πd C = 2 πr
WS

r Worksheet
Homework
Back-to-front
problems

Area of a circle
The area of a circle is:
WS

A = π × (radius)2
Worksheet
Homework
Area and
perimeter
investigations

A = πr2

π = 3.141 592 653 897 93... and can be found by pressing the π key on your calculator. It has
Carpet talk

no exact decimal or fraction value, so like a surd such as 2 , it is an irrational number.

Example 8
For each circle, calculate correct to 2 decimal places:
i  its circumference ii its area.

a b
8.5 m
20 m

Solution
a i C = π × 20 C = πd b i C = 2 × π × 8.5 C = 2πr
= 62.8318… = 53.4070…
≈ 62.83 m ≈ 53.41 m
ii A = π × 102 A = πr2 ii A = π × 8.52 A = πr2
= 314.1592… = 226.9800 …
≈ 314.16 m 2
≈ 226.98 m2

9780170453325 Chapter 10 | Surface area and volume 371

 Example 9
For each sector, calculate correct to one decimal place: A sector is a fraction of a circle
‘cut’ along 2 radii, like a pizza slice.
Perimeter
and area of a
i  its perimeter ii its area.
sector
a b c
120° 5m
3m
80° 4.2 m
O

3m

Solution
1
This sector is a quadrant, a quarter of a circle. × circumference + radius + radius
4
1
a i 
Perimeter = ×2×π×3+3+3
4
= 10.71238...
= 10.7 m
1 1
Area = × π × 32 × area of circle
4 4
ii
= 7.06858...
= 7.1 m2
80 80
b i Perimeter = × 2 × π × 4.2 + 4.2 + 4.2 × circumference + radius + radius
360 360
= 14.26430... There are 360° in a circle, but
= 14.3 m a sector is a fraction of a circle.
80 80
Area = × π × 4.22 × area of circle
360 360
ii
= 12.31504...
= 12.3 m2
240
c i Sector angle = 360° – 120° ii Area of sector = × π × 52
360
= 240° = 52.35987...
240
Perimeter = × 2 × π × 5 + 5 +
5 = 52.4 m2
360
= 30.94395...
= 30.9 m

Arc length, perimeter and area of a sector

For a sector with angle θ:

• Arc length = × 2 πr
360
θ
arc length θ
• Perimeter of the sector = × 2 πr + r + r
360
θ r

(arc length + radius + radius)

• Area of the sector = × πr 2

360
θ

372 New Century Maths 9 Advanced 9780170453325

 Example 10
Find, correct to one decimal place, the area of each shape.

7m
a b

12 m
22 m

10.05

30 m

Solution
a The shape is made up of a rectangle and b This ring shape is called an annulus, it
a quadrant. is the area enclosed by 2 circles with the
Radius of quadrant    = 7 m same centre.
1
Length of rectangle = 22 – 7 Radius of large circle = × 30 m
2
= 15 m = 15 m
1
Area of shape = area of rectangle Radius of small circle = × 12 m
2
+ quadrant =6m
1
= 15 × 7 + × π × 72 Shaded area = large circle – small circle
4
= 143.4845… = π × 152 – π × 62
≈ 143.5 m2 = 593.7610...
≈ 593.8 m2

EXERCISE 10.05 ANSWERS ON P. 593

Circumferences and areas of circular shapes U F PS R
1 For each circle, calculate correct to 2 decimal places:
i  its circumference ii its area.
EXAMPLE
8

a b c d

8 cm 28.2 cm
30 cm
5.2 cm

2 For each sector, calculate correct to one decimal place:

i  its perimeter ii its area.
EXAMPLE
9

a b c d

10 m 210°
120°
36 m 24 m
1m
10 m

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 373

 e f g h

6m 42 m
33 m 8 m 10° 8 m
30° 60°
6m

3 Find the perimeter of this sector. Select the correct answer A, B, C or D.

A 12.6 cm B 18.3 cm 6 cm
C 24.6 cm D 37.1 cm 120°

4 Find the area of this sector. Select the correct answer A, B, C or D.

A 19.6 cm2 B 15.3 cm2 2.5 cm
C 8.5 cm2 D 4.4 cm2
80°
2.5 cm
5 Calculate the perimeter of each shape, correct to 2 decimal places.
a 40 cm b    c   26 mm

40 cm 22 mm

15 cm

12 cm
d  
e    150 cm f

11 m 11 m 90 cm
65 cm 248 m

30 m 248 m
g    
h    15 cm i

8 cm 18 cm

9 cm
7m 7m

18 cm
15 m
6 Calculate the area of the shape. Select the correct answer A, B, C or D.
EXAMPLE
10

7 cm

2 cm 2 cm

A 26.57 cm2 B 20.28 cm2 C 24.28 cm2 D 28.28 cm2

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374 New Century Maths 9 Advanced 9780170453325

 7 Calculate the area of each shape, correct to the nearest square metre.
All measurements are in metres.
a b 15 c
22
12
40 26

40
10.05

d e f

6 3 3
22
7 7 8

22 6

g h i

6 8 30

14
30
16

45

j k l

10 10 90
20
20
150 40

8 A circular playing field has a radius of 80 m. A rectangular cricket pitch measuring 25 m

by 2 m is placed in the middle. The field, excluding the pitch, is to be fertilised. PS R
a Calculate, to the nearest square metre, the area to be fertilised.
b How much will this cost if the fertiliser is $19.95 per 100 square metres?
Give your answer to the nearest dollar.

9 The diameter of the Earth is 12 756 km.

Find the circumference of the Earth
to the nearest kilometre.
a

b The International Space Station

Shutterstock.com/Dima Zel

orbits the Earth at an average height

of 400 km above the Earth’s surface.
Find the distance it travels in one
orbit of the Earth.

