Area of front face ! $12$ % 10 % 11.13 ! 55.65
Area of back face ! 55.65
4
■
Area ofarea
right-side
24.6 % to
12.2
! 300.12
Find the surface
of eachface
solid!(correct
1 decimal
place), given its net:
Area of left-side
face ! 300.12
8.7 m
a
2.4 m
b
31.5 mm
Year 10 Mathematics
Surface
Areac and
Volume
Practice
Test
12.7 mm 5
The
is 2made
up places).
of two circular
faces and a curved
The surface area is 957.54
cm surface
(correct to
decimal
Area of base ! 10 % 24.6 ! 246
18.2 mm
Total surface area ! 55.65 " 55.65 " 300.12 " 300.12 " 246
! 957.54
2
Total
surface area ! 2 " 265.90 # 479.78
Name__________________________
! 1011.58
1.8 m
E x e r c i s e 12A
5
■
S U R F A C E A R E A S 2O F R I G H T P R I S M S
Theprism:
total surface area is 1011.58 m (correct to 2 decimal
Find the surface area of each triangular
1
Find the surface area of each shape:
■
1 Find the surface
area of this solid
a
4.8 cm
2
●
a
4 cm
10 cm
12 cm
15.2 cm
7 cm
cm
For 27.3
thecm
base8.4
area:
open
15 cm
2
A ! πr
6 cm
12 cm
2
4.8 m
!π"7
18.2 cm
Find■
the
of this
prism
6 surface
Find thearea
surface
area trapezoidal
of each trapezoidal
! prism:
153.938
18.2 cm04 (using a calculator)
18 cm
2
b
c
10.3
cm the external surface
A bcylindrical can is open at one
end.
Find
5.2
cm
correct to 1 decimal place.
a
b
11 cm
17.0 m
2m
For
theside
curved
surface area:
Find the surface area of a cube
with
length:
7.7 cm
a6.4 cm
7 cm
b 8.4 cm
c 0.9 m
2
■
A ! 2πrh
3
■
12 cm
31 m
closed
Find the surface area of a rectangular
! 2prism
" π with:
" 7 " 12
6a
cm length ! 4.8 m, width16
!cm
2.4 m and
height
5.2 m
! 527.787!565
8 (using a calculator)
3.0
m
b length
! 14.8 cm, width ! 3.8 cm and height ! 7.6
cm
18 cm
17.2
m
The surface area is made up of one circular face and a curved
3 Calculate
the
surface
area of tis triangular
prism
(correct to 1 decimal76.2place).
You 565
will 8
cm 527.787
7
A glazier
is commissioned
to build
a glass
■
Total
surface area ! 153.938 04
#
need to usedisplay
Pythagoras’
theorem
toacalculate
an unknown1.14
length.
case in the
shape of
trapezoidal
! 681.725
605 8 24 cm88.9 cm
m
prism. The case is made from panes of glass
2
c
Themany
externaldsurface area of metal is 681.7 cm
to 1 d
held together by metal edging.
How
x cm (correct
315
C H A P T E R 1 2
S U R FA C E A R E A A N D V O L U M E
square centimetres of glass are needed to
63.5 cm
x cm
build the display case? (Hint: First change all
63.5 cm
measurements
to centimetres.)
14 cm
19 cm
E x e r c i s e 12B
23 cm
8
■
S U R FA C E A R E A
Calculate the surface
areaFor
of each
(correct
to 1correct
decimaltoplace
where necessary).
eachshape
cylinder,
find
2 decimal
places:
32 cm
16 cm 1
You will need to use Pythagoras’ theorem to calculate an unknown length.
■
the area of a circular base
i
9 a Find the surface area of each composite shape:
■
ii the area of bthe curved surface
x cm
4 For this closed cylinder, find correct to 2 decimal places:
a
12 m
a
8 cm
a) the area of the circular base
2
■
x cm
10 cm
b
20bcm
4 m the
4 mcurved surface area of each cylinder in terms of π:
Find
5 cm
28 cm
b) the area of its acurved2.46surface
m
cm
1.1 m
45 cm
5.6 cm
c
34 cm
14 cm
c) the total surface
area
6 cm
C O N N E C T I O N S
22 cm
16 cm b
13 cm
12 cm
316
38 cm
2.337 m
41 m
M A T H S
9
d
S T A G E
2.7 m
5.2 / 5.1
5 A cylindrical can is open at one cend. Find the
external surface area of metal correct
to 1
d
3.7 m
18
cm
decimal place.
