1. science
2. mathematics
3. geometry

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CHAPTER
11
Geometry and Measurement
Relationships
GOALS
You will be able to
• develop and apply
formulas to calculate the
surface area and volume
of a cylinder
• solve problems that
involve the surface area
and volume of a cylinder
• investigate the properties
of Platonic solids
• determine how the
number of faces, edges,
and vertices of a
polyhedron are related
(Euler’s formula)
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Getting Started
You will need
• a calculator
• centimetre grid
paper
Designing a Juice Container
Hoshi and Tran are designing a container to hold 1 L of juice. They
decide to try the shape of a triangular prism.
dimensions can Hoshi and Tran use for the juice
? What
container?
A. What is the least possible volume, in cubic centimetres, for the
container? (1 cm3 1 mL)
B. About 10% of the space in the container will be empty. How many
cubic centimetres do Hoshi and Tran need for the total volume of the
container?
C. If the container is 10 cm deep, what must the area of the triangular
base be?
D. What dimensions could Hoshi and Tran use for the triangular base?
370 Chapter 11
10 cm
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Do You Remember?
5. Calculate the volume of each prism.
a)
1. What unit (centimetres, square centimetres,
cubic centimetres, litres, or millilitres)
would you use to measure each of the
following?
a)
b)
c)
d)
the quantity of fuel in a car’s gas tank
the length of trim on a blanket
the amount of paint to cover a playhouse
the space inside a storage box
2. Draw a net for each prism using centimetre
grid paper. Calculate the surface area of
each prism to the nearest square centimetre.
a)
12 cm
4.5 cm
6.2 cm
b)
3 cm
4 cm
6 cm
6. How much water would you need to fill
each prism in question 5? (1 cm3 1 mL)
7. List the number of edges, faces, and
vertices on each prism.
b)
4 cm
4 cm
8.4 cm
a)
5 cm
5 cm
4 cm
3. Sarah is covering the faces of this wooden
box with fabric for a gift. How much fabric
will she need to cover the box?
6 cm
3 cm
15 cm
4. Calculate the area and circumference of
each circle.
b)
8. Centimetre cubes need to be packed in a
box that has a volume of 36 cm3. Sketch
three possible boxes that are rectangular
prisms.
9. Think of this cement traffic barrier as a
rectangular prism with a trapezoidal prism
on top.
23.0 cm
80.0 cm
a)
25.5 cm
4.0 cm
66.0 cm
b)
4.5 m
NEL
380.0 cm
a) What is the area of the base of each
prism?
b) What is the volume of each prism?
c) What is the total volume of the cement
needed to make the traffic barrier?
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You will need
• paper
• scissors
• a calculator
11.1 Exploring Cylinders
GOAL
Explore the relationship between the dimensions of a cylinder and the
dimensions of its net.
Explore the Math
Maria and Benjamin are playing a game at their
school math fair. They are shown a lid for a
cylindrical container and four labels. To win a
prize, they have to choose the label that matches
the lid. They can ignore any overlap on the labels.
4.2 cm
13.3 cm
35.7 cm
10.1 cm
Label A
26.4 cm
10.7 cm
Label B
13.4 cm
10.7 cm
27.3 cm
Label C
can you use the size of a lid to predict the width of
? How
the label?
A. Cut out paper models of the lid and the rectangular labels.
B. Use the rectangles to construct cylinders without bases.
C. Find the cylinder that matches the lid. Make a net for this cylinder.
D. Estimate the radius, circumference, and area of the lid.
E. Which measurement of the lid—the radius, circumference, or area—
does the width of the label match?
F. Compare the widths of the labels with the measurement of the lid you
selected in step E. Which label goes with the lid?
Label D
Communication Tip
The curved side of a
cylinder has height
and width. When the
curved side is unrolled
into a rectangle, the
height of the cylinder
becomes the length
of the rectangle.
width
height
Reflecting
1. How did you know which label went with the lid?
2. How would the labels for two cylinders with the same height but
different diameters compare?
width
length
3. If you knew the height and radius of a cylinder, how would that help
you draw its net?
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NETS OF CONES
The net for a cone provides clues that tell you how flat the cone
will be when it is constructed.
Cone A
Cone B
Cone C
Net X
Net Y
Net Z
A. Draw three circles, each with a radius
of 8 cm. Cut out your circles.
B. Cut out a section of one circle, along
two radii of the circle, to create a net for
a bottomless cone. Measure the angle.
You will need
•
•
•
•
•
a compass
scissors
a ruler
a protractor
tape
C. Tape together the cut edges to make
the cone.
D. Repeat steps B and C to make two
different-sized nets and cones.