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 375

 10 A circular plate of diameter 2 m has 250 holes of diameter 10 cm drilled in it. What is
the remaining area of the plate? Answer to the nearest 0.1 m2.

11 A circular pond of diameter 10 m is surrounded by a path one metre wide.

Calculate the area of the path, correct to 2 decimal places.
PS R

a
b If pavers are $165 per square metre laid, what is the cost of the path?

12 A circular patch of grass has a diameter of 8.5 m. How

much further does Arya walk if she walks around the 8.5 m
border instead of directly across it? Give your answer
correct to the nearest 0.1 m. PS R

13 A new tractor tyre has a diameter of 120 cm, while a worn

tyre has a diameter of 115 cm.
Calculate the difference in circumference between a
new and a worn tyre, correct to 3 decimal places.

Shutterstock.com/Anatoliy Kosolapov
a

b Over 1000 revolutions, how much further (to the

­nearest metre) will a new tyre travel compared to
a worn tyre?

14 A square courtyard measuring 5 m by 5 m has a semi-

circular area added to each side. PS R
Calculate the area of the semi-circular additions,
correct to the nearest square metre.
a

b By what percentage has the area of the courtyard increased?

increase in area
(This can be calculated as × 100%)
original area
15 The measurements of a running track are shown below. Each lane is 1 m wide and the
athletes run along the insides of the lanes. PS R
LANE 2
staggered start lane 2 FINISH

start lane 1 LANE 1

36.5 m

85 m

Izabella runs one lap in lane 1. What distance does she cover, correct to
one decimal place?
a

b Noah runs one lap in lane 2. What distance does he cover, correct to one
decimal place?
By how much should the start line be staggered in lane 2 (from lane 1)
so that Izabella and Noah run the same distance in one lap?
c

Foundation Standard Complex

376 New Century Maths 9 Advanced 9780170453325

 16 The wheels of a bicycle have a diameter of 66 cm.
a How far (correct to 2 decimal places) will the bicycle move after one rotation of the wheel?
b How many rotations of the wheel are needed to cover a distance of 5 km?

Did you know?

A piece of pi
π = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820…
10.06

Because π is an irrational number, its decimal digits run endlessly without repeating. Over
history, mathematicians and scientists (and now computer scientists) have tried to calculate
more accurate values of π, using more sophisticated formulas and calculation techniques.
The ancient Greeks, Romans and Chinese first estimated π as 3. This value is also mentioned
in the Bible. Since the first computer, the ENIAC, was invented in 1949, much progress has
been made. Supercomputers have been used to calculate more decimal places of π and, in
2019, Emma Haruka Iwao (Japan), working for Google, calculated π to over 31.4 trillion
places. In 2015, Suresh Sharma (India) memorised π to 70 030 decimal places. It took him
17 hours 14 minutes to recite it.
Research the modern history of π and find out about recent calculations of its value.

Surface area of a prism 10.06

The surface area of a solid is the total area of all the faces of the solid. To calculate the surface
area of a solid, find the area of each face and add the areas together. WS

Worksheet
Homework
Surface area

Example 11
Calculate the surface area of each prism.
WS

b rectangular prism c open rectangular prism

Worksheet
a cube
Homework
Nets of
solids

4 cm 12 cm

4 cm

12 cm 5 cm
Solid

4 cm 4c
shapes

8 cm
m

25 cm Presentation
Surface
area

Solution
a A cube has 6 identical faces in the shape of a square.   6 × area of a square
Surface area = 6 × 42
Technology
Surface area

= 96 cm2
calculator

Surface area
calculator

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 377

 b A rectangular prism has 6 faces that are not all the same. However,
opposite faces, such as the top and bottom, are the same. For the surface area of an
open solid, we only count
Surface area = 2 top and bottom faces + 2 end faces + 2 side faces the external surfaces, not the
= (2 × 5 × 12) + (2 × 5 × 4) + (2 × 12 × 4) internal ones (so that we don’t
count each surface twice).
= 256 cm2
c The open rectangular prism has no top so it has 5 faces that are not
all the same.
Surface area = Bottom face + 2 front and back faces + 2 side faces
= (25 × 8) + (2 × 25 × 12) + (2 × 8 × 12)
= 992 cm2

Example 12
Calculate the surface area of each triangular prism.

5 cm
a b

3 cm
Surface area
of a prism

10 cm
8 cm 24 m
nm

30 m
Surface area

10 m
of prisms

Solution
a This triangular prism has 5 faces: 2 identical triangles 3 cm
5 cm
(front and back), 2 identical rectangles (sides) and
8 cm
another rectangle (bottom). Drawing the net of the
prism may help you see this. 10 cm

1
Surface area = (2 × × 8 × 3) + (2 × 10 × 5) + (8 × 10) Front and back faces +
2
= 204 cm 2 side faces + bottom face

This triangular prism has 5 faces: 2 identical triangles

(front and back), 3 different rectangles. 24
b 
n
To calculate the area of the left face, we need to know
10
the value of n, which can be found using Pythagoras’
30
theorem.
n2 = 102 + 242 10
= 676

n = 676
= 26
1
Surface area = (2 × × 10 × 24) + (30 × 26) Front and back + left + bottom
2
+ (30 × 10) + (30 × 24) + right
= 2040 m2

378 New Century Maths 9 Advanced 9780170453325

 Example 13
Calculate the surface area of this trapezoidal prism. 10 cm
13 cm
15 cm
12 cm
Surface area
of a prism

18 cm
24 cm
10.06
Solution
This trapezoidal prism has 6 faces: 2 identical 10
trapeziums (front and back), 4 different 12
10
rectangles. 24

15

13
18

1
Area of each trapezium = × (10 + 24) × 12
2
= 204 cm2
Surface area = (2 × 204) + (18 × 10) 2 trapeziums + 4 rectangles
+ (18 × 15) + (18 × 24)
+ (18 × 13)
= 1524 cm2

