chapter 6
TRANSCRIPT
Chapter Measurement6What you will learnReview of lengthPythagoras’ theoremAreaSurface area—prisms and cylindersSurface area—pyramids and cones 10A
Volume—prisms and cylindersVolume—pyramids and cones 10A
Spheres 10A
6.16.26.36.46.56.66.76.8
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
363
Monolithic domes
Monolithic domes are round one-piece structures with a smooth spherical-like surface. They offer excellent protection from earthquakes, bushfires and cyclones because of their shape and durability. They are extremely energy efficient because of the minimal surface area for the volume contained within the structure. The first type of monolithic domes used were igloos, which are very strong and provide good insulation in freezing conditions. The minimal surface area of the dome means that there is less surface for heat to be transferred to the outside air. A cube, for example, containing the same volume of air as the dome has about 30% more surface area exposed to the outside air. Volumes and surface areas of spheres and other solids can be calculated using special formulas.
Australian CurriculumM e a s u r e M e n t a n d g e o M e t r y
u s i n g u n i t s o f m e a s u r e m e n t
Solve problems involving surface area and volume for a range of prisms, cylindersand composite solids.
10A Solve problems involving surface area and volume of right pyramids, rightcones, spheres and related composite solids.
P y t h a g o r a s a n d
t r i g o n o M e t r y
Solve right-angled triangle problems.10A Apply Pythagoras’ theorem to solve three-dimensional problems in right-angled triangles.
•
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement364
Pre-
test 1 Evaluate the following.
a 2 × 100 b 5 ÷ 102 c 230 ÷ 102
d 0.043 × 1002 e 62 900 ÷ 1000 f 1.38 × 10002
2 Evaluate the following.
a A = l × w if l = 3 and w = 7
b A = b × h if b = 10 and h = 3
c A = 1
2b × h if b = 2 and h = 3.8
d A = 1
2(a + b)h if a = 2, b = 3 and h = 4
3 Find the perimeter of these shapes.
a
3 cm
b
3 m
1 m c
5 cm
3 cm
4 Use these rules to find the circumference (C) and area (A) of the circles correct to one
decimal place: C = 2pr and A = pr2 where r is the radius.
a
2 m
b
12 cm
c
3.9 km
5 Use Pythagoras’ theorem a2 + b2 = c2 to find the value of the pronumeral in these triangles.
Round to one decimal place where necessary.
a
3
4
c
b
1
2
c
c
13
12
a
6 Find the surface area and volume of these solids. Round to one decimal place
where necessary.
a
2 cm5 cm
4 cm
b
2 m
c
4 cm
6 cm
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
365 Measurement and Geometry
Review of lengthLength measurements are common in
many areas of mathematics, science and
engineering, and are clearly associated
with the basic measures of perimeter and
circumference, which will be studied here.
Let’s start: The simple sectorThis sector looks simple enough but can you describe
how to find its perimeter? Discuss these points to help.
• Recall the rule for the circumference of a circle.
• What is a definition of perimeter?
• What fraction of a circle is this sector?
• Find the perimeter using both exact and rounded numbers.
6.1
100°
10 cm
■■ Converting between metric units of length
×1000
÷1000
km m
×100
÷100
cm
×10
÷10
mm
■■ Perimeter is the distance around the outside
of a closed shape.
■■ The circumference of a circle is the distance
around the circle.
• C = 2pr = pd where d = 2r
■■ Perimeter of a sector
• P = 2r + θ
360 × 2pr
radi
us(r
)
diameter(d)
Key
idea
s
θ r
Engineers at a refinery checking measurements.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement366
Example 1 Finding the perimeter of polygons
Consider the given two-dimensional shape.
a Find the perimeter of the shape if x = 2.6.
b Find x if the perimeter is 11.9 m.
c Write an expression for x in terms of the perimeter P.
Solution Explanation
a Perimeter = 4.5 + 2.1 + 3.4 + 2.6
= 12.6 m
Simply add the lengths of all four sides
using x = 2.6.
b 11.9 = 4.5 + 2.1 + 3.4 + x
= 10 + x
∴ x = 1.9
Add all four sides and set equal to the perimeter 11.9.
Simplify and solve for x by subtracting 10 from both
sides.
c P = 4.5 + 2.1 + 3.4 + x
= 10 + x
∴ x = P - 10
Use P for the perimeter and add all four sides. Simplify
and rearrange to make x the subject.
3.4 m
2.1 m
4.5 m
x m
Example 2 Using the formula for the circumference of a circle
If a circle has radius r cm, find the following, rounding the answer to two decimal places where
necessary.
a The circumference of a circle if r = 2.5
b A rule for r in terms of the circumference C
c The radius of a circle with a circumference of 10 cm
SolUtion Explanation
a Circumference = 2pr
= 2p(2.5)
= 15.71 cm
Write the rule for circumference and substitute r = 2.5,
then evaluate and round as required.
b C = 2pr
∴ =rC
2π
Write the rule for circumference, then divide both sides
by 2p to make r the subject.
c rC=
=
210
2
π
π = 1.59 cm
Substitute C = 10 into the rule from part b and evaluate.
r cm
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 367
Example 3 Finding perimeters of sectors
This sector has a radius of 3 cm.
a Find the sector’s exact perimeter.
b Find the perimeter correct to one decimal place.
Solution Explanation
a P = 2r + θ
360 × 2pr
= 2 × 3 + 240
360 × 2 × p × 3
= 6 + 2
3 × 2 × p × 3
= 6 + 4p cm
The perimeter of a sector consists of two radii and a
fraction 240
360
2
3=
of the circumference of a circle.
6 + 4p is the exact value.
b 6 + 4p ≈ 18.6 cm Round to the required one decimal place using
a calculator.
3 cm240°
Exercise 6A
Unde
rsta
ndin
g
1 Convert the following length measurements into the units given in brackets.
a 4 cm (mm) b 0.096 m (cm) c 0.001 km (m)
d 300 m (km) e 800 cm (m) f 297 m (km)
g 0.0127 m (cm) h 5102 mm (cm) i 0.0032 km (m)
2 What fraction of a circle (in simplest form) is shown in these sectors?
a b c
d
40°
e
105°
f 30°
3 Find the perimeter of these shapes.
a
185 m
220 m
b 0.5 km
2.6 km
c 12 cm
32 cm34 cm
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement368
Flue
ncy
4 Consider the given two-dimensional shape.
a Find the perimeter of the shape if x = 8.
b Find x if the perimeter is 33.7 cm.
c Write an expression for x in terms of the perimeter P.
5 Consider the given two-dimensional shape.
a Find the perimeter of the shape if x = 5.
b Find x if the perimeter is 20 m.
c Write an expression for x in terms of the perimeter P.
6 Find the circumference of these circles, correct to two decimal places.
a
7 cm
b
1.2 m
c
28.4 mm
d
1.1 km
7 If a circle has radius r cm, find the following, rounding to two decimal places,
where necessary.
a The circumference of a circle if r = 12
b A rule for r in terms of the circumference C
c The radius of a circle with a circumference of 35 m
8 Find the perimeter of these sectors, by:
i using exact values
ii rounding the answer to one decimal place.
a
4 m
b
2 m
c
1 km60°
d
6 cm60°
e
5 mm
80°
f
3 cm
205°
Example 1
6.2 cm
10.4 cm
12 cm
x cm
8.4 m
x m
Example 2
r m
Example 3
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 369
Prob
lem
-sol
ving
9 Find the value of x for these shapes with the given perimeters.
a
5
2
7
Perimeter = 17
x
b
5.3Perimeter = 22.9
x
c
2x
Perimeter = 0.072
d 2.8 2x
Perimeter = 16.2
e 11.61
3.72
7.893x
Perimeter = 46.44
f 1.5x
2xx
Perimeter = 10.8
10 A rectangular rose garden of length 15 m and width 9 m is surrounded by a path of
width 1.2 m. Find the distance around the outside of the path.
11 Find the perimeter of these composite shapes correct to two decimal places.
a
1.5
3.6
b 7.9 c
0.3
d
10
e
2.2
3.6 f
6 12
12 Find the perimeter of these shapes, giving answers as exact values.
a 3 cm
b
5 m
2.5 m
c
1 km
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement370
Reas
onin
g
13 A bicycle has wheels with diameter 64 cm.
a Find how far, correct to the nearest cm, the bicycle moves if the wheels turn:
i 1 rotation ii 5 rotations
b How many rotations are required for the bike to travel 10 km? Round to the nearest whole number.
c Find an expression for the number of rotations required to cover 10 km if the wheel has
a diameter of d cm.
14 A square of side length n has the same perimeter as a circle. What is the radius of the circle?
Give an expression in terms of n.
15 A square of side length x just fits inside a circle. Find the exact
circumference of the circle in terms of x.
