mms_g8_se_unit03
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School
The Chinese University of Hong Kong *
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Course
MISC
Subject
Mathematics
Date
Nov 24, 2024
Type
Pages
36
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These students are setting up a tent.
How do the students know how to set up the tent?
How is the shape of the tent
created? How could students find the
amount of material needed to
make the tent? Why might students want to know
the volume of the tent?
Why It’s Important
•
We find out about our environment by looking at objects from different views. When we combine these views, we have a better understanding of these objects.
•
We need measurement and calculation skills to design and build objects, such as homes and parks.
What You’ll Learn
•
Recognize and sketch objects.
•
Use nets to build objects.
•
Develop and use a formula for the
surface area of a triangular prism.
•
Develop and use a formula for the volume of a triangular prism.
•
Solve problems involving prisms.
94
Key Words
• isometric diagram
• pictorial diagram
• triangular prism
• surface area
• volume
95
Drawing Isometric and Pictorial Diagrams
An isometric diagram
shows three dimensions of an object. It is drawn on isometric
(triangular) dot paper. Vertical edges of an object are drawn as vertical line segments.
Parallel edges of an object are drawn as parallel line segments.
Example 1
Make an isometric diagram of this object.
Solution
On isometric paper, join a pair of dots for each vertical edge.
Join a pair of dots diagonally for each horizontal edge that goes up to the right.
Join a pair of dots diagonally for each horizontal edge that goes up to the left.
Shade the faces so the object appears three-dimensional.
In a pictorial diagram
, the depth of an object is drawn to a smaller scale than the
length and width. This gives the object a three-dimensional appearance.
Example 2
Make a pictorial diagram of this cylinder.
96
UNIT 3: Geometry and Measurement
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Skills You’ll Need
97
Solution The top and bottom of a cylinder are circles.
In a pictorial diagram, a circular face is an oval.
One-half of the bottom circular face is drawn as a broken curve. This indicates that this half cannot be seen.
Draw vertical line segments to join the top and bottom ovals.
1.
Make an isometric diagram of each object.
a)
a rectangular prism with dimensions 3 units by 4 units by 5 units
b)
a square pyramid
2.
Make a pictorial diagram of each object.
a)
a rectangular prism with dimensions 3 units by 4 units by 5 units
b)
a regular tetrahedron
Calculating the Surface Area and Volume of a Rectangular Prism
The surface area of a rectangular prism is the sum of the areas of all its faces.
Since opposite faces are congruent,
this formula can be used to find the surface area:
Surface area
2
area of base
2
area of side face 2
area of front face
Using symbols, we write:
SA
2
lw
2
hw
2
lh
Since the congruent faces occur in pairs, this formula can be written as:
SA
2(
lw
hw
lh
)
In this formula,
l
represents length,
w
represents width,
and h
represents height.
The volume of a rectangular prism is the space occupied by the prism.
One formula for the volume is: Volume area of base
height
Using symbols, we write:
V
lwh
✓
Remember to look up any terms that you are unsure of in the Glossary
.
h
w
98
UNIT 3: Geometry and Measurement
Example 3
A rectangular prism has dimensions 4 m by 6 m by 3 m.
a)
Calculate the surface area.
b)
Calculate the volume.
Solution
Draw and label a pictorial diagram.
a)
Use the formula for the surface area of a rectangular prism:
SA
2(
lw
hw
lh
)
Substitute:
l
6,
w
4, and h
3
SA
2(6
4
3
4
6
3) In the brackets, multiply then add.
2(24
12
18) 2(54)
108
The surface area is 108 m
2
.
b)
Use the formula for the volume of a rectangular prism:
V
lwh
Substitute:
l
6,
w
4, and h
3
V
6 4
3 72
The volume is 72 m
3
.
A cube is a regular polyhedron with 6 square faces.
Since all the faces of a cube are congruent,
we can simplify the formulas for surface area and volume. Each edge length is s
.
The area of each square face is:
s
s
s
2
So, the surface area of a cube is:
SA
6
s
2
The volume of a cube is:
V
s
s
s
s
3
3.
Find the surface area and volume of each rectangular prism.
Include a labelled pictorial diagram for each rectangular prism.
a)
12 cm by 6 cm by 8 cm
b)
7 mm by 7 mm by 4 mm
c)
2.50 m by 3.25 m by 3.25 m
d)
5 cm by 5 cm by 5 cm
✓
Area and surface area are
measured in square units (m
2
).
Volume is measured in
cubic units (m
3
).
6 m
4 m
3 m
s
Skills You’ll Need
99
Calculating the Area of a Triangle
The area of a triangle is calculated with either of these formulas:
Area
base
height
2, or Area
one-half
base
height
Using symbols,
we write:
A
b
2
h
or A
1
2
bh
where b
is the length of the base and h
is the corresponding height.