EXERCISE 10.06 ANSWERS ON P. 594

Surface area of a prism U F PS R
1 What is the surface area of this rectangular prism?
Select the correct answer A, B, C or D. 8 cm
EXAMPLE
11

A 320 cm2 B 352 cm2

C 512 cm2 D 768 cm2 8 cm
12 cm
2 Draw a net of each solid.
a cube b rectangular prism c triangular prism d cylinder

3 Match each net of a solid to the name of the solid.

triangular prism cube rectangular prism
square pyramid cylinder trapezoidal prism
a b c

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 379

 d e f

g h

4 Name the prism that each net represents, then calculate the surface area of the prism.
All lengths are in metres.
EXAMPLE
12

a b 15
12

21 6

13
24 26 25
24
c d

13
45
30

51 66

72

5 Calculate the surface area of each prism.

b rectangular prism open cube (no top)
PS R

a cube c

45 cm

6.5 m 144 cm

20 cm
18 cm

Foundation Standard Complex

380 New Century Maths 9 Advanced 9780170453325

 d open rectangular prism e rectangular prism, f cube, open one end
open front and back

35 mm

3.4 m
7.5 m 48 mm
65 mm
5.2 m

44 cm 10.06

6 Find the surface area of each triangular prism.

PS R

10 m 29 cm
10 m
a b c

15 cm
h
8m 6m 40 cm
7m
15 m
16 m
6m

8 cm f 9m
m
.5 c
d e

3
r

3.7 cm 12 m
15 m
10 m
5m
12 m

7 Yasmin made this tent.

Find the surface area of the tent, including the floor.
PS R

19
a
b Yasmin bought material at $12 per square metre. 7c
How much did it cost her for the material to make the tent? m m
0c
21
180 cm

160 cm
8 5 cubes of side length 1 cm are joined to make this solid.
What is the surface area of the solid? Select the correct
answer A, B, C or D.
A 17 cm2 B 18 cm2
C 5 cm2 D 22 cm2

9 An art gallery has a width of 16 metres, a length of

30 metres and a slant height of 12 metres, as shown in
the diagram. PS R
Find the perpendicular height, h, of the building,
correct to one decimal place.
a
12 m

b Find the area of the front wall.

hm

Find the surface area of the art gallery, 30 m

16 m
without the floor.
c

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 381

 10 Calculate the surface area of each prism. Measurements are in centimetres.
b 75 c 35
PS R
EXAMPLE

14
13

52
a

20
20
24 48 2048 15 12
17 40
32 10

11 This swimming pool is 15 m long and 15 m

10 m wide. The depth of the water 10 m 1m
ranges from 1 m to 3 m. PS R
Calculate, correct to 2 decimal 3m
places:
the area of the floor of the
pool
a

b the total surface area of the pool.

Investigation
Surface area of a cylinder r

Collect 5 different cylindrical cans with paper labels.

1 Copy this table with rows for Cans 1 to 5 and complete
height, h

it as you perform the following measurements and calculations.

Circular ends Curved surface Surface area

Height Diameter Radius Area Circumference Length Width Area

Can 1

Can 2

2 Measure the height and diameter of the first can.

3 Calculate the radius, area and circumference of one circular end, correct to 2 decimal
places.
4 Cut the label off the first can and lay it out flat. What shape is this curved surface?
5 Measure the length, width and area of the curved surface (label).
6 Calculate the surface area of the can by adding the areas of the 2 circular ends and the
area of the label.
7 Why is the height of each can and the width of its label the same?
8 Why is the circumference of each circular end and the length of the label the same?
9 Repeat Steps 2 to 6 for the other 4 cans.

Foundation Standard Complex

382 New Century Maths 9 Advanced 9780170453325

 Surface area of a cylinder 10.07
A closed cylinder has 3 faces made up of 2 circles (the circular
ends) and a rectangle (the curved surface). The length of
r

the rectangle is the circumference of the circular end, while

WS

the width of the rectangle is the height of the cylinder.

Worksheet
Homework
Surface area

= 2πr
circumference
Surface area of a cylinder = area of 2 circles
height, h 10.07
WS

+ area of rectangle
SA = 2 × πr2 + 2πr × h
Worksheet
Homework
r Applications
of area 3

= 2πr2 + 2πrh

Surface area of a closed cylinder r

Car song

SA = 2πr2 + 2πrh
where r = radius of circular base
height, h Technology
Surface area

h = perpendicular height
calculator

The area of the 2 circular ends = 2πr2 and the area of the curved surface = 2πrh.
Surface area
calculator

Example 14
Find, correct to one decimal place, the surface area of a cylinder with diameter 12 cm and
Surface

height 20 cm.
area of a
cylinder

Solution
Radius = 1 × 12 cm    1 of diameter
2 2
= 6 cm
20 cm

12 cm
Surface area = a rea of 2 ends + area of the curved surface

SA = 2 πr 2 + 2 πrh
= 2 × π × 62 + 2 × π × 6 × 20
= 980.1769...
≈ 980.2 cm2
12 cm
r = 6 cm

circumference
height
curved surface

end

9780170453325 Chapter 10 | Surface area and volume 383

 Example 15
Find, correct to 2 decimal places, the surface area of this
open half-cylinder with radius 0.5 m and height 3 m. 3m
Note: For the surface area of an open solid, we only count the
external surfaces, not the internal ones (so that we don’t count
0.5 m
each surface twice).