E. What relationship do you notice
between the angles of your nets and the
flatness of your cones?
F. Which net (X, Y, or Z) goes with which
cone (A, B, or C)?
NEL
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You will need
• a calculator
• centimetre grid
paper
• a ruler
• a compass
11.2 Surface Area of a
Cylinder
GOAL
Develop and apply a formula for calculating the surface area of a cylinder.
Learn about the Math
3.5 cm
Toma and Maria are wrapping tea lights to sell
for a school fundraiser. They are wrapping the
tea lights in stacks of five.
1.5 cm
much paper do Toma and Maria
? How
need to wrap each stack of tea lights?
Example 1: Estimating surface area with grid paper
Use a net to estimate the amount of wrapping paper that Toma and Maria need to wrap
each stack of five tea lights.
Toma’s Solution
I imagined unwrapping the package and laying the paper flat. I traced the circular base
and top of the cylinder on grid paper.
curved side
3.5 cm
top
3.5 cm
7.5 cm
1.5 cm
3.5 cm
base
11 cm
The height of each tea light is 1.5 cm,
so the height of the curved side is
5 1.5 cm 7.5 cm.
The width of the curved side is the same
as the circumference of the base. The
diameter of the base is 3.5 cm, so the
circumference is 3.5 cm 11 cm.
I counted the squares in the net. There
are about 10 squares in each circle.
There are about 7.5 11 83 squares in
the curved side (rectangle). Each square
has an area of 1 cm2.
Total surface area area of base area of top area of curved side
10 cm2 10 cm2 83 cm2
103 cm2
Toma and Maria need about 103 cm2 of wrapping paper for each stack of tea lights.
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Example 2: Calculating surface area using a formula
Calculate the amount of wrapping paper that Toma and Maria need to wrap each stack of
five tea lights.
Maria’s Solution
The base and top of each stack are congruent faces, so they have the same area. I just
need to determine the area of one face, and then I can double it.
The diameter is 3.5 cm, so the radius is 3.5 cm 2 1.8 cm.
base
Area of top and base 2 r 2
2 (1.8 cm)2
top
3.5 cm
The area of the top and base is 20.4 cm2. I rounded to one
decimal place because this is how the radius is given in the
problem.
If I laid out the curved side of the package, it would form a rectangle. Its width would be equal
to the circumference of the base. Its length would be equal to the height of the package. So,
its area is equal to the circumference of the base multiplied by the height of the package.
3.5 cm
Area of curved surface
circumference of base height
d height
( 3.5 cm) 7.5 cm
11.0 cm
7.5 cm
7.5 cm
The area of the curved surface is 82.5 cm2.
Total surface area area of top and base area of curved surface
20.4 cm2 82.5 cm2
102.9 cm2
Toma and Maria need 102.9 cm2 of wrapping paper for each stack of tea lights.
Reflecting
1. What part of the net for a cylinder is affected by the height of the
cylinder?
2. What part of the net for a cylinder is affected by the size of the base?
3. Write a formula to calculate the surface area of a cylinder.
NEL
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Work with the Math
Example 3: Calculating surface area using a formula
2.0 cm
Calculate the surface area of this cylinder.
3.0 cm
Solution
Surface area of cylinder area of base and top area of curved surface
Sketch the faces, and calculate the area of each face.
2.0 cm
Area of base and top 2 area of base
2 r 2
2 3.14 (2.0 cm)2
25.1 cm2
3.0 cm
12.6 cm
Width of rectangle circumference of circle
2r
2 3.14 2.0 cm
12.6 cm
Area of curved surface area of rectangle
length width
3.0 cm 12.6 cm
37.8 cm2
Surface area of cylinder area of base and top area of curved surface
25.1 cm2 37.8 cm2
62.9 cm2
The surface area is 62.9 cm2.
A
Checking
B
4. Calculate the surface area of each cylinder.
a)
5 cm
b)
3.6 cm
Practising
5. Use each net to determine the surface area
of the cylinder.
6 cm
a)
b)
4.5 cm
8.0 cm
4 cm
5 cm
14.5 cm
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6. Three cylinders have bases that are the
same size. The area of the base is 10.0 cm2.
Determine the surface area of each cylinder,
given its height.
a) 8.0 cm
b) 6.5 cm
c) 9.4 cm
7. Calculate the surface area of each cylinder.
2.1 m
a)
c)
2.5 cm
8.3 m
10.3 cm
b)
23.0 m
10. Explain why two cylinders that are the same
height can have different surface areas.
11. “Anik” is the name of a series of Canadian
communications satellites. The first Anik,
shown here, was launched in 1972. It was a
cylindrical shape, 3.5 m high and 190 cm
in diameter. All satellites are wrapped with
insulation because the instruments inside
will not work if they become too cold or
hot. What was the approximate area of the
insulation used to wrap the first Anik?