16 Consider a rectangle with perimeter P, length l and width w.
a Express l in terms of w and P.
b Express l in terms of w if P = 10.
c If P = 10 state the range of all possible values of w.
d If P = 10 state the range of all possible values of l.
x
l
w
Enrichment: Rotating circles
17 When a circle rolls around the outside of another circle it will rotate by a certain angle.
For these problems the fixed circle will have radius r. Given the following conditions, by how
many degrees will the moving circle rotate if it rolls
around the fixed circle once?
a Assume the rotating circle has radius r (shown).
b Assume the rotating circle has radius 1
2r.
c Assume the rotating circle has radius 2r.
d Assume the rotating circle has radius 1
3r. r
rRotatingcircle
Fixedcircle
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 371
Pythagoras’ theoremYou will recall that for any right-angled triangle we can link the
length of the three sides using Pythagoras’ theorem. Given two of
the sides, we can work out the length of the remaining side. This has
applications in all sorts of two- and three-dimensional problems.
Let’s start: President Garfield’s proofFive years before he became president of the United States of America
in 1881, James Garfield discovered a proof of Pythagoras’ theorem. It
involves arranging two identical right-angled triangles ( 1 and 2 ) to
form a trapezium as shown.
b
a
b
c
c1
a
3
2
• Use the formula for the area of a trapezium 1
2( )a b h+
to find an expression for the area of
the entire shape.
• Explain why the third triangle 3 is right-angled.
• Find an expression for the sum of the areas of the three triangles.
• Hence prove c2 = a2 + b2.
6.2
US President Garfield discovered a proof of Pythagoras’ theorem.
■■ Pythagoras’ theorem states that:
The sum of the squares of the two shorter sides of a right-angled triangle equals
the square of the hypotenuse.
■■ To write an answer using an exact value, use a square root sign
where possible (e.g. 3).
Key
idea
s
a
b
ca2 + b2 = c2
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement372
Example 4 Finding side lengths using Pythagoras’ theorem
Find the length of the unknown side in these right-angled triangles, correct to two decimal places.
a
9 cm
5 cmx cm
b
1.5 m
1.1 m
y m
Solution ExPlanation
a c2 = a2 + b2
∴ x2 = 52 + 92
= 106
∴ x = 106
= 10.30
The length of the unknown side is
10.30 cm.
x cm is the length of the hypotenuse.
Substitute the two shorter sides
a = 5 and b = 9 (or a = 9 and b = 5).
Find the square root of both sides and round as
required.
b a2 + b2 = c2
y2 + 1.12 = 1.52
y2 = 1.52 - 1.12
= 2.25 - 1.21
= 1.04
∴ y = 1 04.
= 1.02
The length of the unknown side is
1.02 m.
Substitute the shorter side b = 1.1 and the hypotenuse
c = 1.5.
Subtract 1.12 from both sides.
Find the square root of both sides and evaluate.
For centuries builders, carpenters and landscapers have used Pythagoras’ theorem to construct right-angles for their foundations and plots. The ancient Egyptians used three stakes joined by a rope in a triangular shape with side lengths of three, four and five units, which form a right angle when the rope is taut.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 373
Solution Explanation
a
A
E
B
4
7
c2=a2+b2
∴ BE2= 42+72
=65
∴ BE = 65
Drawtheappropriatetriangle.
Substitutea =4andb=7.
SolveforBEexactly.
Leaveintermediateanswersinsurdformtoreducethe
chanceofaccumulatingerrorsinfurthercalculations.
b
E B
H
2
√65
BH 2=HE 2+EB 2
=22+ 652( )
=4+65
=69
∴ BE = 69
≈8.31
Drawtheappropriatetriangle.
SubstituteHE=2andEB= 65.
Note 65 65 65 652( ) × == .
Unde
rsta
ndin
g
Exercise 6B
1 Solve for a in these equations leaving your answer in exact form using a square root sign.
Assume a > 0.
a a2 + 32 = 82 b a2 + 52 = 62 c 22 + a2 = 92
d a2 + a2 = 22 e a2 + a2 = 42 f a2 + a2 = 102
2 Write an equation connecting the pronumerals in these right-angled triangles.
a
zx
y b
ba
c
x
c
Example 5 Using Pythagoras’ theorem in 3D
Consider a rectangular prism ABCDEFGH with the side lengths
AB = 7, AE = 4 and EH = 2. Find:
a BE, leaving your answer in exact form
b BH, correct to two decimal places
10A
D E F
GH
C
A B
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement374
Flue
ncy
3 Use Pythagoras’ theorem to find the length of the hypotenuse for these right-angled triangles.
Round your answers to two decimal places where necessary.
a
4 cm
3 cm
b
5 m
10 m c
15 km
7 km
d 0.2 mm
1.8 mm
e 0.21 km0.37 km f
72.1 cm
27.3 cm
4 Find the length of the unknown side in these right-angled triangles, correct to two
decimal places.
a
2 m 5 m
b
9 m
12 m
c 0.3 m
0.7 m
d 0.71 cm
1.32 cm
e
19.3 cm24.2 cm
f
0.11 cm
0.14 cm
5 For each of the cuboids ABCDEFGH, find:
i BE, leaving your answer in exact form
ii BH, correct to two decimal places.
a
D
A
C
G
B
F2
35
H
E
b DH
GC
BF
EA
84
14
Example 4a
Example 4b
Example 5
10A
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 375
Flue
ncyc
C
GH
E FD
BA 1
7
3
d
F
B8
2
9
A
D
H
E
G
C
6 Use Pythagoras’ theorem to help decide whether these triangles are right-angled. They may
not be drawn to scale.
a
2
3
4
b
5
12 13
c
611
8
d
1210
6
e
1215
9 f
12
√5
Prob
lem
-sol
ving
7 A 20 cm drinking straw sits diagonally in a glass of radius 3 cm and height
10 cm. What length of straw protrudes from the glass? Round to one decimal
place.
8 Use Pythagoras’ theorem to find the distance between points A and B in these
diagrams, correct to two decimal places.
a
A
B
1.2 m
2.6 m
b
A
B
1.7 cm
3 cm
3.5 cm
1.9 cm
c
BA
26 m
35 m
49 m
10A
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement376
Prob
lem
-sol
ving
d
5.3 cm
5.3 cm
A
B
e BA
2.1 km 1.8 km
0.5 km f
AB
17.2 km 19.7 km
14.3 km
9 Find the value of x, correct to two decimal places, in these three-dimensional diagrams.
a x mm
11.4 mm
9.3
mm
b x m
15.5 m
3.7 m
c
5 cm
3 cm
x cm
d
3 m
2 m
1.5 m
x m e
6.2 km
8.2 km
6.7 km
x km f 2.93 cm
4.04 cm5.31 cm
x cm
10 Find the exact distance between these pairs of points
on a number plane.
a (0, 0) and (4, 6)
b (-2, 3) and (2, -1)
c (-5, -3) and (4, 7)
11 a Find the length of the longest rod that will fit inside these objects. Give your answer
correct to one decimal place.
i A cylinder with diameter 10 cm and height 20 cm
ii A rectangular prism with side lengths 10 cm, 20 cm and 10 cm
b Investigate the length of the longest rod that will fit in other solids, such as triangular
prisms, pentagonal prisms, hexagonal prisms and truncated rectangular pyramids. Include
some three-dimensional diagrams.
10A
4
6
2
-4
-6
-2-4-6 -2-8
-8
4 62 80
8
y
x
10A
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 377
Reas
onin
g
12 Two joining chords in a semicircle have lengths 1 cm and 2 cm as
shown. Find the exact radius, r cm, of the semicircle. Give reasons.
13 The diagonals of a rectangle are 10 cm long. Find the exact
dimensions of the rectangle if:
a the length is twice the width
b the length is three times the width
c the length is 10 times the width.
14Streamers are used to decorate the interior of a room that is
4.5 m long, 3.5 m wide and 3 m high, as shown.
a Find the length of streamer, correct to two decimal
places, required to connect from:
i A to H ii E to B
iii A to C iv A to G via C
v E to C via D vi E to C directly
b Find the shortest length of streamer required, correct to
two decimal places, to reach from A to G if the streamer
is not allowed to reach across open space.
4.5 m
3 m
E
AB
C
GH
F
D
3.5 m
10A
1 cm
2 cm
r cm
Enrichment: How many proofs?
15 There are hundreds of proofs of Pythagoras’ theorem.
a Research some of these proofs using the internet and pick one you understand clearly.
b Write up the proof giving full reasons.
c Present your proof to a friend or the class. Show all diagrams, algebra and reasons.
Mathematical proofs are vital to the subject and are considered to be amongst its greatest achievements. Most mathematicians consider a good proof to be a thing of beauty.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement378
AreaArea is a measure of surface and is expressed
as a number of square units.
By the inspection of a simple diagram like
the one shown, a rectangle with side lengths
2 m and 3 m has an area of 6 square metres
or 6 m2.