Solution
1
Surface area = 2 semicircle ends + × curved surface
2
end

SA = 2 × ( 1 × π × 0.52 ) + 1 × (2 × π × 0.5 × 3)
2 2
= 5.49778... surface 3 m
curved

≈ 5.50cm2

0.5 m

EXERCISE 10.07 ANSWERS ON P. 594

Surface area of a cylinder U F PS R
1 Calculate, correct to 2 decimal places, the surface area of a cylinder with:
radius 7 m, height 10 m b diameter 28 cm, height 15 cm
EXAMPLE
14
a
c diameter 6.2 m, height 7.5 m d radius 0.8 m, height 2.35 m

2 Find, correct to one decimal place, the curved surface area of a cylinder with:
a diameter 9 cm, height 32 cm b radius 85 mm, height 16 mm

3 Calculate, correct to one decimal place, the surface area of each solid. All lengths shown
are in metres. PS R
EXAMPLE

closed cylinder b cylinder with closed half-cylinder

15

one open
openend
a c
cylinder with one
closed cylinder end closed half-cylinder
0.37

25 16.2
1.5
29.3
15

d half-cylinder e half-cylinder with f cylinder open
with
openopen
top top top, open top, one end open at both
both ends
half-cylinder with half-cylinder with open cylinder open at

1.2 1.5
one end open ends

2.85 5.75

30
12

Foundation Standard Complex

384 New Century Maths 9 Advanced 9780170453325

 4 A swimming pool is in the shape of a cylinder 1.5 m
deep and 6 m in diameter. The inside of the pool is to
be repainted, including the floor. Find: PS 1.5 m
the area to be repainted, correct to
one decimal place
a

6m
b the number of whole litres of paint needed
if coverage is 9 m2 per litre.

5 Which tent has the greater surface area?

10.08

b
2.24 m
a

2m
5m
2m 5m
2m

(Note: the floor is included for both tents)

6 This lampshade is to be covered. Find the area of

material needed, correct to one decimal place, if 10%
extra is allowed for seams and overlaps. Note:
The lampshade is a cylinder open at both ends. PS
30 cm

20 cm

Volumes of prisms and cylinders 10.08

The volume of a solid is the amount of space it takes up. Volume is measured in cubic units,
for example, cubic metres (m3) or cubic centimetres (cm3).
WS

The capacity of a container is the amount of fluid (liquid or gas) it holds, measured in
millilitres (mL), litres (L), kilolitres (kL) and megalitres (ML).
Worksheet
Homework
Volume and
capacity

Volume and capacity

1 cm contains 1 mL
What is

3
volume?

1 m3 contains 1000 L or 1 kL 1 mL
1 m3 = 1 kL
1 cm3 × 1 000 000 =
Formula
matching
game

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 385

 Volume of a prism
A cross-section of a solid is a ‘slice’ of the solid cut across it, parallel to its end faces, rather
than along it. A prism has the same (uniform) cross-section along its length, and each cross-
section is a polygon (with straight sides).
cross-sections

Triangular prism Square prism

An end face of a prism is called its base. Prisms take their names from their base and cross-
section. For example, this prism is a trapezoidal prism because its base and cross-sections are
trapeziums.

base cross-section

Trapezoidal prism

Because a prism is made up of identical cross-sections, its volume can be calculated by

multiplying the area of its base by its perpendicular height (the length or depth of the
prism).

Volume of a prism
V = Ah
where A = area of base
h = perpendicular height
A h

Example 16
Find the volume of each prism.
b
15 cm 4 cm
a c

30 cm 3m 4c
Volumes of

10 m
3 cm

prisms and

42 cm 6 cm
cylinders m

5m

386 New Century Maths 9 Advanced 9780170453325

 Solution
V = 42 × 30 × 15 For a rectangular prism,
volume = length × width × height (V = lwh)
a
= 18 900 cm 3

1
A  = × 5 × 3 Area of a triangle.
2
b
= 7.5
V = 7.5 × 10 V = Ah, where height h = 10
= 75 m
10.08
3

1
A  = × (4 + 6) × 3 Area of a trapezium
2
c
= 15 cm2
V = 15 × 4 V = Ah, where height h = 4
= 60 cm 3

Example 17
A storage box in the shape of a rectangular
prism is 110 cm long, 65 cm wide and

70 cm
70 cm high.
a Find the volume of the box in cm3.
b What is the volume in m3? 110 cm

cm
65
Solution
a V = 110 × 65 × 70

= 500 500 cm3
Alternatively, you could convert
b 1 m = 100 cm, so 1 m3 = 1003 cm3 = 1 000 000 cm3
110 cm, 65 cm and 70 cm to metres,
∴ V = 500 500 ÷ 1 000 000 then multiply them together for the
volume.
= 0.5005 m3

Volume of a cylinder
A cylinder is like a ‘circular prism’ because its cross-sections are identical
circles. Because of this, we can also use V = Ah to find the volume of a
r

cylinder. But for a circle, A = πr2, so:

Volume = Ah
h

V = πr2 × h
= πr2h

Volume of a cylinder
V = πr2h
where r = radius of circular base
h = perpendicular height

9780170453325 Chapter 10 | Surface area and volume 387

 Example 18
For this cylinder, calculate:
128 cm

a its volume, correct to the nearest cm3

Volumes of
prisms and
cylinders

b its capacity in kL, correct to 1 decimal place. 241 cm

Solution
a Radius = 1 × 128 cm 1 of diameter
2 2
= 64 cm
V = π × 642 × 241 V = πr2h
= 3 101 179.206…
≈ 3 101 179 cm3
b Capacity = 3 101 179 mL 1 cm3 = 1 mL
= (3 101 179 ÷ 1000 ÷ 1000) kL
= 3.101 179 kL ÷ 1000 ÷ 1000
kL L mL

≈ 3.1 kL

EXERCISE 10.08 ANSWERS ON P. 594

Volumes of prisms and cylinders U F PS R C
1 Find the volume of each rectangular prism using the formula V = lwh.