Example 4
The side lengths of
PQR are 12 cm, 5 cm, and 13 cm.
a)
The height from P to QR is about 4.6 cm.
Use this to calculate the area of
PQR.
b)
Triangle PQR is a right triangle with P
90°.
Use this to calculate the area of
PQR a different way.
Solution
a)
Use the formula:
A
b
2
h
Substitute:
b
13 and h
4.6
A
13
2
4.6
59
2
.8
29.9
The area is 30 cm
2
to the nearest square centimetre.
These formulas are equivalent.
Dividing bh
by 2 is the same as
multiplying bh
by 1
2
.
base
height
height
base
13 cm
12 cm
5 cm
4.6 cm
P
R
Q
height
base
4.6 cm
P
R
Q
13 cm
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4.
Calculate the area of each triangle.
a)
b)
c)
Converting among Units of Measure
1 m
100 cm
The area of a square with side length 1 m is:
A
1 m
1 m 1 m
2
The area of a square with side length 100 cm is:
A
100 cm
100 cm 10 000 cm
2
So, 1 m
2
10 000 cm
2
, or 10
4
cm
2
✓
b)
Since PQR is a right triangle,
the two sides that form the right angle are the base and height.
QPR
90°, so QP 5 cm is the height;
and PR 12 cm is the base.
Use the formula:
A
b
2
h
Substitute:
b
12 and
h
5
A
12
2
5
6
2
0
30
The area is 30 cm
2
.
100
UNIT 3: Geometry and Measurement
Any triangle has 3 sets of base and height.
All sets produce the same area.
13 cm
12 cm
5 cm
P
R
Q
10 m
5 m
10 cm
6 cm
8 cm
4.8 cm
1.1 cm
1 m
1 m
100 cm
100 cm
100 cm
1 m
The volume of a cube with edge length 1 m is:
V
1 m
1 m
1 m
1 m
3
The volume of a cube with edge length 100 cm is:
V
100 cm
100 cm 100 cm
1 000 000 cm
3
So, 1 m
3
1 000 000 cm
3
, or 10
6
cm
3
The volume of a cube with edge length 10 cm is:
V
10 cm 10 cm 10 cm
1000 cm
3
Since 1 cm
3
1 mL,
then 1000 cm
3
1000 mL
1 L
Example 5
Convert.
a)
0.72 m
2
to square centimetres
b)
1.05 m
3
to cubic centimetres
Solution
a)
0.72 m
2
to square centimetres
1 m
2
10 000 cm
2
So, 0.72 m
2
0.72
10 000 cm
2
7200 cm
2
b)
1.05 m
3
to cubic centimetres
1 m
3
1 000 000 cm
3
, or 10
6
cm
3
So, 1.05 m
3
1.05
10
6
cm
3
This answer is in scientific notation.
1 050 000 cm
3
This answer is in standard form.
5.
Convert. Write your answers in standard form and in scientific notation,
where appropriate.
a)
726.5 cm to metres
b)
4300 cm
2
to square metres
c)
980 000 cm
3
to cubic metres
d)
4 280 000 cm
3
to litres
e)
8.75 m to centimetres
f)
1.36 m
2
to square centimetres
g)
14.98 m
3
to cubic centimetres
h)
9.87 L to cubic centimetres
To multiply by 10 000, move the
decimal point 4 places to the right.
✓
1 m
1 m
1 m
100 cm
100 cm
100 cm
10 cm
10 cm
10 cm
Skills You’ll Need
101
102
UNIT 3: Geometry and Measurement
When we draw a view of an object, we show internal line segments only where the depth or thickness of the object changes.
Here is an object made with 7 linking cubes.
Here are the views:
The broken lines show how the views are aligned.
3.1
Building and Sketching Objects
Recognize, build, and sketch an object
given different views of the object.
Focus
Work in a group of 3.
Each student needs 8 linking cubes and isometric dot paper.
Each student chooses one
of these views.
You use 8 cubes to build an object that matches the view you chose.
Sketch your object on isometric paper.
Use your cubes to build, then sketch, a different object with the view
you chose.
Reflect &
Share
Compare your objects with those of other members of your group.
Does any object match all 3 views? If not, build and sketch one that does.
What helped you decide the shape of the object?
Are any other views needed to identify the object? Explain.
Left Side
Front
Top
Right Side
Left Side
Front
Top
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3.1 Building and Sketching Objects
103
Example Solution
Each view of an object provides information about the shape of the
object. When an object is built with linking cubes, the top, front, and
side views are often enough to identify and build the object. These
views are drawn with the top view above the front view, and the side
views beside the front view, as they were at the top of page 102.
In this way, matching edges are adjacent.
Which object has these views?
The top view is a regular hexagon.
The side view is 2 congruent rectangles.
The front view is 3 rectangles, 2 of which are congruent.
This object is a hexagonal prism.
Top
Side
Front
104
UNIT 3: Geometry and Measurement
2.