37 m
c
34 cm
19.6 mm
d
20 m
84 m
4m
18cm
mm
11
30 cm
36 cm
12 m1.2 m
39 mm
7.
cylinder, find
significant figures:
■ For each
Surface
areas
ofto 3right
cylinders
the area of the two circular ends
3
i
ii prisms
the area
of the
curved
Cylinders are like
in that
they
havesurface
uniform cross-sections.
C H A P T E R 1 2
S Uplane
R Fcircular
A C face
E A R E A
iii
the
total
surface
area
However, while the faces of a prism are all plane figures (that is, flat),
a cylinder hasa a curved surface.
2.2 m
b
A
2.2 m
15.2 cm
6
■
a What is the volume of a packet
of muesli bars that is 18 cm long, 14.5 cm high and
5 cm
71.6 cm
26 cm
3.5 cm wide?
25.4
b If each packet holds 8 muesli bars, what is
thecm
volume of each bar correct to 1
decimal place (assuming there is no space between the bars)?
19 cm
5 of this
Findtrapezoidal
the volume of
each trapezoidal prism to the nearest square unit:
■
6 Find the volume
prism
7
■
One of the longest hand-squared wooden girders ever cut on the north coast of NSW
20 cm
7.1 m
a
b
c
4.2 m
was made in 1935. It measured 33 cm by 30 cm by 36 m. This blackbutt girder formed
the keel of a boat. What is the volume of this wood, in cubic metres?
5.8 m
6.
3
8
■
What are
the dimensions of a rectangular prism with volume 64 cm that has the
2.1 m
smallest possible surface area? 3.5 m
12 cm
6.4 m
15 cm
3
9
Give the dimensions of three different rectangular prisms with volume 24 m .
■
2.2 m
10 ■
the
volume
each
prismof
(correct
to 1of
decimal
neccessary):
7 Find the volume
of this
prism
6Find
a What
isof
the
volume
a packet
muesliplace
barsifthat
is 18 cm long, 14.5 cm high an
■
E x e r c ias e 3.512D
cm wide?
b
10 cm
VOLUMES OF RIG
holds 8 muesli bars, what is the volume of each bar correct to 1
2 cm between the bars)?
place (assuming there is no space
9decimal
cm
b If
1
■
each8 cm
packet
Find the volume of each cylinder correct to 2 significant
figures:
The height
given is the slant height. D
Pythagoras’
theorem
to find
the
7
One
of
the
longest
hand-squared
wooden
girders
ever
cut
on
the
coast
ofperp
NSW
■4 cm and height 15 cm
cm north
a radius
b radius 7.8 23
cm
and
height
6.5
2
cm
2
2
2
was made in 1935. It 31
measured
33 cm by 30 cm by 36LOm.!
This
blackbutt
girder
form
cm
10 cm
"and
6.1 height 4.7
c radius 1.4themkeel
and
height
1.7 ismthe volume of thisd wood,
radius
9514.5
cm
2 cm
2 cm
of
a
boat.
What
in
cubic
metres?
!
173.04
22 cm
e radius 0.5 m and height 136 cm
f radius 2.5 m and
height 250
3
3.4 m
! LO
! 13.15
to 2the
decim
What are the dimensions of a rectangular prism with
volume
64cm
cm(correct
that has
3.7
cm
5.3 m
1
d
smallest
area?
8 2Find the
volume
of each
cylinder
1surface
decimal
place to
Now:
V !if#3# Ah
Find
thecvolume
ofpossible
eachtoshape,
correct
2 decimal
places
necessary:
8
■
■
2.8 cm 1
Theof
height
is therectangular
slant height.prisms
Draw awith
triangle
and use
! #3#volume
$ 101.26
9
Give1.4
thecm
dimensions
threegiven
different
24$m13.15
.