EXAMPLE
16
a b c
2.5 m

4m 60 mm
40 cm

5m
22 25 mm
40 mm
15 cm
cm

d e f
0.6 m

30 cm

1.2 m 1.8 m
32 mm 14
cm

2 The volume of a cube is 125 000 cm3. Convert this to m3. Select the correct
answer A, B, C or D.
A 1250 m3 B 12.5 m3 C 1.25 m3 D 0.125 m3
EXAMPLE
17

3 The volume of a small jewellery box is 160 000 mm3. Which expression gives its volume
in cm3? Select the correct answer A, B, C or D.
A 160 000 ÷ 103 B 160 000 ÷ 100
C 160 000 ÷ 100 2
D 160 000 ÷ 102

Foundation Standard Complex

388 New Century Maths 9 Advanced 9780170453325

 4 Find the volume of this toolbox in cubic metres by:
calculating the volume in cm , then
PS R C

converting to m3
a

b converting each length to metres first, 40 cm

then calculating the volume.
60 cm
Which method is easier?
120 cm
5 Find the volume of each solid. 10.08

a A = 63.1 m2 b c 64 cm

8m 38 cm
A = 27.5 cm2 A = 312 cm2

6 Find, correct to nearest whole unit, the volume of each cylinder.

8 cm
EXAMPLE
18
a b c

15 cm
1.6 m
35 mm 0.8 m
12
mm

d e f

1.5 m 37
cm
3.5 m

8m 4.8 m
45
cm

7 Calculate, correct to one decimal place, the volume of each solid. All lengths are in
metres.
a 1.8 2.4 b 25 c

3.0 48 0.8 3.7

4.5 2.5

f
5.2
d e

4.2 3.6 4.5

32 9.2
10.1 20 7.9
6.4
i 7.7
2.8
g h

3.5 2.4 5.5

3.5 12.8 11.3
7.2
5.6

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 389

 8 What is the capacity of this cylinder in litres?
2m
Select the closest answer A, B, C or D.
A 33 900 L B 8482 L

2.7 m
C 6280 L D 16 964 L

9 A triangular prism has base length 15 cm, height 8 cm and volume 630 cm3.
What is its length? R

10 A fish tank that is 55 cm long, 24 cm wide and 22 cm high is filled to 4 cm below the top.
Calculate the amount of water in the tank in litres.

11 A wedding cake with 3 tiers rests on a table. Each tier

is 6 cm high. The layers have radii of 20 cm, 15 cm and
10 cm respectively. Find the total volume of the cake,

Shutterstock.com/John Wollwerth
correct to the nearest cm3. 6
10
6
15
6
20

12 Calculate the volume of the skip bin,

correct to the nearest m3. 3.7 m

1.29 m

Shutterstock.com/
2m

Stephen Rees
1.64 m
13 This swimming pool is 15 m long and 15 m
10 m wide. The depth of the water 10 m 1m
ranges from 1 m to 3 m.
Calculate the capacity of this pool in 3m
kilolitres.
14 Robyn and Anthony are planning to make a
raised vegetable garden. If the dimensions
of the garden bed are 7.5 m × 9 m × 60 cm,
calculate how much soil they will need to buy.
Select the correct answer A, B, C or D.
Shutterstock.com/
Alison Hancock

A 405 m3 B 40.5 m3
C 4050 cm3 D 4050 m3

15 This metal pipe has an inner diameter of 8.5 cm and

an outer diameter of 9.5 cm. Calculate, correct to 2 decimal
8.5 cm

9.5 cm

places, the volume of metal needed to make the pipe. PS R

48 cm

Foundation Standard Complex

390 New Century Maths 9 Advanced 9780170453325

 16 A cube has a volume of 2150 mm3. What is its side length, correct to the nearest
millimetre? R

17 The official diameter of a tennis ball is defined as 6.54 cm to 6.86 cm. PS R C

10.08

   
a Find the average diameter of a tennis ball.
b The tennis balls are packed in cylinders as shown. Using the larger diameter,
calculate the radius and height of one cylinder.
i To find the volume of the cylinder, its radius and height are rounded up to one
decimal place. Explain why, and write the radius and height rounded up.
c

ii Hence, calculate the volume of a cylinder, correct to one decimal place.

d For packaging, 12 cylinders are put in a cardboard box, as shown above.
i What are the dimensions of the box?
ii Calculate the volume of the box.
e Calculate the volume of the box not taken up by the cylinders.
What percentage (correct to one decimal place) of the box is not taken up by the
cylinders?
f

Investigation
Volume vs surface area
You will need: At least 20 centicubes
1 Copy the following table.

Length Width Height Volume Surface area

8
8


2 Build as many rectangular prisms as you can with a volume of 8 cubes, but with different
dimensions. Note: 1 × 1 × 8 is the same as 1 × 8 × 1 and 8 × 1 × 1.
3 Record your dimensions in the table and calculate the surface area of each prism.
4 What are the dimensions of the solid that has the smallest surface area?
5 What name do we give to this solid?
6 Repeat the experiment for a volume of:
a 12 cubes b 20 cubes.

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9780170453325 Chapter 10 | Surface area and volume 391

 Mental skills 10B: Maths without calculators ANSWERS ON P. 594
Finding 15%, 2 1 %, 25% and 12 1 %
1
2 2

10
• To find 10% or of a number, divide by 10.
• To find 5% of a number, find 10% first, then halve it (since 5% is half of 10%).
• So to find 15% of a number, find 10% and 5% of the number separately, then add the

1 Study each example.

answers together.