Sketch a different view of two of the objects in question 1.
3.
Use these 4 clues and linking cubes to build an object.
Draw the object on isometric paper.
Clue 1: There are 6 cubes in all. One cube is yellow.
Clue 2: The green cube shares one face with each of the other 5 cubes.
Clue 3: The 2 red cubes do not touch each other.
Clue 4: The 2 blue cubes do not touch each other.
4.
a)
Build the object for the set of views below.
b)
Sketch the object on isometric dot paper.
You will need linking cubes, isometric dot paper, and grid paper.
1.
Match each view A to D with each object H to L.
Name each view: top, bottom, front, back, left side, or right side
A
B
C
D
Front
Top
Right Side
H
I
J
L
K
3.1 Building and Sketching Objects
105
The relationship in part c
is called Euler’s formula
. It
is named for a Swiss
mathematician, Leonhard
Euler, who lived in the
18th century.
Write each number in
scientific notation.
•
3 590 000
•
40 400 000
•
398 759
Number Strategies 5.
a)
Build an object with each number of linking cubes.
For 3 or more cubes, do not
make a rectangular prism.
i)
2
ii)
3
iii)
4
iv)
5
v)
6
b)
For each object you build, count the number of its faces,
edges, and vertices. Record your results in a table.
c)
Look for a pattern in the table in part b.
For any object, how are the numbers of faces, edges,
and vertices related?
d)
Build an object with 7 linking cubes.
Check that the relationship in part c is true.
6.
Assessment Focus
a)
Use these views to build an object.
b)
Sketch the object on isometric dot paper.
c)
Draw the other views of the object.
7.
a)
Use these views to build an object.
A shaded region has no cubes.
b)
Sketch the object.
Explain your work.
How do views help to show a three-dimensional object? Use an example to explain.
Top View
Right Side View
Top View
Front View
Right Side View
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106
UNIT 3: Geometry and Measurement
A net is a pattern that can be folded to make an object.
Here is a net and the rectangular prism it forms.
A polyhedron can have several different nets.
3.2
Sketching and Folding Nets
Sketch nets and use them to build objects.
Focus
Remember that an
internal line segment
on a view shows that
the depth changes.
Set A
Work with a partner.
You will need scissors, tape, and 1-cm grid paper.
For each set of views below:
➢
Identify the object. Draw a net of the object.
➢
Cut out your net. Check that it folds to form the object.
➢
Describe the object.
5 cm
5 cm
6 cm
2.5 cm
6 cm
Top View
Front View
Right Side View
6.5 cm
6 cm
6 cm
6 cm
6 cm
6 cm
Top View
Front View
Right Side View
Back View
Left Side View
Set B
3.2 Sketching and Folding Nets
107
Reflect &
Share
Compare your nets with those of another pair of classmates.
How did you know how many faces to draw?
How do you know which faces share a common edge? How do the faces on the net compare to the views for each object?
Could you have drawn a different net for the same object? Explain.
Some faces are not visible from a particular view.
Look at the views of an object, at the right.
The top view shows that four congruent triangular faces meet at a point.
The front and side view shows an isosceles triangular face.
The bottom view shows the base is a square.
This object is a square pyramid.
We can use a ruler, protractor, and compass to draw the net.
In the centre of the paper,
draw a 5-cm square for the base.
On each side of the base,
draw an isosceles triangle with two equal sides of 6 cm.
Other arrangements of the five faces may produce a net.
Each outer edge must match another outer edge, and no faces must overlap.
The net at the right is constructed so that the 5-cm square base is attached to only one triangle.
5 cm
6 cm
5 cm
6 cm
Top
Front/Side
Bottom
5 cm
6 cm
5 cm
5 cm
5 cm
6 cm
5 cm
108
UNIT 3: Geometry and Measurement
Example Solution
Each net folds to form a square pyramid.
Many views of an object are needed to create its net, especially if the object is made with linking cubes.
These four views show an object made with linking cubes.
Use the views to draw a net for the object.
Use 1-cm dot paper.
Start with the simplest view, which is the bottom view.
Draw the bottom face.
Draw the front face and back face above and below the bottom face.
The top view and side views show a change in depth.
A new face is drawn for each depth change.
A side square is drawn for each depth change.
Each side view is drawn to touch the square that shows the change in depth.
Top View
Bottom View
Left/Right Side View
Front/Back View
Front
Bottom
Side
Back
Side
Top
Top
Top
Side
Side
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3.2 Sketching and Folding Nets
109
The net in the Example
folds to form this object with 10 faces, including the bottom face.
The symmetry of the object made it easier to draw a net. Other arrangements of the 10 faces may be folded to form the object.
1.
Each set of views below represents an object.
a)
Identify each object.
b)
Draw a net for the object.
c)
Cut out the net. Build the object.
d)
Describe the object.
i) ii)
2.
Each set of views represents an object.