2300 cm
8. Describe how you would determine the
surface area of a potato-chip container that
is shaped like a cylinder.
9. a) This railway car is 3.2 m in diameter
and 17.2 m long. Calculate its surface
area.
12. A games shop sells marbles
in clear plastic cylinders.
Four marbles fit across the
diameter of the cylinder, and
10 marbles fit from the base
to the top of the cylinder.
Each marble has a diameter
of 2 cm. What is the area of
the plastic that is needed to
make one cylinder?
C
b) Estimate the cost of painting the outside
of the railway car, if a can of paint
covers an area of 40 m2 and costs $35.
NEL
Extending
13. Gurjit has a CD case that is a cylindrical
shape. It has a surface area of 603 cm2 and
a height of 10 cm. What is the area of the
circular lid of the CD case?
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You will need
11.3 Volume of a Cylinder
GOAL
Develop and apply a formula for calculating the volume of a cylinder.
• centimetre grid
paper
• a compass
• centimetre linking
cubes
• a calculator
Learn about the Math
2.0 cm
4.0 cm
8.0 cm
6.0 cm
4.0 cm
2.0 cm
Benjamin says, “The jar with the greatest volume holds the most
jellybeans. One jellybean has a volume of about 1 cm3. If I can calculate
the volume of each jar, I can estimate the number of jellybeans it holds.
I’ll use models of the jars to estimate their volumes.
I know that the bottom of each jar is a circle. I’ll estimate the number of
centimetre cubes that will cover the bottom. I’ll stack centimetre cubes to
determine the number of layers.”
? Which jar holds the most jellybeans?
A. On centimetre grid paper, draw a circle with a radius of 6.0 cm to
model the base of the first jar. Estimate the area of the base.
6.0 cm
B. Stack centimetre cubes to model the height of the jar. How many
layers of cubes will fit inside the jar?
C. Estimate the volume of the first jar.
D. Repeat steps A to C for the other two jars.
E. Which jar holds the most jellybeans? Explain your answer.
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Reflecting
1. Why do both the radius of a cylinder and its height matter when you
are estimating its volume?
2. How does Benjamin’s strategy show how to use the area of the base
of a cylinder to determine the volume of the cylinder?
3. Write a formula for calculating the volume of a cylinder.
Work with the Math
Example 1: Calculating volume using radius
Calculate the volume of this cylinder.
5.0 cm
6.0 cm
Hoshi’s Solution
Volume of cylinder area of base height
r 2 height
3.14 (5.0 cm)2 6.0 cm
471.0 cm3
The volume of a cylinder is calculated like
the volume of a prism: area of base height.
I calculated the volume using the formula
and 3.14 for .
The volume of the cylinder is 471.0 cm3.
Example 2: Estimating volume using diameter
Estimate which cylinder has the greater volume.
10 cm
14 cm
14 cm
A
B
10 cm
Chad’s Solution
Cylinder A
The diameter of cylinder A is 10.0 cm, so
the radius is 10.0 2 5.0 cm.
For an estimate, I can use rounded
measurements for easier mental math.
Volume of cylinder A area of base height
r 2 height
(5 cm)2 14 cm
25 cm2 16 cm
400 cm3
Cylinder B has the greater volume.
NEL
Cylinder B
The diameter of cylinder B is 14.0 cm, so the
radius is 7.0 cm.
Volume of cylinder B area of base height
r 2 height
(7 cm)2 10 cm
50 10 cm
500 cm3
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Checking
4. Estimate the volume of each cylinder.
5m
a)
7. a) Determine the
volume of
Mandy’s mug.
10.0 cm
b) About how many
millilitres of
liquid will it
hold?
4m
20.0 cm
3.2 cm
b)
8. Calculate the volume of each cylinder.
5 cm
a)
10.5 cm
8 cm
5. Calculate the volume of each cylinder.
14.4 cm
a)
b)
12.6 cm
b)
3.1 m
4.1 m
7.3 cm
9. Cosmo’s family has a pool like this.
2.5 cm
5.4 m
120 cm
B
Practising
6. Estimate the volume of each cylinder.
a)
4 cm
8 cm
b)
10.2 cm
5.8 cm
380 Chapter 11
a) What is the volume of the pool?
b) How many litres of water will the pool
hold?
c) How long will it take to fill the pool at
a rate of 50 L/min?
10. Tennis balls are sold in cylinders.
Each cylinder has a height of
about 22 cm and a diameter of
about 7 cm. Estimate the volume
of the cylinder.