15% × 80 = (10% × 80) + (5% × 80) b 15% × $170 = (10% × $170)

+ (5% × $170)
a
= 8 + 4
= 12 = $17 + $8.50
= $25.50
15% × 3600 = (10% × 3600) d 15% × $28 = (10% × $28)
+ (5% × 3600) + (5% × $28)
c

= 360 + 180 = $2.80 + $1.40

= 540 = $4.20

2 Now find 15% of each amount.

a 120 b $840 c 260 d $202
e $50 f 72 g $180 h 400
i $1600 j $22 k 6000 l $350
To find 2 1 % of a number, first find 5%, then halve it.
2
3 Study each example.
2 1 % × 600 b 2 1 % × $820
2 2
a
10% × 600 = 60 10% × $820 = $82
1
5% × 600 = × 60 = 30 1
2 5% × $820 = × 82 = $41
1 2
2 1 % × 600 = × 30 = 15 1
2 2 2 1 % × $820 = × $41 = $20.50
2 2

4 Now find 2 1 % of each amount.

2
a 400 b 6640 c $2000 d $880
e 1500 $232 g 5400 h $904
1
f

4
To find 25% of a number, halve the number twice as 25% = .
5 Study each example.
a 25% × 700 b 25% × $86
1 1
50% × 700 = × 700 = 350 50% × $86 = × $86 = $43
2 2
1 1
25% × 700 = × 350 = 175 ∴ 25% × $86 = × $43 = $21.50
2 2

392 New Century Maths 9 Advanced 9780170453325

 6 Now find 25% of each of each amount.
a 2000 b $80 c 18 d $25
e $324 f $140 g 66 h 298
i $780 j $1700 k $126 l 1160

To find 12 1 % of a number, find 25% first, then halve it.

2 1
In other words, halve 3 times because 12 1 % = .
2 8
7 Study each example.
10.08

12 1 % × 400 12 1 % × $144
2 2
a b
1 1
50% × 400 = × 400 = 200 50% × $144 = × $144 = $72
2 2
1 1
25% × 400 = × 200 = 100 25% × $144 = × $72 = $36
2 2
1 1
12 1 % × 400 = × 100 = 50 12 1 % × $144 = × $36 = $18
2 2 2 2

8 Now find 12 1 % of each amount.

2
a 1280 b $12 c 60 d $260
e $540 f $250 g 304 h 1360

Technology
Approximating the volume of a pyramid Technology
Measuring

We can approximate the volume of a pyramid by dividing it into of layers of

pyramids

prisms as shown below.

Technology
Approximating
the volume of
a cone

10
6
8
  
We will create a spreadsheet that approximates the volume of a rectangular pyramid with a
base of length 8 units and width 6 units, and a perpendicular height of 10 units.
The volume of each layer can be easily calculated using V = lwh. Finding the sum of the
layers will then give an approximation of the volume of the pyramid.
Let n be the number of layers. Then the height of each layer is 10 .
The length and width of each layer decreases from 8 units and 6 units by a constant amount
n

of n8 and n6 respectively from layer to layer.

9780170453325 Chapter 10 | Surface area and volume 393

 1 Create this spreadsheet for calculating the volue if each layer and the sum of the
volumes.

A B C D E F
1 Number of
layers
2
3 Height Length Width Thickness of Volume of Sum of
layer layer volumes
4 10 8 6 =$A$4/$D$2 =B4*C4*D4 =E4
5 =B4-$B4/$D$2 =C4-$C$4/$D$2 =E5+F4
6


13

2 a To divide the volume of the pyramid into 10 layers, enter 10 in cell D2.
b Copy each formula down to row 13.
c Explain the results in cells E13 and F13.
d How accurate was your result in F13? Explain.
e Print out your spreadsheet.
To divide the pyramid into 40 layers to calculate a better approximation, enter 40 in
cell D2 and copy each formula down to row 43.
3 a 

b In one or 2 sentences, compare your volume approximation in F43 with the previous
approximation in F13.
4 a Enter each of these values in cell D2, copy the formulas down to the appropriate
row and write down the approximation for the volume of the pyramid.
i 100 (copy down to row 104)
ii 200 (copy down to row 204)
iii 400 (copy down to row 404)
1
b Use the formula V = Ah to calculate the exact volume of the pyramid.
3
c Write a brief report about your results in parts a and b.

394 New Century Maths 9 Advanced 9780170453325

 Volumes of pyramids and cones 10.09
A pyramid is named by the shape of its base. STAGE 5.3

WS

Worksheet
Homework
Back-to-front
10.09
problems
(Advanced)

Square pyramid Triangular pyramid Rectangular pyramid

Volume of a pyramid
1
3
V= Ah h
where A = area of base

h = perpendicular height
A

Example 19
Find the volume of each pyramid.
a b

30 mm
8m

20 mm
25 mm 7m

10 m

Solution
A = 25 × 20 Area of rectangular base
= 500 mm2
a

1 1
V  = × 500 × 30 V= Ah, where height h = 30
3 3
    = 5000 mm3
1 Area of triangular base
× 10 × 7
2
b A=

= 35 mm2
1 1
V  = × 35 × 8 V= Ah, where height h = 8
3 3
1
     = 93 m3
3

9780170453325 Chapter 10 | Surface area and volume 395

STAGE 5.3
Example 20
Find the capacity (to the nearest millilitre) of a square
pyramid with base edge 64 mm and slant height 40 mm.

h 40 mm

Solution
First find h, the perpendicular height of the pyramid. 64 mm

h2 = 402 – 322 Using Pythagoras’ theorem

= 576
h = 576
= 24 mm
A = 642 Area of square base
= 4096 mm2
1 1
V= × 4096 × 24 V= Ah, where height h = 24
3 3
= 32 768 mm3
= 32.768 cm3 1 cm3 = 1000 mm3
Capacity = 32.768 mL 1 cm3 = 1 mL
≈ 33 mL

Example 21
The volume of a square pyramid is 100 cm3. If its height is 12 cm, calculate the length of its base.