■
b Pythagoras’
d3 (using
theorem to find c
the perpendicular!height,
LO.
443.856
333
a
24.7 cm
3.6 m
6.6
m
10
Find
the
volume
of
each
prism
(correct
to
1
decimal
place
if
neccessary):
■
315.7
1c
The volume of theLpyramid is
443.86
LO ! 14.5 " 6.1
OP
! ## $
12.1 m
a
2
2
2
Calculate the volume of each solid correct
to 3 significant figures:
a
8.7 m
3.1 cm b
a
2
1
3.010
cm
cm
E xtoe2rdecimal
c i s e places)
12F
8.1 cm ! LO !c 13.15 cm (correct
9 cm
324
b
! 173.04
8 cm
Now:
3
1.2
h m
2 cm
1
V ! #3# Ah
! #2# $
! 6.1
14.5 cm
■ Find the volume of each pyramid (correct to
! #3# $ 101.26 $ 13.15
b23 cm
The height given is the slant height. Draw a triangle
and333
use3a(using 2acm
O 6.1 cm P
!
443.856
calculator)
52
cm
23.5 cm
2.4
m
31 cm
10 cm
Pythagoras’
theorem in
to find
the perpendicular
height,
LO. to 1 decimal
3
the volume
cubic
centimetres
correct
place
of a soft-dri
14 m
9 3FindFind
the volume
of this solid
to
1
decimal
place
to 2 decimal places).
The
volume
of the pyramid is 443.86 cm 2 (correct
cm4 m 2 cm
10.8
cm
2
2
2
1
22 cm
L
! 14.5
" 6.1
LO 120
OP ! #2# $ HI
height
mm
and
radius
33 mm.
3.8 m
e
f
3.4 m
1
! 173.04
! ## $ 12.2
3.7 cm
C O N N E C T I O N S
M A T H S
9
S T A G E 1 5 . 2 / 5 . 11
32 m
■
d
E x e r c i s e 12F
4
■
2
5.3 m
VOLUMES OF RIGH
48
d ! volume?
a !Which
ofc the
following
cylinders
LO ! 13.15
cmcm
(correct
to 2 decimal
places)has the larger
6.1 cm 24 m
14.5 cm
1.6 m
2.8 cmnecessary)
1
1
Find the volume of eachhpyramid (correct to 1 decimal place where
Now:
V ! #3# Ah ■
i
ii
10 cm
a
c
1
25.4 cm
2b Find the volume of each pyramid
to Athe nea
■
! #3# $ 101.26 $ 13.15
2
4 cm
2.4 m
20 cm
O 6.1 cm P a
6.6 m 333 3 (using a calculator)
! 443.856
12.1
4mm
3
b
(hole cut
0.4 m
to 2 decimal
places).
The volume of the pyramid
is through
443.86 cm (correct(cylinder
of diameter
1.3 m
2 cm
centre8.7
of m
cylinder)
24 m
2
0.25 m
AO !
3.0 cm
10 cm
8.4 cm cut through cube)
20 cm
E x e r c i s e 12F
7.3 cm
2
V O L U M E S O F R I G H T 26.5
P Y R Acm
MIDS
24 cm
O
The cross-sections below are for some wood mouldings available from a
101 Find
the
volume
cubic
38 cm
C O of
N of
N each
E Cpyramid
T Ipyramid
O N (correct
S Mto
A the
T H1 nearest
S
9 S place
T A Gwhere
Ecentimetre
5.2
/ 5.1
the volume
each
decimal
necessary):
■ Find324
hardware store:
2
Find the to
volume
of each pyramid
to
the nearest cubic centimetre:
■
e
bb)a
c b c)d A 20.1
a) a
0.25cm
m
c
i
ii
iii
0.14 m
40 0.4
cm
b Are the4 msurface areas
ofmthe7.3cylinders
the same?AOExplain.