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8 cm
11. A cylindrical candle
is sold in a gift box
that is a squarebased prism. What
is the volume of
empty space in
16 cm
the box?
16. Which container holds more, the cylinder or
the triangular prism? Justify your answer.
2.1 cm
6.0 cm
3.5 cm
10.0 cm
12. Copy and complete this chart for cylinders.
4.0 cm
Radius
of base
Diameter
of base
Height
12.0 m
11 cm
5.0 m
4 cm
Volume
307.7 cm3
3.5 cm
2.0 m
17. Which holds more, a cylinder with a height
of 10.0 cm and a diameter of 7.0 cm or a
cylinder with a height of 7.0 cm and a
diameter of 10.0 cm? Explain your answer.
226.1 m3
C
13. The area of the base of a cylinder is
50.2 cm2. The volume of the cylinder is
502.4 cm3. Determine the height of the
cylinder.
14. A lipstick tube has a volume of 25.1 cm3
and a diameter of 2.0 cm. What is the height
of the tube?
15. The volume of each cylinder below is
0.3040 m3. Solve for the unknown measure.
4.0 cm
a)
■ cm
■ cm
b)
8.8 cm
NEL
Extending
18. These two metal cans both hold the same
quantity of soup.
10.0 cm
7.5 cm
8.0 cm
a) What is the height of the can of
mushroom soup? Show your solution.
b) Which can uses more metal? Explain.
19. Suppose that the radius of a cylinder is the
same as its height. What would happen to
the volume of the cylinder if its radius were
doubled and the height stayed the same?
20. These two containers
each hold 1 L of
liquid. What might
their dimensions be?
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You will need
• a calculator
11.4 Solve Problems Using
Diagrams
GOAL
Use diagrams to solve measurement problems.
Learn about the Math
For a babysitting course, Toma is designing a toy that will be filled
with water.
? How many millilitres of water will Toma’s toy hold?
1 Understand the Problem
To determine the number of millilitres of water the toy will hold, Toma
needs to determine its volume in cubic centimetres. (1 mL 1 cm3)
3.0 cm
6.0 cm
2 Make a Plan
Toma will use diagrams and a formula to determine the volume of each of
the three figures that make up the toy. Then she will add the three volumes.
3.5 cm
3 Carry Out the Plan
Figure
Area of base of figure
3.0 cm
top
6.0 cm
4.0 cm
middle
5.0 cm
bottom
3.5 cm
4.0 cm
4.0 cm
6.0 cm
4.0 cm
5.0 cm
6.0 cm
Volume of figure
Area of base of top figure
r 2
(1.5 cm)2
7.1 cm2
Volume of top figure
area of base height
7.1 cm2 6.0 cm
42.6 cm3
Area of base of middle figure
r 2
(2.0 cm)2
12.6 cm2
Volume of middle figure
area of base height
12.6 cm2 5.0 cm
63.0 cm3
The area of the base of the bottom
figure is the area of six triangles.
Area of one triangle
1
base height
2
1
(4.0 cm) 3.5 cm
2
7.0 cm2
Volume of bottom figure
area of base height
42.0 cm2 6.0 cm
252.0 cm3
Area of six triangles
6 7.0 cm2
42.0 cm2
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Total volume of toy 42.6 cm3 63.0 cm3 252.0 cm3
357.6 cm3
The volume of Toma’s toy is about 358 cm3. Since 1 cm3 1 mL,
the toy will hold about 358 mL of water.
4 Look Back
Toma estimates that the top two cylinders are smaller than a 4 cm by
4 cm by 11 cm square prism that has a volume of 176 cm3. She thinks
the bottom is smaller than a 6 cm cube that has a volume of 216 cm3.
Her toy volume calculation is less than 176 cm3 216 cm3 392 cm3,
so her answer is reasonable.
Reflecting
1. How did the strategy of using diagrams help Toma solve this problem?
2. How did using diagrams to compare the sizes of the figures help
Toma check her answer?
Work with the Math
Example: Using a tree diagram to determine probability
Maria and Tran are playing a game with two standard dice. To win the game, they need to
roll a sum of 8. What is the probability of rolling a sum of 8?
Tran’s Solution
1
Understand the Problem
I have to determine the number of different ways that I can roll a sum of 8 and compare
this number with the number of possible outcomes.
2
Make a Plan
I’ll draw a tree diagram to list all the possible outcomes. Then I’ll count the number of
outcomes that give a sum of 8.
3
Carry Out the Plan
Roll 1
1
2
3
4
Roll 2
Sum
123456
234567
123456
345678
123456
456789
12345 6
5 6 7 8 9 10
5
1234 5 6
6 7 8 9 10 11
6
123 4 5 6
7 8 9 10 11 12
There are 36 possible outcomes. The probability of rolling a sum of 8 is 5.