Solution
Let the length of the square base of the pyramid be x cm.
1
   V = Ah

100 = × x2 × 12 x = 25
3
1

100 = 4x 2 =5
3

∴ Length of base = 5 cm.

  x2 =
100

  = 25
4

Volume of a cone
A cone is like a ‘circular pyramid’ so:
1
Volume = Ah
3
1
V = × πr2 × h
3
1 2
= πr h
3

396 New Century Maths 9 Advanced 9780170453325

 Volume of a cone STAGE 5.3

1 2
V = πr h
3
where r = radius of circular base and h = perpendicular height
h

10.09

Example 22
Calculate the volume of the cone, correct to the nearest cm3.
30 cm

21 cm

Solution
V = × π × 10.52 × 30 V = πr2h, where r = × 21 = 10.5
1 1 1

= 3463.6059…
3 3 2

≈ 3464 cm3

Example 23
A cone has a slant edge of 61 mm and a base radius of 61 mm
11 mm. Find its volume, correct to one decimal place. 11 mm
h

Solution
First, find the height, h.
h2 = 612 – 112 Using Pythagoras’ theorem
= 3600
h = 3600
= 60

1 1 2
V= × π × 112 × 60 V= πr h
3 3
= 7602.6542…
≈ 7602.7 mm2

9780170453325 Chapter 10 | Surface area and volume 397

STAGE 5.3 EXERCISE 10.09 ANSWERS ON P. 594
Volumes of pyramids and cones U F R
1 Calculate the volume of each pyramid (correct to one decimal place where necessary).

7 cm
EXAMPLE

10 cm
19 a b c
9 cm

8 cm
9 cm
10 cm
8 cm 8 cm
d e f
15 cm 8m
8m
14 m
12 cm 8m
20 cm
5m
18 m

2 This grain hopper is in the shape of an inverted square pyramid. 4.5 m

4.5 m
What is its capacity? Select the correct answer A, B, C or D.
A 15 m3 B 33.75 m3
C 50.625 m3 D 101.25 m3 5m

3 For each pyramid, find correct to 2 decimal places:

EXAMPLE i its perpendicular height, h ii its volume iii its capacity.
20

41 mm 41 mm
a b c
52 m
26 cm 25 m
30 m
h

9 mm 9 mm
h

20 cm 96 m
20 cm

160 cm
126 cm
d e f
68 mm 61 mm 8.5 m 8.5 m
116 cm
11 mm
11 mm 3.6 m 3.6 m 105 cm
32 mm 32 mm 3.6 m 3.6 m

Foundation Standard Complex

398 New Century Maths 9 Advanced 9780170453325

 4 The great pyramid of Khufu (or Cheops) in
STAGE 5.3

Egypt was built on a square base with side

lengths approximately 230 m.

Shutterstock.com/Jeremy Red
147 m
Find the volume in cubic metres if the
original height of the pyramid was 147 m.
a

b There are an estimated 2.3 million stone

230 m
blocks in the pyramid. Calculate the
average volume of each block.
10.09

5 The area of the base of a pyramid is 40 m2. If its volume is 360 m3, calculate its
perpendicular height. R

6 The volume of a square pyramid toy is 1620 mm3. If the length of its base is 8 mm,
EXAMPLE
21

calculate, correct to the nearest millimetre, the height of the pyramid. R

7 A square pyramid has a volume of 80 cm3 and a height of 10 cm. Calculate, correct to
one decimal place, the length of the base of the pyramid. R

8 Calculate the volume of each cone, correct to one decimal place.

EXAMPLE
a b c 22

15 mm
14 cm
8m
18 mm
17 cm
5m

7 cm 10 cm
d e f
12 cm
15 cm 30 mm

18 mm

9 Calculate the volume of each cone, correct to one decimal place.

44 m
13.5 cm
EXAMPLE
a b c 23

8 cm 10.8 cm
35 m
4 cm

0.8 m
68 m
d e f
83 cm
247 m
3.6 m
83 cm

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 399

 10 Calculate the volume of each cone, leaving your answer in terms of π.
STAGE 5.3

12 cm
6m
a b c

20 cm

15 cm

3m
8 cm

11 A cone has a volume of 1468 cm3 and a base radius of 12 cm. Find its height, correct to
one decimal place. R

12 A cone has a volume of 820 m3 and a perpendicular height of 10 m. Find its radius,
correct to one decimal place. R

13 A cone has a volume of 150 m3. If the height and radius of the cone are equal in length,
calculate the radius of the cone. Give your answer to 2 decimal places. R

14 A cone has a base radius of 20 cm and a perpendicular height of 48 cm.

a Find the volume of the cone.
b The top of the cone is removed at half its height. What percentage of
the cone remains? R

Power plus ANSWERS ON P. 595

1 A spiral is formed from 4 semicircles as shown.
+

The diameter of the smallest semicircle is 10 cm,

10
and the semicircles are 10 cm apart. Find the
total length of the spiral:
10
a correct to 2 decimal places b in terms of π.
2 In this pattern, all lengths are in metres.
Calculate, correct to 2 decimal places:
a its perimeter 4 4 4 4 4 4

b its area
3 A rhombus has diagonals of length 24 cm
D C

and 18 cm. Find its perimeter.

A B

Foundation Standard Complex

400 New Century Maths 9 Advanced 9780170453325

 4 To qualify for the next round of the discus trials, 40 m arc
Bronte must throw the discus beyond the 40 m arc.
Find, correct to the nearest square metre, the shaded 1m 45°
disqualifying area of the sector, given that the

qualifying area
disqualifying
small circle has a radius of 1 m. area
40 m

5 For each composite shape calculate, correct to one decimal place:

10.09

i its area ii its perimeter.