! 12.5 cm
cm
P
How many times larger than the volume of cylinder i is the volume of cylindO
a i
24 cm
24 m2
23.7 cm
2
26.5 cm
38 cm
31 cm 16.8 cm
39.4 cm2
B
ii
b i O
ii
cmnearest cubic centimetre:
AC !pyramids
1.2 20
m cm have the sam
3e 10Show
D these three
d
cm of each pyramid 10
cm that
■
Find the 10
volume
to the
20.1 cm
a
5 cm
b
24 cm
6
■
d
c
0.25 m
10 0.14
cmm
0.4 m
1 cm
63 cm
A
5
■
2
■
D
23.7 cm
0.5 m
BP ! 43 cm
rectangular pyramids.
DO ! 68 cm
5 cm
a
P
63
A cm 5 m
b
C
O
16.8 cm
31 cm
B
6m
a Find the volume of each cylindrical can, leaving your answers
in terms of
8
m
5m
38 cm
3
Show that these three pyramids have the same volume. Note that a and
b are
■
AC
!
1.2
m
D
rectangular
pyramids.
e
334 iii C O N N E0.5C mT I O N S M A T H S 9 S T A G E 5 .iv2
20.1 cm
i
ii
2 cm
BP ! 43 cm
Give your
answer in scientific notation correct to
21.5 cm
calculation very approximate?
E x e r c i s e 12G
1.3 m
VOLUMES OF RIGHT CONES
of each solid correct to 1 decimal place:
6 ■
Pythagoras’
theorem
to find
or h, then
2 Use
Find
the volume of each
solid correct
to 1rdecimal
place: calc
7of these
Suppose
thetodimensions
a cylinder are doubled. What changes will have to be ma
11b Find the volume
1 decimal of
place
■
c cones
1
■
■
Find the volume of each cone correctto
toa32 significant
decimal places:
figures:
b
c
to the dimensions of a cone that just fits inside the original
cylinder so that the ratio
bthe
c still beb 1 : 3?
m new b)
1.8 mthe volumes of3.5
acone and the new cylinder will
16.8 cm
a)a
12.2 cm
8
■
Find the maximum volume of this
funnel. 2.4 m
2.4 m
12.1 cm
12 cm
12.2 cm
6 cm
3.2 m
18.7 cm
21.5 cm
4 cm
brothers of
3
The Montgolfier
brothers of
6.4 cm
■
e of the first
France made one
1.3ofmthe first
to carry
20.4 cm
1 cm
hot-air balloons to carry
they sent up a
people.
1783
they cubic
sent upcentimetre:
a of a cylinder are double
7 eachSuppose
the
dimensions
9of this
Find
the volume
of
solid
to In
the
nearest
■
moke-filled 12
the volume
solid
to3.6
1 decimal
place
2 Find
A cone
has base
diameter
m and height
2.8 m. Find
its volume correct to
large
spherical
smoke-filled
to thebdimensions of a cone that just fits inside th
a
m across.
1 decimal place.
cloth bag 10.6 m across.
lume of gas in
cm new cone and the new cylind
the
volumes
of10the
Calculate
the volume
of gas in
8 cm
he nearest
14.7
cm
3
A cone’s base diameter is equal to its height.
If itstoheight
is 6.6 m, what is its volume?
this balloon
the nearest
8
Find
maximum volume of this funnel.
cubicthe
metre.
Answer correct to 2 decimal places.
■
■
■
■
4
What happens to the volume of a 10.2
cone
if:
■
cm
nce of Earth at
the equator is about
20 cm
16 cm
4
■
The circumference of Earth at the equator is about
a its height is doubled?
40 000 km.
ula C ! 2πr to find the
of Earth
b radius
its radius
is doubled?
a Use the formula C ! 2πr to find the radius of Earth
r cm
e nearest 100 km. c both the radius and height
16.4
are doubled?
correct to the nearest 100 km.
us to find the volume of Earth correct
equ at o r
b Use this radius to find the volume of Earth correct
10of these
A piece
of circular
filter paper
has diameter 10.0 cm. A quadrant was cut out and
nt figures. Write
answer
in
■
13 your
Find
the volume
solids
to 1 decimal
place
to 3 significant figures. Write your answer in
discarded. The remaining piece was joined together along the cuts to form a cone.
ation.