Draw 2 different nets for each object.
Build the object.
a)
b)
3.
Choose one set of views from question 2.
Describe the steps you used to draw the net.
Top View
Side View
Front View
8 cm
8 cm
6 cm
2 cm
2 cm
Top View
2 cm
3 cm
3 cm
2 cm
Front View
Side View
Top View
3 cm
3 cm
Front View
Side View
Top View
Front/Side View
7 cm
3.4 cm
7 cm
4 cm
3 cm
4 cm
110
UNIT 3: Geometry and Measurement
Describe how a set of views of an object relates to the figures on a net of the object. Is there one correct net for a set of views?
Explain with an example.
Officials mark the
perimeter of a playing
field with spray paint.
One can of paint covers 50 m of perimeter.
How much paint is
needed to spray the
perimeter of each field?
• Lacrosse field:
100 m by 55 m
• Soccer field:
109 m by 73 m
Number Strategies
4.
A chocolate box has the shape of a prism with a base that is a
rhombus. Each side length of the rhombus is 3.6 cm.
The angles between adjacent sides of the base are 60° and 120°.
The prism is 10.8 cm long.
a)
Draw a net for this box.
b)
How is your net different from the cardboard net from which the box is made? Explain.
5.
These views represent an object.
Each shaded region is an opening in a face.
Draw a net for the object.
Build the object.
6.
Assessment Focus
These views represent an object.
a)
Identify the object.
b)
Draw two different nets.
c)
Use the two different nets to build the object.
d)
Is one net easier to draw or fold? Explain.
18 cm
5 cm
5 cm
7.1 cm
5 cm
5 cm
8 cm
9 cm
Top View
Front View
5 cm
4 cm
Right Side View
6 cm
Top View
Side View
Front/Back View
8 cm
6 cm
8 cm
2 cm
120
°
Mid-Unit Review
111
1.
a)
Use linking cubes.
Use the views below to build the object.
Remember that internal lines show where the depth of the object changes.
b)
Sketch the object on isometric dot paper.
2.
a)
Use the views below to describe
the object. Remember that a
shaded area shows an opening in the face.
b)
Sketch a pictorial diagram of
the object.
3.
Each set of views that follows represents an object.
a)
Identify each object.
b)
Draw a net for each object.
c)
Cut out the net.
Build the object.
d)
Describe the object.
i)
ii)
iii)
LESSON
3.1
3.2
Top View
Left Side View
Front View
Right Side View
Top View
Front/Side View
Top View
3 cm
7 cm
Front View
Right Side View
6 cm
Top View
1 cm
2 cm
2 cm
Front View
Right Side View
155
°
155
°
3 cm
25
°
25
°
Top View
Front View
Side View
8 cm
12 cm
8 cm
12 cm
5 cm
13 cm
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112
UNIT 3: Geometry and Measurement
3.3
Surface Area of a Triangular Prism
Work with a partner.
You will need 1-cm grid paper.
➢
For each triangular prism below:
Draw a net.
Find the surface area of the prism.
➢
Write a formula you can use to find the surface area of any
triangular prism.
Reflect & Share
Compare your nets and formula with those of
another pair of classmates.
Did you write the same formula?
If not, do both formulas work? Explain.
Did you write a word formula or use variables? Explain.
Focus
Develop a formula for finding the surface
area of a triangular prism. The surface area of an
object is the sum of the
areas of its faces.
6 cm
5 cm
4 cm
3 cm
4 cm
6 cm
3.2 cm
2.5 cm
Prism A
Prism B
A triangular prism is formed when a triangle is translated in the air so that each side of the triangle is always parallel to its original position.
The two triangular faces are the bases of the prism.
3.3 Surface Area of a Triangular Prism
113
Here is a triangular prism and its net.
Both the prism and net are drawn to scale.
The two triangular faces of the prism are congruent.
Each triangular face has base 11 cm and height 3 cm.
So, the area of one triangular face is:
1
2
11 3
The surface area of a triangular prism can be expressed using a word formula:
SA
sum of the areas of three rectangular faces
2 area of one triangular face
Use this formula to find the surface area of the prism above.
SA
(5
2)
(11
2)
(8
2)
2
1
2
11
3 10
22
16
33
81
The surface area of the prism is 81 cm
2
.
We can use variables to write a formula for the surface area of a
triangular prism.
To avoid confusion between the height of a triangle and the height of
the prism, we now use
length
instead of
height
to describe the edge
that is perpendicular to the base.
For the triangular prism above:
The length of the prism is l
.
Each triangular face has side lengths a
,
b
, and c
.
The height of a triangular face is h
and its base is b
.
The area of a rectangle
is base
height.
a, b, c, h,
and l
are
variables.
Use the order of
operations. Multiply
before adding.