Area = 6 m2
2 m
3 m
For rectangles and other basic shapes, we can use area formulas to help us calculate the number of
square units.
Some common metric units for area include square kilometres (km2), square metres (m2), square
centimetres (cm2) and square millimetres (mm2).
Let’s start: Pegs in holesDiscuss with reasons relating to the area of the shapes which is the better fit:
• a square peg in a round hole
• a round peg in a square hole
6.3
■■ Conversion of units of area
×10002
÷10002
km2 m2
×1002
÷1002
cm2
×102
÷102
mm2Key
idea
s
The appropriate unit for the area of a desert is the square kilometre.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 379
Example 6 Converting between units of area
Convert these areas to the units shown in the brackets.
a 2.5 cm2 (mm2) b 2 000 000 cm2 (km2)
Solution Explanation
a 2.5 cm2 = 2.5 × 100 mm2
= 250 mm2
10 mm
10 mm
1 cm2
1 cm2 = 10 × 10 mm2 = 100 mm2
b 2 000 000 cm2 = 2 000 000 ÷ 1002 m2
= 200 m2
= 200 ÷ 10002 km2
= 0.0002 km2
km m cm2 2 2
1000÷ 2 ÷1002
10002 = 1 000 000 and 1002 = 10 000
■■ The area of a two-dimensional shape can be defined as the number of square units contained
within its boundaries. Some common area formulas are given.
Square
Area = l2
l
Rectangle
Area = lw
w
l
Triangle
Area = 1
2bh
h
b
Rhombus
Area = 1
2xy
yx
Parallelogram
Area = bh
h
b
Trapezium
Area = 1
2(a + b)h
h
b
a
■■ The rule for the area of a circle is:
Area = pr2, where r is the radius
■■ The rule for the area of a sector is A r= θ360
2πr
θ
Key
idea
s
x
y
Area = 1
2xy
Kite
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement380
Example 7 Finding the area of basic shapes
Find the area of these basic shapes, correct to two decimal places where necessary.
a
2 cm
3 cm
5 cm
b
5.8 m
3.3 m
c
1.06 km
Solution Explanation
a A = 1
2(a + b)h
= 1
2(3 + 5)2
= 8 cm2
The shape is a trapezium, so use this formula.
Substitute a = 3, b = 5 and h = 2.
Simplify and include the correct units.
b A = 1
2bh
= 1
2(5.8)(3.3)
= 9.57 m2
The shape is a triangle.
Substitute b = 5.8 and h = 3.3.
Simplify and include the correct units.
c A = pr2
= p (0.53)2
= 0.88 km2
The shape is a circle.
The radius r is half the diameter.
Round to the required number of decimal places.
Example 8 Using area to find unknown lengths
Find the value of the pronumeral for these basic shapes, rounding to two decimal places where
necessary.
a
Area = 11 mm2
l mm2.3 mm
b
Area = 0.5 m2
1.3 m
0.4 ma m
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 381
Example 9 Finding areas of sectors and composite shapes
Find the area of this sector and composite shape. Write your answer as an exact value and as a decimal
correct to two places.
a
80°3 cm
b
5 m
Solution Explanation
a A r= ×
= × ×
=
≈
θ π
π
π
360280
3603
7
21 99
2
2
2. cm
Write the formula for the area of a sector.
Sector angle = 360° − 80° = 280°.
Simplify to express as an exact value (7π) then round
as required.
b A = × − × ×
= −
≈
2 51
45
5025
4
30 37
2 2
2.
π
π
m
The area consists of two squares minus a quarter circle
with radius 5 m.
50 − 25
4
π is the exact value.
SoLuTion ExPLAnATion
a A = lw
11 = l × 2.3
∴ =.
l11
2 3 = 4.78
Use the rectangle area formula.
Substitute A = 11 and w = 2.3.
Divide both sides by 2.3 to solve for l.
b A = 1
2(a + b)h
0.5 = 1
2(a + 1.3) × 0.4
0.5 = 0.2(a + 1.3)
2.5 = a + 1.3
∴ a = 1.2
Use the trapezium area formula.
Substitute A = 0.5, b = 1.3 and h = 0.4.
Simplify then divide both sides by 0.2 and solve for a.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement382
Exercise 6C
Unde
rsta
ndin
g
1 Write the formula for the area of these shapes.
a circle b sector c square d rectangle
e kite f trapezium g triangle h rhombus
i parallelogram j semicircle k quadrant (quarter circle)
2 Convert the following area measurements into the units given in brackets.
a 3000 mm2 (cm2) b 29 800 cm2 (m2) c 205 000 m2 (km2)
d 0.5 m2 (cm2) e 5 km2 (m2) f 0.0001 km2 (m2)
g 0.023 m2 (cm2) h 537 cm2 (mm2) i 0.0027 km2 (m2)
j 10 m2 (mm2) k 0.00022 km2 (cm2) l 145 000 000 mm2 (km2)
Example 6
Flue
ncy
3 Find the area of these basic shapes, rounding to two decimal places where necessary.
a
5 cm
b 5.2 m
10.5 m
c
2.8 km
1.3 km
d
0.2 mm
0.1 mm
0.3 mm e
7 m
f
15 cm10 cm
g 64 m 23 m h 0.4 mm
0.25 mm
i km23
km14
j
√3 km
k 0.82 m
1.37 m
l
√6 cm√2 cm
√10 cm
Example 7
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 383
Flue
ncy4 Find the value of the pronumeral for these basic shapes, rounding to two decimal places where
necessary.
a
Area = 15 cm2
w cm
5.2 cm b
l m
Area = 206 m2
c
Area = 1.3 km2
1.8 km
h km
d
1.3 m
2.8 m
a mArea = 2.5 m2
e 5.34 mm
h mm
Area = 10.21 mm2
f
18 m
x mArea = 80 m2
g r cm
Area = 5 cm2
h d m
Area = 0.21 m2
i d m
Area = 26 km2
5 Find the area of each sector. Write your answer as an exact value and as a decimal rounded to
two places.
a
6 cm
b 10 m c
120°
7 m
d
100°2 m
e
150°
6 km
f
3 mm
325°
Example 8
Example 9a
6 A lawn area is made up of a semicircular region with diameter
6.5 metres and a triangular region of length 8.2 metres, as
shown. Find the total area of lawn, to one decimal place.
Prob
lem
-sol
ving
6.5 m
8.2 m
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement384
7 Find the area of these composite shapes. Write your answers as exact values and as decimals
correct to two places.
a
5 cm
b
4 m
7 m
10 m
c
1.7 m1.8 m
1.6 m
d
4.2 mm
e 28 km
18 km
26 km
f 0.25 m 0.3 m
0.7 m
8 An L-shaped concrete slab being prepared for the foundation of a new house is made up of
two rectangles with dimensions 3 m by 2 m and 10 m by 6 m.
a Find the total area of the concrete slab.
b If two bags of cement are required for every 5 m2 of concrete, how many whole bags of
cement will need to be purchased for the job?
Example 9b
Prob
lem
-sol
ving
9 1 hectare (1 ha) is 10 000 m2 and an acre is 1
640 square miles (1 mile ≈1.61 km).
Find how many:
a hectares in 1 km2
b square metres in 20 hectares
c hectares in 1 acre (round to one decimal place)
d acres in 1 hectare (round to one decimal place)
Reas
onin
g
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 385
10 Consider a trapezium with area A, parallel side lengths a and b and height h.
b
a
h A = 1
2 (a + b)h
a Rearrange the area formula to express a in terms of A, b and h.
b Hence, find the value of a for these given values of A, b and h.
i A = 10, b = 10, h = 1.5 ii A = 0.6, b = 1.3, h = 0.2 iii A = 10, b = 5, h = 4
c Sketch the trapezium with the dimensions found in part b iii above. What shape have
you drawn?
11 Provide a proof of the following area formulas, using only the area formulas for rectangles
and triangles.
a parallelogram b kite c trapezium
Reas
onin
g
Enrichment: Percentage areas
12 Find correct to one decimal place the percentage areas for these situations.
a The largest square inside a circle. b The largest circle inside a square.
c The largest square inside a right d The largest circle inside a right isosceles
isosceles triangle. triangle.
x
x
x
x
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement386
Surface area—prisms and cylinders Knowing how to find the area of simple shapes combined with some knowledge about
three-dimensional objects helps us to find the total surface area of a range of solids.
A cylindrical can, for example, has two circular ends and a curved surface that
could be rolled out to form a rectangle. Finding the sum of the two circles and the
rectangle will give the total surface area of the cylinder.
You will recall the following information about prisms and cylinders.
■■ A prism is a polyhedron with a uniform cross-section and two congruent ends.
• A prism is named by the shape of the cross-section.
• The remaining sides are parallelograms.