36
4
Look Back
I see a pattern in the sums. In each group, they increase by 1. So, I’m sure that my
diagram shows all the possible combinations.
NEL
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6. Erik rolls two standard dice. Determine the
probability that the sum will be 6, 7, or 8.
Checking
3. Sketch the prisms that make up this cabin.
Then calculate the volume and surface area
of the whole cabin.
0.5 m
7. Fritz is making a stained-glass window. It
consists of a rectangle that is 0.5 m wide
by 2.5 m long, with a semicircle above the
rectangle.
a) Sketch an outline of the window. Label
the dimensions.
b) How much glass does Fritz need?
2.2 m
2.0 m
8. To copy a poster, Sohel reduces each
dimension by 70%. The width of the
original poster is 0.4 m. What is the width
of the reduced copy?
9. How many squares are in figure 100?
1.8 m
B
Practising
4. Sketch the prisms that make up this
skateboard ramp. Then calculate the
volume and surface area of the whole
skateboard ramp.
Figure 1
1.6 m
Figure 3
10. How many different ways can 360 players
in a marching band be arranged in a
rectangle?
1.6 m
4.2 m
2.1 m
1.2 m
5. Perdita has a red shirt, a black shirt, a
yellow shirt, and a white shirt. She also has
a pair of white shorts, a pair of red shorts,
and a pair of blue shorts.
a) Determine the number of different
outfits that Perdita could wear.
b) What is the probability that she will wear
at least one piece of clothing that is red?
384 Chapter 11
Figure 2
11. A city park is a square with 600 m sides.
Diane started walking from a point 150 m
south of the northwest corner, straight to a
point 150 m north of the southwest corner.
How far did she walk?
12. James is estimating the amount of paint he
needs for the walls of his 3.4 m by 2.6 m
bedroom. His bedroom is 2.7 m high.
One litre of paint covers about 10 m2.
About how much paint does James need?
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CALCULATING SURFACE AREA
OF CUBE STRUCTURES
Use nine linking cubes to build this shape.
You will need
• linking cubes
Each linking cube has a surface area of 6 square units. So, the total surface area of nine
unattached cubes is 9 6 54 square units. In your shape, however, some of the cubes are
attached to each other. Therefore, the surface area of your shape is less than 54 square units.
1. How many pairs of faces are attached to each other?
2. How can you use your answer to question 1 to calculate the surface area of your shape?
3. Build each shape below, and calculate its surface area.
NEL
a)
c)
b)
d)
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Mid-Chapter Review
Frequently Asked Questions
Q: How do you calculate the surface area of a cylinder?
A: Sketch a net of the cylinder. The surface area is the sum of the
areas of the faces. Like the base and top of a rectangular prism,
the base and top of a cylinder are congruent. Therefore, they have
the same area. Calculate the area of the base, and double it. Add
this to the area of the curved surface.
top
4.0 cm
4.0 cm
12.0 cm
curved surface
12.0 cm
d
base
Surface area 2 area of base area of curved surface
2 r 2 (d h)
2 (4.0 cm)2 ( 8.0 cm 12.0 cm)
402.1 cm2
Q: How do you calculate the volume of a cylinder?
A: The base of a cylinder is a circle. Calculate the volume of a
cylinder in the same way you would calculate the volume of a
prism—multiply the area of the base by the height.
Volume area of base height
r 2 h
(4.0 cm)2 12.0 cm
603.2 cm3
386 Chapter 11
4.0 cm
12.0 cm
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Practice Questions
(11.2)
1. Sketch a net for each cylinder, and label its
dimensions. Then calculate the surface area.
a)
b)
c)
d)
(11.2)
(11.2)
(11.2)
Item
Radius of
base (cm)
Height of
cylinder (cm)
potato-chip
container
coffee can
CD case
oil barrel
4
7.5
8.5
25.0
8
15.0
20.5
80.0
2. Karim is painting a design on a cylindrical
barrel. The height of the barrel is 1.2 m.
The radius of its base is 0.3 m. What area
will the paint have to cover? (Remember to
include the bottom and lid of the barrel.)
3. Write step-by-step instructions for
determining the surface area of an empty
paper-towel roll.
4. Determine the surface area of each tin.
a)
170 mm
b)
12.0 cm
(11.3)
8. Determine the volume of this figure.
(11.3)
Explain what you did.
2.5 m
8.0 m
6.5 m
4.0 m
12.0 m
9. A soup can has a radius of 4 cm and a height
of 11 cm. There are 24 cans in one case. How
many litres of soup are in one case? (11.3)
12.0 cm
60 mm
(11.2)
7. Deirdre is buying birdseed for the class
bird feeder. The bird feeder is a cylinder
with a diameter of 25 cm and a height of
45 cm. How many millilitres of seed
(11.3)
should she buy?