11 cm
3 cm
5 cm
8 cm
2 cm
Area = 10 cm
2
Area = 16.5 cm
2
Area = 22 cm
2
Area = 16.5 cm
2
Area = 16 cm
2
5 cm
11 cm
11 cm
8 cm
3 cm
2 cm
b
a
h
c
a
h
h
b
c
A = a
A = b
A = c
A
= bh
1
2
A
= bh
1
2
114
UNIT 3: Geometry and Measurement
Example Solution
The surface area of the prism is:
SA
sum of the areas of the 3 rectangular faces
2
area of one
triangular face
SA
al
bl
cl
2
1
2
bh
SA
al bl
cl
bh
Find the surface area of the prism below.
Each dimension has been rounded to the nearest whole number.
Write the surface area in square centimetres and in square metres.
Identify the variable that represents each dimension.
Sketch, then label the prism with these variables.
The length of the prism is:
l
40
The 3 sides of a triangular face are:
a
20,
b
31,
c
29
The height of a triangular face is:
h
18
Substitute for each variable in the formula for surface area.
SA
al
bl
cl bh
(20
40) (31
40) (29
40) (31
18)
800 1240 1160 558
3758
The surface area of the prism is 3758 cm
2
.
To convert square centimetres to square metres, divide by 10 000.
3758 cm
2
1
3
0
7
0
5
0
8
0
m
2
0.3758 m
2
The surface area of the prism is 0.3758 m
2
.
When you use this
formula, you must
identify what each
variable represents.
There are 10 000 cm
2
in 1 m
2
.
40 cm
31 cm
18 cm
29 cm
20 cm
= 40
b = 31
h = 18
c = 29
a = 20
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3.3 Surface Area of a Triangular Prism
115
1.
Calculate the area of each net.
a)
b)
2.
Calculate the surface area of each prism.
Draw a net first if it helps.
Write the surface area in square metres.
a)
b)
3.
Calculate the surface area of each prism.
The shaded region indicates that the face is missing.
Write the surface area in square centimetres.
a)
b)
4.
a)
What area of wood, in square metres, is needed to make the
ramp at the left? The ramp does not have a base.
b)
Suppose the ramp is built against a stage.
The vertical face that is a rectangle is against the stage.
How much wood is needed now? Number Strategies
Use up to four 3s. Use any
of
,
,
, or
.
Write an expression for
each number from 1 to 5.
4 cm
4 cm
4.5 cm
2 cm
4 m
5 m
3 m
3 m
5 m
3 m
9 m
4 m
11 m
26 cm
24 cm
50 cm
10 cm
9.00 m
4.00 m
11.72 m
7.00 m
7.20 m
6 cm
5 cm
10 cm
13 cm
12 cm
750 cm
250 cm
200 cm
776 cm
When a face of a prism
is missing, the prism is
a shell, not a solid.
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116
UNIT 3: Geometry and Measurement
Take It Further
Write to explain how to find the surface area of a triangular
prism. Include an example and a diagram.
5.
A plastic container company designs a container with a lid to
hold one piece of pie.
a)
Design the container as a triangular prism.
Explain your choice of dimensions.
b)
Calculate the area of plastic in your design.
6.
The total area of the 3 rectangular faces of an equilateral triangular prism is 72 cm
2
.
a)
No dimension can be 1 cm. What are the possible whole-number dimensions of the edges?
b)
Sketch the prism with the greatest length.
Include dimensions. Explain your choice.
7.
Assessment Focus
How much metal, in square metres,
is needed to build this water trough?
8.
A right triangular prism has a base with perimeter 12 cm and
area 6 cm
2
.
a)
Find the whole-number dimensions of the base.
b)
The length of the prism is 6 cm.
Calculate the surface area of the prism.
c)
Sketch the prism. Include its dimensions.
9.
Use the variables below to sketch and label a triangular prism.
The lengths have been rounded to the nearest whole number.
Calculate the surface area of the prism.
a
7 cm,
b
17 cm,
c
11 cm,
h
3 cm,
l
12 cm
23 cm
5 cm
120 cm
40 cm
60 cm
50 cm
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3.4 Volume of a Triangular Prism
117
Recall that the area of a triangle is one-half the area of a rectangle that has the same base and height.
That is,
the area of
DEC
1
2
the area of rectangle ABCD
3.4
Volume of a Triangular Prism
Develop and use a formula for finding the
volume of a triangular prism.
Focus
Work in a group of 4.
You will need 4 identical cereal boxes, one for each group member,
and markers.
➢
Find the volume of your cereal box, which is a rectangular prism.
➢
Use a ruler to draw a triangle on one end of the cereal box.
The base of each triangle should be along one edge of the box.
The third vertex of the triangle should be on the opposite edge.
Make sure you draw different triangles.
What is the volume of a triangular prism with this triangle as its base, and with length equal to the length of the cereal box?
➢
Compare your answer with those of other members of your
group. What do you notice?
➢
Work together to write a formula for the volume of a triangular prism.