All measurements are in centimetres.
a b 15 c

8 12
8 6

9
6 The 3 faces of a rectangular prism have areas as shown.
Calculate the volume of the prism. 40 cm2

20 cm2 32 cm2

7 A cube opened at one end has an external surface area of 1125 cm2. Find its volume.
8 A 10 m flat square roof drains into a cylindrical 10 m
rainwater tank with a diameter of 4 m.
If 5 mm of rain falls on the roof, by how much
(to the nearest mm) does the level of water 10 m
in the tank rise?

4m
9 Calculate, correct to one decimal place, the volume of a tetrahedron with a side
length of 24 cm.

9780170453325 Chapter 10 | Surface area and volume 401

 Qz
CHAPTER 10 REVIEW
Measurement

Language of maths
arc length base capacity circumference
cross-section curved surface diagonal diameter
Surface

giga- kilobyte limits of accuracy mega-

area and
volume
CHAPTER 10 REVIEW

crossword

megabyte micro- (µ) net perimeter

(Advanced)

perpendicular height pi (π) prefix radius/radii

rectangular prism sector surface area triangular prism
1 Draw a circle and label on it:
a a sector b a diameter c an arc length
2 In the formula V = Ah, explain what V, A and h stand for.
3 Which metric prefix means ‘one-millionth’?
4 Write the definition of a prism using the word cross-section.
5 How do you find the surface area of a solid?
6 Name a solid that has a curved surface.

Topic summary
• Write 10 questions (with solutions) that could be used in a test for this chapter. Include some
WS

questions that you have found difficult to answer.

Worksheet
Homework
Mind map:

• Swap your questions with another student and check their solutions against yours.
Surface area
and volume

• List the sections of work in this chapter that you did not understand. Follow up this work
(Advanced)

with your friend or teacher.

Print (or copy) and complete this mind map of the topic, adding detail to its branches and using
pictures, symbols and colour where needed. Ask your teacher to check your work.

Limits of accuracy of
measuring instruments

Metric units Perimeters and areas

of composite shapes

Pyramids and cones SURFACE AREA

AND VOLUME
Areas of quadrilaterals

Circumferences and areas

Prisms and cylinders
of circular shapes

402 New Century Maths 9 Advanced 9780170453325

 TEST YOURSELF 10 ANSWERS ON P. 595

1 Convert:
8 km to m b 15 min to s 0.7 m to mm
10.01
a c
d 250 000 g to kg e 8.4 GB to MB f 9.5 t to kg
g 300 min to h h 34 000 B to kB i 125 kL to L
2500 cm to km k 17 000 000 kg to t 8900 GB to TB

TEST YOURSELF 10
j l
m 10 500 ms to s n 9.1 millennia to years o 2.6 Mm to m
p 3 ly to km q 0.000 000 4 s to μs r 75 000 000 μm to m

2 For each measuring instrument, state:

i the size of one unit on the scale its limits of accuracy.
10.02
ii
a

Shutterstock.com/DG-Studio

iStock.com/AlonzoDesign
c d

Shutterstock.com/Uros Jonic
iStock.com/AndrewScherbackov

3 Calculate the perimeter of each shape.

b 18 mm 24 cm
10.03
a 13 m c d
7m

18 mm
25 cm
13 m

17 cm

7m 15 cm
9 cm

18 cm
12 mm

4 Calculate the area of each shape.

18 mm 17 m c d
10.03
a b

15 m
12
8mcm 20 cm
75 mm
18 mm

17 m

34 mm
15 m

16 cm
18 mm

80 mm
12 mm

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 403

 5 Calculate the area of each quadrilateral.
b
10.04

12 mm 9 mm 9m
a c
4 cm 40 mm
10.3 m
3 cm 10 cm 9 mm

5.7 m

8 mm e
TEST YOURSELF 10

12 m 5 m
d f P Q

PR = 20 cm
12 mm 12 m SQ = 26 cm
5m
9 mm 16 mm
S R

6 For each shape calculate, correct to one decimal place:

i  its perimeter its area.
10.05
ii
a b c

11 cm 16 m
24 mm

48 mm 12 m
d e f

14 m 120° 1.2 m 1.6 m 60°

6 cm
60° 60°
2.5 m

7 Calculate, correct to 2 decimal places, the area of each shape.

a 6m
10.05

20 mm
b c

9 mm 11 cm
15 cm

8 Calculate the surface area of each prism.

closed cube b open rectangular d 
10.06
a c 

15 m
prism
4m 24 m
16 m
8m 30 m
5.5 m 14 m
3.2 m 8m

9 Calculate the volume of each prism above.

10.08

Foundation Standard Complex

404 New Century Maths 9 Advanced 9780170453325

10 Calculate, correct to 2 decimal places, the surface area of each cylinder.
15 mm
10.07
a b c

2.7 cm

Cylinder, 4.8 cm
21 m

23 mm
open at
35 m one end

TEST YOURSELF 10
11 Calculate, correct to 2 decimal places, the volume of each cylinder above. 10.08

12 A rectangular fish tank that is 75 cm long by 55 cm wide by 35 cm deep is filled to 5 cm

from the top. How many litres of water is in the tank?
10.08

13 Calculate the volume of this lunchbox. 28 cm 10.08

13 cm
35 cm
24 cm
14 Find the volume of each pyramid. STAGE 5.3

a b c
12 cm
10.09

15 cm
8m

11 m 15 cm 18 cm 9 cm 14 cm
11 m

15 Find, correct to one decimal place, the volume of each cone.

32 cm
10.09
a b c
20 mm
20 cm
48 mm
cm
37

8 cm

16 A cone has a volume of 115 cm3 and a base radius of 3.2 cm. Find, correct to one decimal
place, the height of the cone.
10.09

Foundation Standard Complex

9780170453325 Chapter 10 | Surface area and volume 405