2 of
Find
the
volume
of each
solid correct
■
olume
each
solid
correct
to 1 decimal
place:to 1 decimal place:
scientific notation.
of the
a)b is 20 cm, and itsb a Whatcis the circumference
b)c
9
Find
the
volume of each solid to the nearest cubi
a
shell’s outer
diameter
5
A
spherical
steel
20 cm,
you
seeEho
■
circular
base
of
the
cone
(correct
to C shell’s
C
H
A
P
T
E
R
1
2
S
U
R
F
A
E Aouter
R E diameter
A A Nis D
VToand
Ohelp
Lits
U M
s 18 cm.
a
b
answer
this
questio
inner
diameter
is
18
cm.
2 decimal places)?
3.5 m
inner and outer radii of the shell?
3.5 m
construct a cone
a height
What are
the
b
What
are
the
radius
and
of
thisinner and outer radii of the shell?
cm
12.2
cm
2.4
m
hickness
of
steel
in
the
shell?
from10
a circular
12.2 cm
m
b What is the thickness of steel in the shell?
cone (correct to 2 decimal places)?
piece of paper.
volume of steel in the shell to the nearest
14.7
cm in the shell to the
c Calculate the volume
of steel
nearest
c Calculate its volume correct to
etre.
cubic centimetre.
3
f 1 cm of steel is 7.2 g, what is the mass of 1 decimal place.
3
d If the mass of 1discard
cm of steel is 7.2 g, what is the mass of
20 cm
ll, in kilograms correct to 2 decimal places?
18 cm
this steel shell, in kilograms correct to 2
places?
20decimal
cm
14 ofFind the
volume
3 brothers
The Montgolfier
brothers
of of these solids to 1 decimal place
■
golfier
of each solid correct
to 3 of
significant
10.2
cmcorrect to 3 significant figures:
made one
the first figures:
6
Find the volume of each
solid
ade one France
of the first
338
C O N N E C T I O N S M A■
T H S 9 S T A G E 5.2 / 5.1
a)
b)
hot-air
balloons
to
carry
b
lloons to carry
a
b
people.
In 1783
they sent up a
1783 they
sent up
a
1.3 m
large
spherical smoke-filled
1.3
rical smoke-filled
11.8 cm
11.8 cm
16.4
cm
cloth
bag
10.6
m
across.
2.1
m
10.6 m across.
2.1 m
Calculate
the volume
of gas the
in volume of gas in
balloon to the nearest
10
A piece of circular filter paper has diameter 10.0
on to thethis
nearest
m
cubic metre.
6.7 cm
re.
■
■
discarded. The remaining piece was joined toget
a What is the circumference of the
4
The
circumference
of
Earth
at
the
equator
is
about
■ of Earth at the equator is about
circular base of the cone (correct to
mference
M A T H S 9 S T A G E 5.2 / 5.1
340 C O N N E C T I O N S M A T H S 9 S T A G E 5 . 2 / 5 . 1
40 000 km.
.
2 decimal places)?
a CUse
theto
formula
C radius
! 2πr of
to Earth
find the radius of Earth
e formula
! 2πr
find the
b Whatr are the radius and height of this
r
correct
the nearest 100 km.
to the nearest
100 to
km.
radius tooffind
thecorrect
volume of Earth correct
s radiusbto Use
find this
the volume
Earth
cone (correct
to 2 decimal places)?
equ at o r
equ at o r
to 3 significant
nificant figures.
Write yourfigures.
answerWrite
in your answer in
c Calculate its volume correct to
fic notation.scientific notation.
1 decimal place.
5 shell’s
A spherical
steel shell’s
outer
al ■
steel
outer diameter
is 20
cm,diameter
and its is 20 cm, and its
discard
inner
meter is 18
cm.diameter is 18 cm.
a
are theradii
inner
outer radii of the shell?
re the innerWhat
and outer
of and
the shell?
b What
the in
thickness
of steel in the shell?
s the thickness
ofissteel
the shell?
C O N N E C T I O N S
c Calculate
the
the shell to the nearest
ate the volume
of steel
in volume
the shelloftosteel
the in
nearest
entimetre. cubic centimetre.
3
3
massisof
1 g,
cmwhat
of steel
7.2 g,of
what is the mass of
mass of 1dcmIf the
of steel
7.2
is theismass
338
M A T H S
9
S T A G E
5.2 / 5.1