Reflect &
Share
How is the volume of a triangular prism related to the volume of a
rectangular prism? Compare your formula for the volume of a triangular prism with
that of another group.
Did you use variables in your formula?
If not, work together to write a formula that uses variables.
The volume of a rectangular prism is:
V
base area
length
h
h
A
E
B
C
D
base area
length
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Example Suppose we draw a triangle on the base of the prism so that the base
of the triangle is one edge, and the third vertex of the triangle is on
the opposite edge.
The volume of a triangular prism with this base, and with length equal to the length of the rectangular prism, is one-half the volume
of the rectangular prism.
Since the base area of the triangular prism is one-half the base area
of the rectangular prism, the volume of a triangular prism is also:
V
base area
length
The base is a triangle, so the base area is the area of a triangle.
We can use variables to write a formula for the volume of
a triangular prism.
For the triangular prism below:
The length of the prism is l
.
Each triangular face has base b
and height h
.
The volume of the prism is:
V
base area
length
V
1
2
bh
l
V
1
2
bhl
How much water can the water trough hold? Give the answer in litres.
118
UNIT 3: Geometry and Measurement
When we use this
formula, we must
identify what each
variable represents.
base area
length
h
b
120 cm
40 cm
60 cm
50 cm
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3.4 Volume of a Triangular Prism
119
Solution
Capacity is the
amount a container
will hold, commonly
measured in litres (L)
or millilitres (mL).
Volume is the amount
of space an object
occupies, commonly
measured in cubic units.
The amount of water the trough can hold is the capacity of
the triangular prism.
Sketch the prism.
Identify the variable that represents each dimension.
The base of the triangle is:
b
60
The height of the triangle is:
h
40
The length of the prism is:
l
120
Substitute for each variable into the formula for volume.
V
1
2
bhl
V
1
2
60
40
120
V
144 000
The volume of the trough is 144 000 cm
3
.
1000 cm
3
1 L
So, 144 000 cm
3
144 L
The trough can hold 144 L of water.
1.
The base area and length for each triangular prism are given.
Find the volume of each prism.
a)
b)
c)
2.
Find the volume of each triangular prism.
a)
b)
c)
A = 9.2 cm
2
2.3 cm
5 cm
A = 43.5 cm
2
15 m
A = 3 m
2
13 cm
21 cm
7 cm
12 m
5 m
8 m
1.75 m
2.50 m
1.75 m
120
h
40
b
60
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120
UNIT 3: Geometry and Measurement
Use an isometric
diagram or a pictorial
diagram for the sketch
of the prism.
4.4 cm
6.8 cm
11.2 cm
15.5 cm
15.6 cm
12 cm
6 cm
3 cm
6.7 cm
2 m
4 m
3 m
5 m
3.
Calculate the volume of each prism.
a)
b)
4.
Find possible values for b
,
h
, and l
for each volume of a triangular prism. Sketch one possible prism for each volume.
a)
5 cm
3
b)
9 m
3
c)
8 m
3
d)
18 cm
3
5.
What is the volume of glass in this prism?
6.
Any face can be used as the base of a rectangular prism.
Can any face be used as the base of a triangular prism? Explain.
7.
The volume of a triangular prism is 30 cm
3
.
Each triangular face has an area of 4 cm
2
.
How long is the prism?
8.
a)
Calculate the surface area and volume of this triangular prism.
b)
What do you think happens to the surface area and volume
when the length of the prism is doubled? Justify your answer.
Calculate the surface area and volume to check your ideas.
c)
What do you think happens to the surface area and volume
when the base and height of the triangular faces are doubled?
Justify your answer.
Calculate the surface area and volume to check your ideas.
d)
What do you think happens to surface area and volume when
all the dimensions are doubled? Justify your answer.
Calculate the surface area and volume to check your ideas.
2 cm
9 cm
2 cm
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3.4 Volume of a Triangular Prism
121
9.
Assessment Focus
Jackie uses this form to build a concrete pad.
a)
How much concrete will Jackie need to mix to fill the form?
b)
Suppose Jackie increases the equal sides of the form from 3 m to 6 m.
How much more concrete will Jackie need to mix? Include a diagram.
10.
A chocolate company produces different sizes of chocolate bars that are packaged in equilateral triangular prisms.
Here is the 100-g chocolate bar.
a)
Calculate the surface area and volume of the box.
b)
The company produces a 400-g chocolate bar.
It has the same shape as the 100-g bar.
i)
What are the possible dimensions for the 400-g box?
ii)
How are the dimensions of the two boxes related?
11.
The volume and surface area of a prism, with a base that is not a triangle or a rectangle, can be found by dividing the prism into smaller prisms.
Find the volume and surface area of each prism.
a)
b)
Take It Further
The number 12 has six
factors: 1, 2, 3, 4, 6, and 12.
Which number less than 50
has the most factors?
List the factors of this
number.