■■ A cylinder has a circular cross-section.
• A cylinder is similar to a prism in that it has a
uniform cross-section and two congruent ends.
Let’s start: Drawing netsDrawing or visualising a net can help to find total surface
area of a solid. Try drawing a net for these solids.
Square prismCylinder
6.4
Right triangular
prism
Oblique pentagonal
prism
■■ The total surface area (TSA) of a three-dimensional object can be found by finding the sum
of the areas of each of the shapes that make up the surface of the object.
■■ A net is a two-dimensional illustration of all the surfaces of a solid object.
■■ The net and surface area of a cylinder
h
rDiagram
h
r
r
2πr
NetTSA = 2 circles + 1 rectangle
= 2πr2 + 2πrh
= 2πr (r + h)
Key
idea
s
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 387
Example 10 Finding the surface area of prisms and cylinders
Find the total surface area of these objects. Round to two decimal places where necessary.
a 3 cm
5 cm8 cm
b
5.3 m
1.7 m
Solution Explanation
a TSA = 2 × (8 × 3) + 2 × (5 × 3) + 2 × (8 × 5)
= 158 cm2
Draw the net of the solid.
3 cm
5 cm
8 cm
5 cm
3 cm
b TSA = 2pr2 + 2prh
= 2p(1.7)2 + 2p(1.7) × 5.3
= 74.77 m2
5.3 m2πr
1.7 m
1.7 m
■■ Composite solids are solids made up of two or more basic solids.
• To find a total surface area do not include any common faces.
Common face Key
idea
s
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement388
Example 11 Finding the surface area of composite solids
A composite object consists of a square-based prism and a cylinder as shown. Find the total surface
area correct to one decimal place.
8 cm
10 cm
20 cm
6 cm
Solution Explanation
TSA = 4 × (20 × 10) + 2 × (20 × 20) + 2 × p × 6 × 8
= 1600 + 96p= 1901.6 cm2
The common circular area, which should
not be included, is added back on with the
top of the cylinder. So the total surface area of
the prism is added to the curved area only of
the cylinder.
Unde
rsta
ndin
g
Exercise 6D
1 Draw an example of these solids.
a cylinder b rectangular prism c triangular prism
2 Draw a net for each of these solids.
a
10 cm
4 cm b
2 cm
c
2 cm
1 cm3 cm
Flue
ncy
3 Find the total surface area of these solids. Round your answers to two decimal places where
necessary.
a 3 cm
3 cm6 cm
b
2.7 mm
1.3 mm5.1 mm
c 2.1 cm
4.5 cm7 cm
Example 10
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 389
Flue
ncyd 8 m
17 m
15 m
20 m
e 12.8 m
9.2 m
f 5.2 m
6.4 m
4 Find the total surface area of these solids.
a
1.2 cm
b 5 mm
3 mm
4 mm
7 mm
c 25 m
10 m
8.66 m
d
16 mm
9 mm
11.5 mm
12 mm
e 1.2 cm
1.5 cm
0.8 cm0.76 cm
0.5 cm f
2.3 m
4.8 m
2.1 m
1.4 m
1.9 m
5 Find the total surface area in square metres of the outside of an open pipe of radius 85 cm and length
4.5 m, correct to two decimal places.
6 What is the minimum area of paper required to wrap a box with dimensions 25 cm wide, 32 cm long
and 20 cm high?
Prob
lem
-sol
ving
7 The cross-sections of these solids are sectors. Find the total surface area, rounding to
one decimal place.
a
3 cm10 cm
b
2 m
1.6 m
c
6 m
4 m d
20 cm
40 cm
40°
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement390
Prob
lem
-sol
ving
8 Use Pythagoras’ theorem to determine any unknown side lengths and find the total surface
area of these solids, correct to one decimal place.
a
7
3
2
b
1.4
1.8 3.2 c
8
7
10
d 8
10
e
33
26
f
1.85.2
9 Find the total surface area of these composite solids. Answer correct to one decimal place.
a
2 cm
6 cm
4 cm
4 cm b
10 m
8 m
c 10 cm
20 cm
d
15 cm
40 m20 m
10 Find the total surface area of this triangular prism correct to one decimal place.
5 m2 m
Example 11
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 391
11 Find a formula for the total surface area of these solids using the given pronumerals.
a
x
b
a b
c
c
h
d
d
rh
12 Find the exact total surface area for a cylinder with the given dimensions. Your exact answer will
be in terms of p.
a r = 1 and h = 2 b r = 12
and h = 5
13 If the total surface area of a cylinder is given by the rule Total surface area = 2pr(r + h), find the
height, to two decimal places, of a cylinder that has a radius of 2 m and a total surface area of:
a 35 m2 b 122 m2
14 Can you find the exact radius of the base of a cylinder if its total surface area is 8p cm2 and
its height is 3 cm?
Reas
onin
g
Enrichment: inventing formulas for special solids
15 Derive the formulas for the total surface area of the following solids.
a A cylinder with its height b A square-based prism with square side
equal to its radius r length x and height y
r
r
y
x
c A half cylinder with radius r and d A solid with a sector cross-section, radius
height h r, sector angle q and height h
r
h
θh
r
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement392
Surface area—pyramids and cones 10A Pyramids and cones are solids for which we can also
calculate the total surface area by finding the sum of the
areas of all the outside surfaces.
The total surface area of a pyramid involves finding the
sum of the areas of the base and its triangular faces. The
rule for the surface area of a cone can be developed after
drawing a net including a circle (base) and sector (curved
surface), as you will see in the following examples.
Let’s start: The cone formulaUse compasses to construct a large sector. Use any sector
angle q you like. Cut out the sector and join the points A and
B to form a cone of radius r.
• Give the rule for the area of the base of the cone.
• Give the rule for the circumference of the base of the cone.
• Give the rule for the circumference of a circle with radius s.
• Use the above to find an expression for the area of the base of the cone as a fraction of the area ps2.
• Hence explain why the rule for the surface area of a cone is given by Total surface area = pr2 + prs.
6.5
s
rθ
sB
A
■■ A cone is a solid with a circular base and a curved surface that reaches from
the base to a point called the apex.
• A right cone has its apex directly above the centre of the base.
• The pronumeral s is used for the slant height and r is the radius of the base.
• Cone total surface area.
r
s rs
2πr
Area (base) = πr 2
and
Area = × =2
2
2π
ππ π
r
ss rs
∴ TSA (cone) = πr2 + πrs = πr(r + s)
■■ A pyramid has a base that is a polygon and remaining sides that are
triangles that meet at the apex.
• A pyramid is named by the shape of its base.
• A right pyramid has its apex directly above the centre of the base.
Apex
Right square-based pyramid
Key
idea
s
s
Apex
Right coner
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 393
Example 12 Finding the total surface area of a cone and pyramid
Find the total surface area of these solids, using two decimal places for part a.
4.5 m
2 m
a Cone with radius 2 m and slant height 4.5 m
b Square-based pyramid with square-base length 25 mm and triangular
face height 22 mm
Solution Explanation
a TSA = pr 2 + prs
= p(2)2 + p(2) × (4.5)
= 40.84 m2
The cone includes the circular base plus the curved
part. Substitute r = 2 and s = 4.5.
b TSA = l2 + 4 × 1
2bh
= 252 + 4 × 1
2 × 25 × 22
= 1725 mm2 25 mm
22 mm
Example 13 Finding the slant height and vertical height of a cone
A cone with radius 3 cm has a curved surface area of 100 cm2.
a Find the slant height of the cone correct to one decimal place.
b Find the height of the cone correct to one decimal place.
Solution Explanation
a Surface area = prs
100 = p × 3 × s
s = 100
3π = 10.6 cm
Substitute the given information into the rule for the
curved surface area of a cone and solve for s.
b h2 + r2 = s2
h2 + 32 = 100
3
2
π
h2 = 100
3
2
π
− 9
h = 100
39
2
π
−
= 10.2 cm
sh
r
Identify the right-angled triangle within the cone
and use Pythagoras’ theorem to find the height
h. Use the exact value of s from part a to avoid
accumulating errors.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement394
Exercise 6E
Unde
rsta
ndin
g
1 Write the rule for the following.
a Area of a triangle
b Surface area of the base of a cone with radius r
c Surface area of the curved part of a cone with slant height s and radius r
2 Find the exact slant height for these cones, using Pythagoras’ theorem. Answer exactly, using
a square root sign.
a 5 cm
2 cm
b
14 m
5 m c
6 cm
10 cm
3 Draw a net for each of these solids.
a
2 cm
b
4 cm2 cm
c
3 cm
4 Find the total surface area of these cones, correct to two decimal places.
a 9 mm
12 mm
b
0.5 m
0.8 m
c 11 km
15 km
5 Find the total surface area of these pyramids.
a
6 m
4 m
b
8 cm
5 cm
c
0.3 m
0.2 m
6 Find the area of the curved surface only of each cone, correct to two decimal places.
a
10 m
2 m
b 1.1 cm
1.5 cm
c
26 mm
48 mm
Example 12a
Example 12b
Flue
ncy
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 395
Flue
ncy 7 A cone has height 10 cm and radius 3 cm.
a Use Pythagoras’ theorem to find the slant height of the cone, rounding your answer to
two decimal places.
b Find the total surface area of the cone, correct to one decimal place.