5. A cylindrical candle has a radius of 6 cm
and a height of 20 cm. How much waxed
paper will Jake need to wrap the candle?
10. The height of each cylinder in a set of
food-storage containers is 30 cm. The
radius of the largest container is 10 cm.
The volume of the smallest container is 1
3
the volume of the largest container. The
volume of the middle-sized container is 2
3
the volume of the largest container. What
(11.4)
is the volume of each container?
10 cm
6. Determine the volume of a cylinder that is
20 cm high and has the radius or diameter
below.
a) radius 13 cm
b) radius 6.5 cm
c) diameter 20 cm
NEL
30 cm
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You will need
11.5 Exploring the Platonic
Solids
GOAL
Investigate properties of the Platonic solids.
• congruent
equilateral triangles
• congruent squares
• congruent regular
pentagons
• congruent regular
hexagons
• tape
• a protractor
Explore the Math
A Platonic solid is a polyhedron with faces that are all congruent
regular polygons. The same number of faces meet at all the vertices.
There are only five Platonic solids.
tetrahedron
cube
octahedron
dodecahedron
polyhedron
a 3-D shape that has
polygons as its faces
icosahedron
? Why are there only five Platonic solids?
A. Look at the five Platonic solids. What shapes are the faces? How
many faces meet at each vertex?
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B. A Platonic solid can have faces that are equilateral triangles. What is
the measure of the interior angles of an equilateral triangle?
C. Using only equilateral triangles, draw the nets for as many polyhedrons
as you can. Make sure that the same number of triangles meet at each
vertex. Cut out and fold your nets to make sure that they work.
D. For each polyhedron you made, how many faces meet at each vertex?
What is the sum of the angles at each vertex?
E. What is the least number of equilateral triangles you can join at a
vertex and still fold the net to make a Platonic solid? What is the
greatest number?
F. Repeat steps B to E using only squares.
G. Repeat steps B to E using only regular pentagons.
H. Copy and complete the following chart.
Face information
Platonic solid
Polygon
tetrahedron
cube
octahedron
dodecahedron
icosahedron
equilateral triangle
square
equilateral triangle
regular pentagon
equilateral triangle
Measure of interior
angles on one face
Number of faces
at each vertex
Sum of angles
at each vertex
Explain why you cannot use any other regular polygons as the faces
of a Platonic solid. Explain why you cannot use any more of the
regular polygons you have already used.
I.
Reflecting
1. Why does a Platonic solid look the same no matter which vertex you
position at the top?
2. Refer to your chart. Look at the measures of the interior angles in
each Platonic solid and the number of faces. How does the measure
of the interior angles determine the number of faces that can join at
a vertex?
3. A Platonic solid cannot be made from regular polygons that have
more than five sides. How can you use the measures of the interior
angles of regular polygons to show this is true?
NEL
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You will need
11.6 Polyhedron Faces, Edges,
• pipe cleaners or
straws
• modelling clay
and Vertices
GOAL
Determine how the number of faces, edges, and vertices of a polyhedron are related.
Learn about the Math
Toma and Benjamin noticed that whenever you make a 2-D polygon, the
number of vertices is the same as the number of edges. They wondered
whether the number of faces, vertices, and edges of 3-D polyhedrons are
related.
pattern links the number of faces, edges, and
? What
vertices of a polyhedron?
Benjamin tried building an unusual polyhedron first. “I’ll start building it
from the top. The number of vertices and number of edges are the same,
and there is one face.”
Part built
Number of faces
Number of vertices
top
1
4
390 Chapter 11
Number of edges
4
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“Next, I’ll add squares. For each square, I add one new face, two new
vertices, and three new edges. The total number of new faces and new
vertices is equal to the number of new edges.
“Now I’ll add triangles. For each triangle, I add one new face and one
new edge, but no new vertices. Again, the total number of new faces and
vertices is the same as the number of new edges.”
Part built
top
4 new squares
4 new triangles
Number of
faces
Number of
vertices
Number of
edges
1
4 more
4 more
4
8 more
0 more
4
12 more
4 more
A. Construct the parts that Benjamin constructed. Add the next set of
squares. Explain why the number of edges is 1 less than the total
number of faces and vertices.
B. Add the bottom of the polyhedron. Explain why you have added no
new edges or vertices, but one new face.
C. Why is the number of edges 2 less than the total number of faces and
vertices?
D. Choose one of the following shapes. Compare the number of edges
with the total number of faces and vertices. What do you observe?