Number Strategies
Describe the relationships among the dimensions, faces,
and volume of a triangular prism.
Include an example in your description.
0.25 m
3 m
3 m
5 m
4 m
7 m
6 m
6 m
1.5 cm
1.5 cm
1.8 cm
5 cm
2.1 cm
1 cm
5 cm
The form is not
drawn to scale.
3.1 cm
3.5 cm
21 cm
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122
UNIT 3: Geometry and Measurement
Features of Word Problems
Context
– may describe who, what, when,
where, and why, like the setting of a story.
Math Information
– may include words,
numbers, figures, drawings, tables, graphs,
and/or models.
Problem Statement
– tells you to do
something with the information. It may be
a question or an instruction. Key words are
sometimes used to suggest how the
response is to be communicated.
Suppose you have an old refrigerator and
are thinking about buying a new one.
The new refrigerator costs $900.
It costs $126 a year to run your old fridge, and $66 a year to run the new, more energy-efficient fridge.
Is it more economical to buy the new
fridge? Justify your answer.
Problem
Statement
Math Information
Cumulative Costs
Year
Old Fridge Cost ($)
New Fridge Cost ($)
1
126
900 + 66 = 966
2
252
1032
3
378
1098
4
504
1164
5
630
1230
6
756
1296
7
882
1362
In Unit 2
, you wrote problem statements as questions.
Math problems include information to help you understand and solve the problem.
Here are some features of math word problems.
To solve the problem, we need to compare the costs of buying the
new fridge and running it with the costs of running the old fridge.
We can use a table and a graph.
In the second column of the table, add $126 each year.
In the third column of the table, add $66 each year.
Here is part of the table and a graph.
Continue the table until the costs are equal.
Draw a graph for the data.
Use the table and graph to solve the problem.
Cumulative Costs of Fridges
0
2 3
1
4 5
Year
Cost ($)
6 7
200
400
600
800
1000
1200
1400
Old fridge
New fridge
Context
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Reading and Writing in Math: Features of Word Problems
123
CARPET SALE
Regular $9.99
per square metre
Now on Sale
20
For
%
off
✓
With a partner, identify the context,
math information, and problem statement for each of questions 1 to 4.
Then solve the problem.
1.
Hori is buying carpet for his living room.
It is rectangular with dimensions 4 m by 5 m.
How much will Hori save if he buys the carpet at the sale price shown at the left? 2.
Suppose you are in charge of setting up the cafeteria for a graduation dinner. One hundred twenty-two people will attend.
The tables seat either 8 or 10 people.
You do not want empty seats.
How many of each size table will you need to use to make sure
everyone has a seat? List all the combinations.
3.
To win a contest, you have to find a mystery number.
The mystery number is described this way:
Sixteen more than 2
3
of the mystery number is equal to two times the mystery number.
What is the mystery number?
4.
Alicia and Chantelle are playing a game.
There are six tiles in a box: three red and three blue.
A player picks two tiles without looking.
Alicia gets a point if the tiles do not match; Chantelle gets a point
if they do match. The tiles are returned to the box each time.
What is the probability that both tiles have the same colour? Is this a fair game? Explain.
5.
Write a word problem using the numbers 48, 149, and 600.
Remember to include a context, math information,
and a problem statement.
Trade problems with a classmate.
Identify the features of your classmate’s problem.
Solve the problem.
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124
UNIT 3: Geometry and Measurement
What Do I Need to Know?
Review any lesson with
✓
Euler’s Formula
For any polyhedron, the numbers of faces, edges, and vertices are related by this formula:
vertices
faces
edges
2
✓
Surface Area of a Triangular Prism
Surface area
sum of the areas of 3 rectangular faces
2
area of one triangular face
SA
al
bl
cl
bh
The side lengths of a triangular face are a
,
b
, and c
.
The height of a triangular face is h
.
The length of the prism is l
.
✓
Volume of a Triangular Prism
Volume
area of triangular base
length of prism
V
1
2
bhl The base and height of a triangular face are b
and h
, respectively.
The length of the prism is l
.
h
a
b
c
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LESSON
3.1
3.2
1.
You will need linking cubes and isometric dot paper.
a)
Build the object that matches the views below.
b)
Sketch the object on isometric dot paper.
2.
Sketch as many objects as possible that have two different rectangles as two of its views.
3.
This set of views represents an object.
a)
Identify the object.
b)
Draw a net for the object on 1-cm grid paper.
c)
Cut out the net.
Build the object.
d)
Describe the object.
4.
Here is a net of a triangular prism.
a)
Calculate the surface area of
the prism in square centimetres.
b)
Calculate the volume of the prism in cubic centimetres.
5.
a)
Calculate the surface area of
this prism. Sketch a net first,
if it helps.
b)
Calculate the volume of
the prism.
Unit Review
125
What Should I Be Able to Do?
For extra practice, go to page 490.