Prob
lem
-sol
ving
8 A cone with radius 5 cm has a curved surface area of 400 cm2.
a Find the slant height of the cone correct to one decimal place.
b Find the height of the cone correct to one decimal place.
9 A cone with radius 6.4 cm has a curved surface area of 380 cm2.
a Find the slant height of the cone correct to one decimal place.
b Find the height of the cone correct to one decimal place.
10 This right square-based pyramid has base side length 4 m and vertical
height 6 m.
a Find the height of the triangular faces correct to one decimal place.
b Find the total surface area correct to one decimal place.
11 Party hats A and B are in the shape of open cones with no base. Hat A has
radius 7 cm and slant height 25 cm and hat B has radius 9 cm and slant height 22 cm. Which hat has the
greater surface area?
12 Find the total surface area of these composite solids correct to one decimal place as necessary.
a
4 cm
5 cm
2 cm
b 3 cm
3 cm
c 4.33 cm
8 cm
5 cm
4.33 cm
Example 13
6 m
4 m
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement396
Prob
lem
-sol
ving
d
10 m
3 m
8 m
e
10 mm
6 mm
9 mm
f
5 m
7 m2 m
Enrichment: Carving pyramids from cones
16 A woodworker uses a rotating lathe to produce a cone with radius
4 cm and height 20 cm. From that cone the woodworker then cut
slices off the sides of the cone to produce a square-based pyramid
of the same height.
a Find the exact slant height of the cone.
b Find the total surface area of the cone correct to two decimal
places.
c Find the exact side length of the base of the square-based pyramid.
d Find the height of the triangular faces of the pyramid correct to three decimal places.
e Find the total surface area of the pyramid correct to two decimal places.
f Express the total surface area of the pyramid as a percentage of the total surface area of the
cone. Give the answer correct to the nearest whole percentage.
20 cm
8 cm
13 Explain why the surface area of a cone with radius r and height h is given by the expression
πr r r h+ +( )2 2 .
14 A cone has a height equal to its radius (h = r). Show that its total surface area is given by the
expression pr2(1 + 2).
15 There is enough information in this diagram to find the total surface
area but the side length of the base and the height of the triangular
faces are not given. Find the total surface area correct to one decimal
place.
Reas
onin
g
10 cm
10 cm
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 397
Volume—prisms and cylindersVolume is the amount of space contained within the outside surfaces of a three-dimensional object and is
measured in cubic units.
The common groups of objects considered in this section are the prism and the cylinder.
Let’s start: Right and oblique prismsRecall that to find the volume of a right prism, you would first find the area of the base and multiply by the
height. Here is a right rectangular prism and two oblique rectangular prisms with the same side lengths.
10
63
10
63
10
63
• What is the volume of the right rectangular prism?
• Do you think the volume of the two oblique prisms would be calculated in the same way as for the
right rectangular prism and using exactly the same lengths?
• Would the volume of the oblique prisms be equal to or less than that of the rectangular prism?
• Discuss what extra information is required to find the volume of the oblique prisms.
• How does finding the volume of oblique prisms (instead of a right prism) compare with finding the
area of a parallelogram (instead of a rectangle)?
6.6
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement398
Example 14 Finding the volume of right prisms and cylinders
Find the volume of these solids, rounding to two decimal places for part b.
a
4 m
3 m2 m
b
6 cm
2 cm
Solution Explanation
a V = lwh
= 2 × 3 × 4
= 24 m3
Write the volume formula for a rectangular prism.
Substitute l = 2, w = 3 and h = 4.
b V = pr2h
= p(2)2 × 6
= 75.40 cm3
The prism is a cylinder with base area pr2.
Substitute r = 2 and h = 6.
Evaluate and round.
■■ Metric units for volume include cubic kilometres
(km3), cubic metres (m3), cubic centimetres (cm3)
and cubic millimetres (mm3).
■■ Units for capacity include megalitres (ML),
kilolitres (kL), litres (L) and millilitres (mL).
• 1 cm3 = 1 mL
■■ For right and oblique prisms and cylinders the volume is given by:
V = Ah where
• A is the area of the base
• h is the perpendicular height.
w
h
l
h
xx r
h
Right rectangular prism Oblique square prism Right cylinder
V = Ah V = Ah V = Ah
= lwh = x2h = pr2h
×10003
÷10003
km3 m3
×1003
÷1003
cm3
×103
÷103
mm3
×1000
÷1000
ML kL
×1000
÷1000
L
×1000
÷1000
mL
Key
idea
s
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 399
Example 15 Finding the volume of an oblique prism
Find the volume of this oblique prism.
Solution Explanation
V = 1
2bh × 8
= 1
2 × 5 × 3 × 8
= 60 cm3
The base is a triangle so multiply the area of the base
triangle by the perpendicular height.
3 cm 8 cm
5 cm
Example 16 Finding the volume of a composite solid
Find the volume of this composite solid correct to one
decimal place.
Solution Explanation
Radius of cylinder = 6
2 = 3 cm
V = lwh + pr2h
= 6 × 6 × 2 + p × 32 × 2
= 72 + 18p = 128.5 cm3
First find the radius length, which is half the side length
of the square base.
Add the volume of the square-based prism and the
volume of the cylinder.
6 cm2 cm
2 cm
Exercise 6F
Unde
rsta
ndin
g
1 Find the volume of these solids with the given base areas.
a
A = 8 cm2
10 cm
b 2 m
A = 16 m2
c
12 mm
A = 9 mm2
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement400
Flue
ncy
3 Find the volume of each rectangular prism.
a
2 cm
4 cm5 cm
b
30 m10 m
35 m
c
7 mm
3.5 mm10.6 mm
4 Find the volume of each cylinder, correct to two decimal places.
a
5 m
10 m
b
1.5 cm
2 cm c
12 m
14 m
5 Find the volume of these oblique solids. Round to one decimal place for part b.
a
3 cm
2 cm
b
20 m
5 m c 1.2 mm
2.4 mm
1.7 mm
6 Find the volume of these solids, rounding your answers to three decimal places where necessary.
a
2 km10 km
3 km
b
3.5 cm
8 cm
7 cm
c
7 m3 m
3 m
2 m
Example 14a
Example 14b
Example 15
2 Convert these volume measurements into the units given in brackets.
a 2 cm3 (mm3) b 0.2 m3 (cm3) c 0.015 km3 (m3)
d 5700 mm3 (cm3) e 28 300 000 m3 (km3) f 762 000 cm3 (m3)
g 0.13 m3 (cm3) h 0.000 001 km3 (m3) i 2.094 cm3 (mm3)
j 2.7 L (mL) k 342 kL (ML) l 35 L (kL)
m 5.72 ML (kL) n 74 250 mL (L) o 18.44 kL (L)
Unde
rsta
ndin
g
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 401
Flue
ncyd
5 cm2 2 cm
e
0.12 m
0.38 m
f 21.2 cm
18.3 cm
24.5 cm
g 0.5 m
0.8 m
h 12 cm
12 cm
i
2 m
3 m
2 m
6 m
Prob
lem
-sol
ving7 How many containers holding 1000 cm3 (1 L) of water will you need to fill 1 m3?
8 How many litres of water are required to fill a rectangular fish tank that is 1.2 m long, 80 cm wide
and 50 cm high?
9 Find the volume of these composite objects, rounding to two decimal places where necessary.
a
8 cm
2 cm
5 cm
b
5 m
5 m
c
2.3 m
1.6 m
1.1 m 2.6 m
d
5 mm
8 mm11 mm
e
9 cm
3 cm
4 cm
5 cm
f 4 cm
6 cm
10 Find the exact volume of a cube if its surface area is:
a 54 cm2 b 18 m2
Example 16
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement402
11 Use the rule V = pr2h to find the height of a cylinder, to one decimal place, with radius 6 cm and
volume 62 cm3.
12 A fellow student says the volume of this prism is given by V = 4 × 3 × 5.
Explain his error.
13 Find a rule for the volume of a cylindrical portion with angle
q, radius r and height h as shown.
14 Decide whether there is enough information in this diagram
of a triangular prism to find its volume. If so, find the volume
correct to one decimal place.
5
34
h
r θ
10 m4 m
Reas
onin
g
Enrichment: Concrete poles
15 A concrete support structure for a building is made up of a cylindrical base and a square-based
prism as the main column. The cylindrical base is 1 m in diameter and 1 m high and the square
prism is 10 m long and sits on the cylindrical base as shown.