E. Compare your results with the results of students who chose different
shapes. What do you notice?
Reflecting
1. The relationship you described in step C is called Euler’s formula
(pronouced “oiler”). Explain why it can be written as F V E 2,
where F is the number of faces, V is the number of vertices, and E is
the number of edges of the shape.
2. How does Euler’s formula allow you to predict the number of edges,
faces, or vertices of a shape if you know two of these values?
NEL
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Work with the Math
Example 1: Checking whether a polyhedron is possible
Is it possible to make a polyhedron with 6 faces, 7 vertices, and 10 edges?
Tran’s Solution
FVE2
F V E 6 7 10
3
I used Euler’s formula. If it is possible to make a polyhedron
like this, the result should be 2 when I substitute the values into
Euler’s formula.
I substituted the values into the formula. The result is 3, not 2,
so it is not possible to make such a polyhedron.
Example 2: Using Euler’s formula to determine a missing value
If a polyhedron has 10 faces and 18 edges, how many vertices should it have?
Benjamin’s Solution
FVE2
10 V 18 2
V82
V8828
V 10
A
I used Euler’s formula.
I substituted 10 for the number of faces and 18 for the number
of edges.
I used balancing to solve the equation.
The polyhedron should have 10 vertices.
Checking
3. A student used 10 pipe cleaners to make the
edges of a polyhedron. If the polyhedron has
6 vertices, how many faces must it have?
4. Show that Euler’s formula works for a
tetrahedron.
392 Chapter 11
B
Practising
5. Show that Euler’s formula works for the
other four Platonic solids: a cube, an
octahedron, a dodecahedron, and an
icosahedron.
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6. Copy and complete the chart for some
polyhedrons.
Number of
faces
Number of
edges
Number of
vertices
9
12
5
6
6
20
16
7
10. Make another cube using modelling clay.
Then make a pyramid on each face of the
cube. Show that Euler’s formula works for
this polyhedron.
11. Imagine that you drilled a rectangular hole
through a cube. Does Euler’s formula work
for the new shape?
30
12
10
6
12. a) Construct a triangular prism.
b)
c)
d)
e)
7. The following crystals and gemstones have
been cut to form polyhedrons. Show that
Euler’s formula works for each polyhedron.
a)
How many faces does the prism have?
How many edges does the prism have?
How many vertices does the prism have?
Show that Euler’s formula works for
the prism.
13. Repeat question 12 using a pentagonal
pyramid.
C
b)
Extending
14. Make a cube using modelling clay. Mark a
point in the centre of each face. Imagine that
you joined these points with string inside the
cube to form a polyhedron. Show that
Euler’s formula works for this polyhedron.
15. A prism has a base with n sides.
8. Show that Euler’s
formula works for
this cuboctahedron.
a)
b)
c)
d)
How many faces does the prism have?
How many edges does the prism have?
How many vertices does the prism have?
Show that Euler’s formula works for
the prism.
16. A pyramid has a base with n sides.
9. Make a cube using
modelling clay. Cut
the corners off the
cube. Show that
Euler’s formula
works for the new
shape.
NEL
a) How many faces does the pyramid
have?
b) How many edges does the pyramid
have?
c) How many vertices does the pyramid
have?
d) Show that Euler’s formula works for
the pyramid.
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THE VOLUMIZER GAME!
In this game, you will calculate the volume of a cylinder using
a radius given on a card and a height obtained by rolling a die
or spinning a spinner.
You will need
• index cards
• a die or spinner
• a calculator
Number of players: 2 or more
Rules
1. To create a deck of Volumizer cards, use about 20 blank
index cards. On each card, sketch a cylinder and
6
label the radius of the base in centimetres.
2. To play the game, each player
5
Volumizer
1
2
• selects a Volumizer card from the pile
4
3
• rolls the die or spins the spinner
• calculates the volume of the cylinder, using the number on
the die or spinner as the height of the cylinder in centimetres
All players take their turns at the same time. Players may
check each other’s calculations.
r 12.0 cm
3. The player who has the cylinder with the greatest volume keeps the card. The other
players return their cards to the deck.
4. The game is finished when the number of cards left in the pile is less than the number
of players.
5. The player with the most cards at the end of the game is the winner.
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Chapter Self-Test
1. Draw a net for the paper needed to wrap
each candle. Label the dimensions.
4. This railway car is 320 cm in diameter and
17.2 m long. Calculate its volume.
4 cm
4 cm
14 cm
11 cm
2. Which parts of this
net of a cylinder
are equal to the
circumference of
the base?
c
a
b
b
a
d
5. Suppose that you increase the height of this
cylinder by 10 cm. By how much does the
volume increase?