3.3
3.4
Top View
Left Side View
Front View
Right Side View
Back View
5.7 cm
4 cm
14 cm
6 cm
6 cm
6 cm
6 cm
Top View
Right Side View
Back View
6 cm
4 cm
4 cm
Front View
1.0 cm
2.4 cm
0.8 cm
2.6 cm
3.4 m
0.9 m
1.5 m
1.2 m
2.8 m
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6.
The horticultural society is
building a triangular flower bed at
the intersection of two streets.
The edges of the bed are raised
0.25 m. How much soil is needed
to fill this flower bed? Justify your answer.
7.
Find the possible values of
b
,
h
,
and
l
for a triangular prism with
volume 21 m
3
.
How many different ways can you
do this? Sketch a diagram of one
possible prism.
8.
Alijah volunteers with the horticultural society.
He wants to increase the size but not the depth of the flower bed in
question 6.
a)
How can Alijah change the dimensions so that:
• the flower bed remains
triangular, and
• the area of the ground
covered by the bed doubles?
b)
Sketch the new flower bed.
Label its dimensions.
c)
How does the change in size affect the volume of soil needed? Explain.
9.
The bucket on the front of
a lawn tractor is a triangular prism.
a)
Find the volume of soil, in
cubic metres, the bucket can hold. What assumptions do
you make?
b)
Suppose the dimensions of the triangular faces are doubled.
How much more soil do you expect the new bucket to hold? Explain.
c)
Calculate the new volume.
Sketch the new bucket and include the new dimensions.
10.
A tent has the shape of a triangular prism.
Its volume is 25 m
3
.
a)
Find possible dimensions for this prism.
b)
Choose one set of dimensions
from part a. Sketch the tent.
Label the dimensions.
c)
A larger tent has volume 100 m
3
. It is also a triangular prism.
i)
What could the dimensions
of this larger tent be? Justify your answer.
ii)
How are the dimensions of
the two tents related?
LESSON
3.4
126
UNIT 3: Geometry and Measurement
6 m
8 m
10 m
48 cm
85 cm
84 cm
80 cm
178 cm
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Practice Test
127
1.
You will need linking cubes and isometric dot paper.
a)
Build the object that has these views.
b)
Sketch the object on isometric dot paper.
c)
Draw a net for the object.
Cut out and fold the net to make the object.
2.
Calculate the surface area and volume of this prism.
3.
Look at the triangular prism in question 2.
Suppose the base and height of the triangular faces are tripled.
a)
How does this affect the volume of the prism? Explain.
b)
Sketch the larger prism.
c)
Calculate the volume of the larger prism.
4.
The triangular faces at the left are the bases of four triangular
prisms. All the prisms have the same length.
a)
Which prism has the greatest volume? Explain.
b)
Which prism has the least surface area? Explain.
5.
The volume of a triangular prism is 210 cm
3
.
a)
The length of the prism is 7 cm.
What are the possible base and height of the triangular faces?
b)
On 1-cm grid paper, draw two possible triangular faces for this prism. Measure to find the lengths of any sides you do not know.
c)
Calculate the surface area of each prism whose face you drew in part b.
Top View
Front View
Right Side View
3.50 m
1.82 m
1.75 m
1.00 m
12.0 m
A
10.4 m
15.9 m
16 m
7.8 m
11.2 m
B
16.1 m
19.7 m
15.6 m
C
8.0 m
12.0 m
12.0 m
10.4 m
D
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128
UNIT 3: Geometry and Measurement
Under the Big Top
Suppose your class is responsible for building a circus tent.
The organizers have given you the views below.
Work in a group.
Part 1
Prepare a presentation for the organizers.
Your presentation must include:
•
a 3-D sketch •
a net for the tent
•
a list of steps for building the net
•
a model of the tent
Unit Problem
Top View
Front/Back View
Side View
4.5 m 4 m
7.2 m
20 m
3 m
3 m
3 m
6 m
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Part 2
Some members of a local service club will perform in the circus.
They need a prop built for their act.
It is a triangular prism in which one performer can hide,
and which can also be used as a ramp for a bicycle jump.
Prepare an estimated cost to build this prop.
Your estimate must include:
•
a diagram of the prop with appropriate dimensions
•
an explanation of the dimensions you chose
•
calculations for the amount of materials used
•
cost of materials if the building material is $16.50/m
2
Part 3
The volume or capacity of the tent depends on its dimensions.
Research situations where volume and capacity are used in your home.
Write a report on your findings.
Unit Problem: Under the Big Top
129
Write a paragraph on what you have learned about representing three-dimensional objects and triangular prisms.
Try to include something from each lesson in the unit.
Your work should
show:
a correctly
constructed model
all diagrams,
sketches, and
calculations in detail
clear explanations of your choices,
procedures, and results
a description of
situations at home where volume and
capacity are used
Check List
✓
✓
✓
✓
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