10 m
1 m
1 m
a Find the exact side length of the square base of the prism.
b Find the volume of the entire support structure correct to one decimal place.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 403
Volume—pyramids and cones 10A
The volume of a cone or pyramid is a certain fraction of the volume of a prism with the same base area.
This particular fraction is the same for both cones and pyramids and will be explored in this section.
The volume of sand needed to construct these garden features can be calculated from the formula for the volume of a cone.
Let’s start: is a pyramid half the volume of a prism?Here is a cube and a square pyramid with equal base side lengths and equal heights.
• Discuss whether or not you think the pyramid is half the volume of the cube?
• Now consider this diagram of the cube with the pyramid inside.
The cube is black.
The pyramid is green.
The triangular prism is blue.
• Compared to the cube, what is the volume of the triangular prism (blue)? Give reasons?
• Is the volume of the pyramid (green) more or less than the volume of the triangular prism (blue)?
• Do you know what the volume of the pyramid is as a fraction of the volume of the cube?
6.7
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement404
Example 17 Finding the volume of pyramids and cones
Find the volume of this pyramid and cone. Give the answer for part b correct to two decimal places.
a
1.4 m
1.2 m
1.3 m
b
29 mm
23 mm
Solution Explanation
a V Ah
l w h
=
= × ×
= × ×
=
1
31
31
31 4 1 2 1 3
0 728 3
( )
( . . ) .
. m
The pyramid has a rectangular base with area l × w.
Substitute l = 1.4, w = 1.2 and h = 1.3.
Evaluate the answer.
b V Ah
r h
=
=
= ×
=
1
31
31
311 5 29
4016 26
2
2
3
π
π( . )
. mm
The cone has a circular base of area pr2.
Substitute r = 23
2 = 11.5 and h = 29.
Evaluate and round the answer.
■■ For pyramids and cones the volume is given by: V = 1
3Ah
where A is the area of the base and h is the perpendicular height.
Right square pyramid Right cone Oblique cone
xx
h
h
r
h
r
V = 13
Ah V = 1
3Ah V =
1
3Ah
= 1
3x2h =
1
3pr2h =
1
3pr2h
Key
idea
s
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 405
Exercise 6G
Unde
rsta
ndin
g
1 A cylinder has volume 12 cm3. What will be the volume of a cone with the same base area and
perpendicular height?
2 A pyramid has volume 5 m3. What will be the volume of a prism with the same base area and
perpendicular height?
3 Find the volume of these solids with the given base areas.
a
A = 5 m2
6 m
b
A = 2 cm2
4 cm
c
7 mm
A = 25 mm2
Flue
ncy
4 Find the volume of the following pyramids. For the oblique pyramids (parts d, e, f) use the given
perpendicular height.
a
2 cm
3 cm b
18 m
15 m
13 m
c
12 km
5 km
5 km
d 2 cm
e 4 cm
3 cm
f 1.4 mm
0.6 mm
Example 17a
Perpendicular
height = 2 cm
Perpendicular
height = 4 cmPerpendicular
height = 1.2 mm
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement406
Prob
lem
-sol
ving
5 Find the volume of the following cones, correct to two decimal places.
a
2.6 m
1.1 m
b
3.5 mm
1.6 mm c
60 m
20 m
d
6 m
2 m
e
1.3 m
0.6 m f
5 cm
20 cm
6 A medicine cup is in the shape of a cone with base radius 3 cm and height 5 cm. Find its capacity in
mL correct to the nearest mL.
7 Find the volume of these composite objects, rounding to two decimal places where necessary.
a
8 m
6 m
3 m
b
5 cm
6 cm
c
4 m
3 m6 m
2 m
d
4.8 mm
1.5 mm
5.2 mm
e
3.96 m
2.8 m
f
7.1 m
10 m
2.6 m
Example 17b
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 407
9 A wooden cylinder is carved to form a cone that has the same base area and same height as the
original cylinder. What fraction of the wooden cylinder is wasted? Give a reason.
10 A square-based pyramid and a cone are such that the diameter of the cone is equal to the length
of the side of the square base of the pyramid. They also have the same height.
a Using x as the side length of the pyramid and h as its height, write a rule for:
i the volume of the pyramid in terms of x and h
ii the volume of the cone in terms of x and h
b Express the volume of the cone as a fraction of the volume of the pyramid. Give an exact
answer.
11 a Use the rule V = 1
3pr2h to find the base radius of a cone, to one decimal place, with height
23 cm and volume 336 cm3.
b Rearrange the rule V = 1
3pr2h to write:
i h in terms of V and r ii r in terms of V and h
Reas
onin
gPr
oble
m-s
olvi
ng
8 The volume of ice-cream in the cone is half the volume of
the cone. The cone has a 3 cm radius and 14 cm height. What
is the depth of the ice-cream correct to two decimal places?
14 cmIce-cream
depth ofice-cream
3 cm
Enrichment: Truncated cones
12 A truncated cone is a cone that has its apex cut off by an intersecting plane.
In this example the top has radius r2, the base has radius r1 and the two
circular ends are parallel.
a Give reasons why rr
h
h1
2
1
2
= .
b Find a rule for the volume of a truncated cone.
c Find the volume to one decimal place of a truncated cone if r1 = 2 cm, h1 = 5 cm and
h2 equals:
i 1
2 1h ii 2
3 1h
r1
h1
r2
h2
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement408
Spheres 10A
Planets are spherical in shape due to the effects of
gravity. This means that we can describe a planet’s
size using only one measurement: its diameter or
radius. Mars, for example, has a diameter of about
half that of the Earth, which is about 12 756 km. The
Earth’s volume is about 9 times that of Mars and this
is because the volume of a sphere varies with the
cube of the radius. The surface area of the Earth is
about 3.5 times that of Mars because the surface of a
sphere varies with the square of the radius.
Let’s start: What percentage of a cube is a sphere?A sphere of radius 1 unit just fits inside a cube.
• First guess the percentage of space occupied by the sphere.
• Draw a diagram showing the sphere inside the cube.
• Calculate the volume of the cube and the sphere. For the sphere use V = 4
3pr3.
• Now calculate the percentage of space occupied by the sphere. How close was your guess?
6.8
Mars is approximately spherical in shape.
Example 18 Finding the surface area and volume of a sphere
Find the surface area and volume of a sphere of radius 7 cm, correct to two decimal places.
Solution Explanation
TSA = 4pr 2
= 4p(7)2
= 615.75 cm2
V = 4
3pr3
= 4
3p(7)3
= 1436.76 cm3
Write the rule for the surface area of a sphere and
substitute r = 7.
Evaluate and round the answer.
Write the rule for the volume of a sphere and
substitute r = 7.
Evaluate and round the answer.
■■ The surface area of a sphere depends on its radius r and is given by:
Surface area = 4pr 2
■■ The volume of a sphere depends on its radius r and is given by:
Volume = 4
3pr 3
r
Key
idea
s
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 409
Example 19 Finding the radius of a sphere
Find the radius of a sphere with volume 10 m3 correct to two decimal places.
V = 10 m3
Solution Explanation
V = 4
3pr3
10 = 4
3pr3
30 = 4pr3
15
2π = r3
∴ r=15
23
π
= 1.34
∴ the radius is 1.34 m.
Substitute V = 10 into the formula for the volume of a
sphere.
Solve for r3 by multiplying both sides by 3 and then
dividing both sides by 4p. Simplify 30
4
15
2π π= .
Take the cube root of both sides to make r the subject
and evaluate.
r m
Example 20 Finding the surface area and volume of composite solids with sphere portions
This composite object includes a hemisphere and cone as shown.
a Find the surface area, rounding to two decimal places.
b Find the volume, rounding to two decimal places.
Solution Explanation
a Radius r = 7 - 5 = 2
s2 = 52 + 22
s2 = 29
s = 29
TSA = 1
2 × 4pr 2 + prs
= 1
2 × 4p(2)2 + p(2) 29( )
= 8p + 2 29p = 58.97 cm2
First find the radius, r cm, of the
hemisphere.
Calculate the slant height, s, of the
cone using Pythagoras’ theorem.
Write the rules for the surface area of
each component and note that the top
shape is a hemisphere (half sphere). Only the curved
surface of the cone is required.
Substitute r = 2 and h = 5.
Simplify and then evaluate, rounding as required.
5 cm 7 cm
5 cm
2 cm
√29
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement410
b V r r h= × +
= × +
=
1
2
4
3
1
31
2
4
32
1
32 5
16
3 2
3 2
π π
π π
π
( ) ( ) ( )
33
20
336
312
37 70 3
+
=
=
=
π
π
π
. cm
Volume (object) = 1
2Volume(sphere) + Volume(cone)
Substitute r = 2 and h = 5.
Simplify and then evaluate, rounding as required.