5 cm
3. Calculate the surface area and volume of
each cylinder.
a)
5.1 cm
c)
6 cm
7 cm
2.6 cm
9 cm
b)
4.1 cm
d)
5.0 cm
7.0 cm
16.0 cm
NEL
6. No more than three congruent squares can
meet at each vertex of a Platonic solid.
Explain why.
7. A soccer ball is a polyhedron made from
20 regular hexagons
and 12 regular
pentagons. It has
60 vertices.
Determine the
number of edges
on a soccer ball.
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Chapter Review
Frequently Asked Questions
Q: What is a Platonic solid?
A: A Platonic solid is a polyhedron with faces that are all congruent
regular polygons. The same number of faces meet at all the vertices
in a Platonic solid. The five Platonic solids are shown below.
tetrahedron
cube
octahedron
dodecahedron
icosahedron
Q: Why are there only five Platonic solids?
A: The total of the interior angles that meet at each vertex of a
Platonic solid must be less than 360°. At least three faces must
meet at each vertex.
Regular polygons with more than five sides have angles that
measure at least 120°. When you build a polyhedron, at least
three faces have to meet at each vertex to make the polyhedron
3-D. If you tried to use the faces of polygons with more than five
sides as the faces of a polyhedron, the sum of the angles that
meet at each vertex would be at least 360°. This is not possible.
Q: How are the number of edges, vertices, and faces of a polyhedron
related?
A: The relationship among the number of edges, vertices, and faces
can be represented using the equation F V E 2, where
F is the number of faces, E is the number of edges, and V is the
number of vertices. This equation is known as Euler’s formula.
For example, the following cube has 6 faces, 8 vertices, and
12 edges.
FVE2
6 8 12 2
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Practice Questions
(11.2)
1. Calculate the surface area of this cylinder.
3.3 m
6.2 m
(11.3)
2. Calculate the volume of the cylinder in
question 1.
3. Mohammed is choosing a bass drum to buy
for his band. The “Bashmaster” is 71.1 cm
in diameter and 35.6 cm high. The “Crash”
is 91.5 cm in diameter and 66.0 cm high.
The “Boomalot” is 81.3 cm in diameter
and 45.7 cm high.
4. A glass in the shape of a cylinder is 10.0 cm
high and has a diameter of 3.5 cm. How
many millilitres of juice will the glass hold
(11.3)
if it is filled to the top?
5. What might be the dimensions of a
cylindrical container that holds 750 mL
(11.3)
of juice?
6. Sketch a shape made up of a cylinder and a
triangular prism that has a total volume
(11.4)
between 100 cm3 and 200 cm3.
7. Why is it impossible to have a Platonic
solid in which six or more equilateral
(11.5)
triangles meet at each vertex?
8. Show that Euler’s formula works for a
(11.6)
pentagonal prism.
a) Which drum has the greatest surface
area? Justify your answer.
b) Which drum has the greatest volume in
cubic centimetres? Justify your answer.
(11.2)
(11.3)
9. A polyhedron has 9 edges and 6 vertices.
a) Calculate the number of faces.
b) Sketch the polyhedron.
(11.6)
10. A polyhedron has 6 faces and 6 vertices.
(11.6)
Calculate the number of edges.
11. A polyhedron has 8 faces and 12 edges.
(11.6)
Calculate the number of vertices.
NEL
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Chapter Task
Storage Capacity of a Silo
In this task, you will design a silo that can be
used to store corn for animal feed. The outside
of the silo will be painted to make it rust
resistant and more attractive.
Keep in mind:
• The paint comes in 3.8 L cans. Each can
covers an area of 40 m2 and costs $35,
including taxes.
• In 2003, corn for animal feed was sold for
about $120 per tonne.
can you design a silo and
? How
report on the costs?
A. Sketch the silo you recommend. Show its
diameter and height.
B. Calculate the surface area of your silo.
C. Calculate the cost to cover your silo with
one coat of paint.
D. What is the volume of the corn that can be stored in your silo?
E. What mass of corn can be stored in your silo? Use the height of the
corn in your silo and the following table to estimate the mass of the
corn. (1 t 1000 kg)
Height of corn (m)
Mass (kg) of 1 m3 of corn
9
12
15
18
21
24
570
610
660
700
740
770
F. Estimate the value of the corn that can be stored in your silo.
G. Prepare a written report that shows your calculations and
explains your thinking.
398 Chapter 11
Task Checklist
✓
❏
Did you show all your
steps?
✓
❏
Did you explain your
thinking?
✓
❏
✓
❏
Did you draw and label
your diagram neatly and
accurately?
Did you use appropriate
math vocabulary?
NEL