Exercise 6H
Unde
rsta
ndin
g
1 Evaluate and round to two decimal places.
a 4 × p × 52 b 4 × p × 2.22 c 4 × p × 1
2
2
d 4
3 × p × 23 e
4
3 × p × 2.83 f
4 7
3
3π( )
2 Rearrange S = 4pr 2 to write r in terms of S.
3 Rearrange V = 4
3pr 3 to write r in terms of V.
4 What fraction of a sphere is shown in these diagrams?
a b c
5 Find the surface area and volume of the following spheres, correct to two decimal places.
a
2 cm
b 0.5 m
c
38 mm
d
18 cm
e 0.92 km
f 1.36 m
Example 18
Flue
ncy
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 411
Flue
ncy
6 Find the total surface area and volume of a sphere with the given dimensions. Give the answer
correct to two decimal places.
a radius 3 cm b radius 4 m c radius 7.4 m
d diameter 5 mm e diameter 7 m f diameter 2.2 km
7 a Find the radius of these spheres with the given volumes, correct to two decimal places.
i
V = 15 cm3
r cm
ii
V = 180 cm3
rcm
iii
V = 0.52 km3
rkm
b Find the radius of these spheres with the given surface area, correct to two decimal places.
i
S = 10 m2
rm
ii
S = 120 cm2
rcm
iii
S = 0.43 mm2
rmm
Example 19
Prob
lem
-sol
ving
8 A box with dimensions 30 cm long, 30 cm wide and 30 cm high holds 50 tennis balls of radius
3 cm. Find:
a the volume of 1 tennis ball, correct to two decimal places
b the volume of 50 tennis balls, correct to one decimal place
c the volume of the box not taken up by the tennis balls, correct to one decimal place.
9 An expanding spherical storage bag has 800 cm3 of water pumped into it. Find the diameter of the
bag correct to one decimal place, after all the water has been pumped in.
10 A sphere just fits inside a cube. What is the surface area of the sphere as a percentage of the
surface area of the cube? Round to the nearest whole percentage.
11 Find the volume of these portions of a sphere correct to two decimal places.
a 5 cm b 4 cm c 1 m
12 Two sports balls have radii 10 cm and 15 cm. Find the difference in their total surface areas,
correct to one decimal place.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement412
Prob
lem
-sol
ving
13 A monolithic structure has a cylindrical base of radius 4 m
and height 2 m and a hemispherical top.
a What is the radius of the hemispherical top?
b Find the total volume of the entire monolithic structure
correct to one decimal place.
14 Find the total surface area for these solids, correct to two
decimal places.
a 5 m b
1.3 cm
0.7 cm c
4 mm
8 mm
d
1 m
5 m3 m
e
2 m
f 3 cm
2 cm
√2 cm
15Find the volume of the following composite objects correct to two decimal places.
a
3 m
1 m
b
13 m
19 m
c
2 cm
d 28 cm
20 cm
e
2 m
2 m
2 m
f
1 cm
Hollow
2 m4 m
Example 20a
Example 20b
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 413
Prob
lem
-sol
ving
16 A spherical party balloon is blown up to help decorate a room.
a Find the volume of air, correct to two decimal places, needed for the balloon to be:
i 10 cm wide ii 20 cm wide iii 30 cm wide
b If the balloon pops when the volume of air reaches 120 000 cm3, find the diameter of the
balloon at that point, correct to one decimal place.
17 A hemisphere sits on a cone and two height measurements are
given as shown. Find:
a the radius of the hemisphere
b the exact slant height of the cone in surd form
c the total surface area of the solid, correct to one decimal place.
15 cm10 cm
18 a Find a rule for the radius of a sphere with surface area S.
b Find a rule for the radius of a sphere with volume V.
19 A ball’s radius is doubled.
a By how much does its surface area change? b By how much does its volume change?
20 Show that the volume of a sphere is given by V = 1
6 pd3 where d is the diameter.
21 A cylinder and a sphere have the same radius r and volume V. Find a rule for the height of the
cylinder in terms of r.
Reas
onin
g
Enrichment: Comparing surface areas
22 Imagine a cube and a sphere that have the
same volume.
a If the sphere has volume 1 unit3, find:
i the exact radius of the sphere
ii the exact surface area of the sphere
iii the value of x (the side length of the cube)
iv the surface area of the cube
v the surface area of the sphere as a percentage of the surface area of the cube correct to one
decimal place.
b Now take the radius of the sphere to be r units. Write:
i the rule for the surface area of the sphere
ii the rule for x in terms of r, given the volumes are equal
iii the surface area of the cube in terms of r.
c Now write the surface area of the sphere as a fraction of the surface area of the cube using your
results from part b.
Simplify to show that the result is π6
3 .
d Compare your answers from part a v and part c.
xx
x
r
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement414
Similar objects
Enlarging a rectangular prism
The larger rectangular prism shown here is an enlargement of the smaller prism by a factor of 1.5.
4
26
6
3 9
a Write down the ratios of the side lengths.
b Find the surface areas of the prisms and write these as a ratio in simplest form.
c Find the volumes of the prisms and write these as a ratio in simplest form.
d What do you notice about the ratios for length, area and volume? Explain.
e Can you write all three ratios using indices with the same base?
Using similarity
From the previous section, we note that if two similar objects have a length ratio a : b the following
apply.
Length ratio = a : b Scale factor = b
a
Area ratio = a2 : b2 Scale factor = b
a
2
2
Volume ratio = a3 : b3 Scale factor = b
a
3
3
a A sphere of radius 1 cm is enlarged by a length scale factor of 3.
i Find the ratio of the surface areas of the two spheres.
ii Find the ratio of the volumes of the two spheres.
iii Use your ratio to find the surface area of the larger sphere.
iv Use your ratio to find the volume of the larger sphere.
b This rectangular prism is enlarged by a factor of 1
2, which means that the prism is reduced in size.
10 m
2 m12 m
i Find the ratio large : small of the surface areas of the prisms.
ii Find the ratio large : small of their volumes.
iii Use your ratios to find the surface area of the smaller prism.
iv Use your ratios to find the volume of the smaller prism.
inve
stig
atio
n
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 415
DensityDensity is defined as the mass or weight of a substance per cubic unit of volume.
Densitymass
volume= or Mass = density × volume
Finding mass
Find the total mass of these objects with the given densities, correct to one decimal place where
necessary.
a
1 m2 m
3 m
Density = 50 kg per m3
b
Density = 100 kg per m3
2 m
1 m c
Density = 0.05 kg per m3
10 cm
5 cm
6 cm
Finding density
a Find the density of a compound with the given mass and volume measurements, rounding to two
decimal places where necessary.
i Mass 30 kg, volume 0.4 m3 ii Mass 10 g, volume 2 cm3
iii Mass 550 kg, volume 1.8 m3
b The density of a solid depends somewhat on how its molecules are packed together. Molecules
represented as spheres are tightly packed if they are arranged in a triangular form. The following
relates to this packing arrangement.
1 cmA
C
i Find the length AC for three circles, each of radius 1 cm, as shown. Use exact values.
ii Find the total height of four spheres, each of radius 1 cm, if they are packed to form a
triangular-based pyramid. Use exact values.
First note that AB = 2BC for an equilateral triangle (shown). Pythagoras’
theorem can be used to prove this, but this is trickier.
C
A
B
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement416
ConcentrationConcentration is associated with the purity of dissolved substances and will be considered here
using percentages.
Concentration (%)volume of substance
total v=
oolume× 100
1
Finding concentration
Find the concentration of acid as a percentage, if 10 cm3 of pure acid is mixed into the given containers
that are half full (half the volume) of water. Give your answers correct to two decimal places.
a
10 cm
5 cm
b
2 cm
12 cm
c 6 cm
8 cm
Finding volumes
a Find the volume of pure acid required to give the following concentrations and total volumes.
i Concentration 25%, total volume 100 cm3
ii Concentration 10%, total volume 90 m3
iii Concentration 32%, total volume 1.8 L
b A farmer mixes 5 litres of a chemical herbicide into a three-quarter full tank of water. The tank is
cylindrical with diameter 1 metre and height 1.5 metres. Find:
i the number of litres of water in the tank, correct to three decimal places
ii the concentration of the herbicide after it has been added to the water, correct to two decimal
places
iii the possible diameter and height measurements of a tank that would make the concentration
of herbicide 1%.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 417
1 A cube has a surface area that has the same value as its volume. What is the side length of this
cube?
2 The wheels of a truck travelling at 60 km/h make 4 revolutions per second. What is the diameter
of each wheel in metres to one decimal place?
3 A sphere fits exactly inside a cylinder, and just touches the top, bottom and curved surface. Show
that the surface area of the sphere equals the curved surface area of the cylinder.
4 A sphere and cone with the same radius r have the same volume. Find the height of the cone in
terms of r.
5 Four of the same circular coins of radius r are placed such that they are just
touching as shown. What is the area of the shaded region enclosed by the
coins in terms of r?
6 Find the exact ratio of the equator to the distance around the Earth at latitude 45° north. (Assume
that the Earth is a perfect sphere.)
45°
7 A ball just fits inside a cylinder. What percentage of the volume of the cylinder is taken up by the
ball? Round to the nearest whole percentage.
Chal
leng
es
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement418
Chap
ter s
umm
ary
Measurement
Pythagoras’ theorem
For prisms and pyramids draw thenet and add the areas of all the faces.
e.g.
Surface areaVolume
r
r
h
rh
h2πr
s
r
TSA = 4 × triangles + square base
TSA = 2πrh + 2πr2
curved base surface ends
TSA = 4πr2
TSA = πrs + πr2
curved + base
Cylinder
Cone
h
r
Cone:
Sphere:
Sphere
For composite solids considerwhich surfaces are exposed.
×10003
÷10003
km3 m3
×1003
÷1003
cm3
×103
÷103
mm3
×1000
÷1000
megalitres(ML)
kilolitres(kL)
millilitres(mL)
Litres(L)
×1000
÷1000
×1000
÷1000
h
wl
V = πr2hV = lwh
V = Ah13
V = πr2h13
V = πr343
LengthUnits of length
Perimeter is the distance aroundthe outside of a closed shape.Circumference of a circle,C = 2πr = πdPerimeter of a sector
×1000
÷1000
km m
×100
÷100
cm
×10
÷10
mm
P = 2r + × 2πrθ360
θ
r
cb
a
c2 = a2 + b2
Can occur in 3D shapes
AreaUnits of area
Formulas
×10002
÷10002
km2 m2
×1002
÷1002
cm2
×102
÷102
mm2
r
Circle
Sector Rhombus Parallelogram Trapezium Kite
A = πr2
l
Square
A = l 2
w
l
Rectangle
A = lw
h
b
Triangle
A = bh12
A = xy12A = xy1
2A = × πr2θ360
A = h (a + b)A = bh 12
θ h
b
h
b
ax
yxy
+
where A is the area of the base
Units of volume
Rectangular prism
For pyramids and cones:
Cylinder
Units of capacity
1 cm3 = 1 mL
For right and oblique prisms and cylindersV = Ah, A is the area of the base h is the perpendicular height
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 419
Multiple-choice questions 1 If the perimeter of this shape is 30.3 cm the value of x is:
A 12.8
B 6.5
C 5.7
D 6.4
E 3.6
2 The perimeter of the sector shown, rounded to one decimal place, is:
A 8.7 m
B 22.7 m
C 18.7 m
D 56.7 m
E 32.7 m
3 The value of x in this triangle is closest to:
A 4.9 B 3.5 C 5.0
D 4.2 E 3.9
4 The exact shaded area in square units is:
A 32 − 72p B 120 − 36p C 32 + 6pD 120 − 18p E 48 + 18p
5 0.128 m2 is equivalent to:
A 12.8 cm2 B 128 mm2 C 1280 cm2 D 0.00128 cm2 E 1280 mm2
6 A cube has a total surface area of 1350 cm2. The side length of the cube is:
A 15 cm B 11 cm C 18 cm D 12 cm E 21 cm
7 A cylindrical tin of canned food has a paper label glued around its curved surface. If the
can is 14 cm high and has a radius of 4 cm, the area of the label is closest to:
A 452 cm2 B 352 cm2 C 126 cm2 D 704 cm2 E 235 cm2
8 The exact surface area of a cone of diameter 24 cm and slant height 16 cm is:
A 216p cm2 B 960p cm2 C 528p cm2 D 336p cm2 E 384p cm2
9 A cone has a radius of 7 m and a slant height of 12 m. The exact height
in metres of the cone is:
A 52 B 193 C 85
D 137 E 95
10 The volume of the hemisphere shown correct to the nearest cubic
centimetre is:
A 1257 cm3 B 628 cm3 C 2094 cm3
D 1571 cm3 E 4189 cm3
10A
10A
10A
x cm
7.1 cm10.4 cm
100°5 m
x km7 km
12
10
7 m
12 m
10 cm
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement420
11 The volume of this oblique square-based prism is:
A 72 cm3 B 48 cm3 C 176 cm3
D 144 cm3 E 120 cm3
12 The volume of air in a sphere is 100 cm3. The radius of the sphere
correct to two decimal places is:
A 1.67 cm B 10.00 cm C 2.82 cm D 23.87 cm E 2.88 cm
Short-answer questions1 Convert the given measurements to the units in the brackets.
a 0.23 m (cm) b 270 mm2 (cm2) c 2.6 m3 (cm3)
d 8.372 litres (mL) e 638 250 mm2 (m2) f 0.0003 km2 (cm2)
2 Find the perimeter of these shapes correct to one decimal place where necessary. Note
Pythagoras’ theorem may be required in part b.
a
8 m
8 m
b
9 m
7 m
4 m
c
80°6 cm
3 A floral clock at the Botanic Gardens is in the shape of a circle and has a circumference of 14 m.
a Find the radius of the clock in exact form.
b Hence, find the area occupied by the clock. Answer to two decimal places.
4 For the rectangular prism with dimensions as shown, use
Pythagoras’ theorem to find:
a AF, leaving your answer in exact form
b AG to two decimal places.
9 cm
4 cm
10A
D
H G
C
B
FE
A
2
47
10A
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 421
5 Find the area of these shapes. Round to two decimal places where necessary.
a
4.2 m
6.2 m
b
12 m
18 m
13 m
c
7 m11 m
d
10 cm
e
6.4 m
9.8 m
f 13 m
7 m4.2 m
8.1 m
6 A backyard deck as shown has an area of 34.8 m2.
w m
9.2 m
5.3 m
deck
a Find the width w metres of the deck.
b Calculate the perimeter of the deck correct to two decimal places. Pythagoras’ theorem will
be required to calculate the missing length.
7 For each of these solids find, correct to two decimal places where necessary:
i the total surface area ii the volume
a
6 m
5 m8 m
b
6 cm10 cm
5 cm4 cm
c 8 cm
20 cm
d 13 cm
10 cm
10 cm
12 cm
e
4 m
f
12 cm
10 cm
13 cm
8 A cone has a radius of 6 cm and a curved surface area of 350 cm2.
a Find the slant height of the cone in exact form.
b Find the height of the cone correct to one decimal place.
10A
10A
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Chapter 6 Measurement422
9 A papier mache model of a square-based pyramid with base length 30 cm has volume 5400 cm3.
a What is the height of the pyramid?
b Use Pythagoras’ theorem to find the exact height of the triangular faces.
c Hence, find the total surface area of the model correct to one decimal place.
10 A cylinder and a cone each have a base radius of 1 m. The cylinder has a height of 4 m.
Determine the height of the cone if the cone and cylinder have the same volume.
11 For each of the following composite solids find, correct to two decimal places where necessary:
i the total surface area ii the volume
a
10 cm
3 cm
3 cm
b
7 m
4 m
5 m c
2.5 mm 1.6 mm
0.9 mm
12 For each of the following composite solids find, correct to two decimal places where necessary:
i the total surface area ii the volume
a
4 cm
5 cm
b 3.9 cm
5 cm
3 cm
2 cm
c
3.2 cm
1.8 cm
13 A water buoy is in the shape shown.
a Find the volume of air inside the buoy in exact form.
b Find the surface area of the buoy in exact form.
10A
10A
10A
10A
39 cm
51 cm
36 cm
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press
Measurement and Geometry 423
Extended-response questions1 A water ski ramp consists of a rectangular floatation container and a triangular angled section as
shown.
6 m
2 m 8 m
1 m
c m
a What volume of air is contained within the entire ramp structure?
b Find the length of the angled ramp (c metres) in exact surd form.
The entire structure is to be painted with a waterproof paint costing $20 per litre. One litre of
paint covers 25 square metres.
c Find the total surface area of the ramp, correct to one decimal place.
d Find the number of litres and the cost of paint required for the job. Assume you can only
purchase 1 litre tins of paint.
2 A circular school oval of radius 50 metres is marked with spray paint to
form a square pitch as shown.
a State the diagonal length of the square.
b Use Pythagoras’ theorem to find the side length of the square in exact
surd form.
c Find the area of the square pitch.
d Find the percentage area of the oval that is not part of the square pitch. Round to the nearest
whole percentage.
Two athletes challenge each other to a one-lap race around the oval. Athlete A runs around the
outside of the oval at an average rate of 10 metres per second. Athlete B runs around the outside
of the square at an average rate of 9 metres per second. Athlete B’s average running speed is less
because of the need to slow down at each corner.
e Find who comes first and the difference in times, correct to the nearest hundredth of a second.
ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.
© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press