UNIT 2
Measurement
Topics Covered in this Unit Include: Pythagorean Theorem, Perimeter, Area and Volume and
Optimization Problems
Evaluations Given this Unit (Record Your Marks Here)
Mastery Test – Measurement
Assignment – Down by the Bay
Unit Test
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25
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Area and Perimeters of Circles
1. Find Area & Perimeter first then cost to paint
at a rate of $0.23/cm²
2. Find Area & Perimeter first then the cost to
fence at a rate of $1269/km
8.5 km
4.5 cm
Perimeter
Area
Perimeter
3. Find Area & Perimeter first then cost to paint
at a rate of $11.68/m²
Area
4. Find Area & Perimeter first then the
cost to fence at a rate of $1310/km
6 km
5m
Perimeter Problem
Area Problem
Perimeter Problem
Area Problem
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Additional Work
a) A circular patio of radius 8 m is to be created. Find the total cost of the patio if the
patio stones cost $18.99 per m² and fencing costs $16.59 per m.
b)Sand will be laid down below and a fence put around a circular pool of diameter 10 m.
Find the total cost sand and fencing if the sand cost $3.99 per m² and fencing costs
$17.99 per m.
c) A window in the shape of a half-circle must be replaced.
The glass costs $0.16 per cm² and the trim costs $1.12 per
cm. Find the total cost of replacing the window.
d) The backyard of a home is in the shape of a half-circle.
Grass costs $1.39 per m² and a fence costs $7.12 per m.
Find the total cost of preparing the yard.
26 cm
13 m
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More Areas and Perimeters
Determine the area and perimeter of each of the following shapes.
1.
8.0 cm
2.
6 cm
5 cm
6.5 cm
10 cm
Area =
3.
Perimeter =
6.3 m
Area =
Perimeter =
4.
10 m
6m
4.8 m
7m
14 m
Area =
Perimeter =
Area =
Perimeter =
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Calculating the Area and Perimeters of Composite Shapes
ex. 1
15cm
6cm
9cm
7cm
ex. 2
5cm
5cm
12cm
9cm
8cm
ex. 3 (the shaded area)
2.5 cm
8cm
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Areas and Perimeters of Composite Shapes Worksheet
Find the area and perimeter of each of the following shapes.
1.
2.
5.8m
7cm
8.7 m
7cm
10cm
6cm
4.5m
8m
3.
.
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4.
5.
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EQAO Sample Question – Composite Areas
Dominique is helping her uncle install paving stones on the pathway in front of her house.
This is the plan for the pathway.
Determine the total area of the pathway. Show your work.
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On Frozen Pond
The town planners have hired you to design a rectangular ice rink for the local park.
They will provide 122 metres of fencing. Your design should enclose the greatest
possible area for the skaters.
Explore
It is possible to build a long, narrow ice rink, as shown.
56 m
5m
Area = length  width
Area = 5  56
Area = 280 m2
Area = 280 m2
56 m
On a separate page, sketch three more ice rinks that have a larger area than this ice rink.
Label the dimensions on the sketch and calculate the area.
Hypothesize
Based on your exploration, predict the length and the width of the largest rectangular ice
rink.
Model
Complete the table with all possible combinations of width and length for the ice rinks.
Perimeter (m)
Width (m)
Length (m)
122
122
122
122
122
122
122
122
122
122
122
122
122
0
5
10
61
56
Area (m2)
l×w
0
280
Describe what happens to the area when the width of the ice rink increases.
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Construct a scatter plot of Area vs. Width.
Manipulate
Circle the region on the scatter plot where the area of the rink is the largest.
On a separate page, construct two more sketches of rinks with lengths in this region.
Add these points to the scatter plot.
Conclude
Write a report to the town advising them of the dimensions that would be best for the new ice rink. Justify your
recommendation. Include a sketch and the area of the ice rink that you are recommending.
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Measurement Performance Task
Down by the Bay
The city planners would also like you to design a swimming area
at a local beach. There is 100 m of rope available to enclose
the swimming area. The shore will be one side of the swimming
area; so only three sides of the rectangle will be roped off. It
is your job to design the largest rectangular swimming area.
Explore
It is possible to build a long, narrow swimming area.
90 m
5m
5m
Area = length  width
Area = 5  90
Area = 450 m2
Sketch three more swimming areas that have a larger area than this swimming area and use
the same 100m of rope. Label the dimensions on the sketch and calculate the area as shown
above.
1.
2.
3.
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Hypothesize
Based on your exploration, predict the dimensions of the largest rectangular swimming area.
Model
Complete the table with possible combinations of width and length for the swimming pools.
Perimeter (m)
Width (m)
100
0
Length (m)
Area (m2)
l × w
5
Describe what happens to the area when the width of the swimming area increases.
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Construct a scatter plot of Area vs. Width.
Area
Width
Manipulate
Look at the scatter plot. Circle the region on the scatter plot where the area of the
swimming area is the largest.
Construct two more sketches of swimming areas with widths and areas in this region.
Add these points to the scatter plot.
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Conclude
Write a report to the town advising them of the dimensions that would be best for the new
swimming area.
Justify your choice.
Include a sketch and the area of the swimming
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Optimizing Perimeter and Area Worksheet
1. A farmer’s field is to be fenced in with 800m of fencing. What is the maximum area that
can be enclosed using this fencing? Draw a sketch of the field and label it with the
dimensions.
2. A local community centre is building a skateboard park adjacent to one of the walls of
the centre. The have 60m of fencing to enclose the area. They only need to fence in 3
sides since it is next to the wall. What is the maximum area of the skateboard park.
Sketch the skateboard park and label it with the dimensions that you chose.
3. A rectangular shaped skating surface has an area of 324m2. The city wants to put a
fence around it. What are the dimensions of the skating area that will use the least
amount of fencing? What is the total amount of fencing required? (hint: work backwards
to find dimensions that will give the area specified)
4. What will the dimensions be of the skating surface in question 4 if they only need to
fence in 3 sides of the skating surface? (hint: this one is tricky … use a table like you did
on the performance task)
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Pythagorean Theorem Worksheet #1
Finding the Long Side of a Right Angled Triangle (S2 + M2 = L2)
4 cm
x
S =
M=
L=
6 cm
5.5 cm
y
S =
M=
L=
5 cm
x
S =
M=
L=
12 cm
3 cm
Finding the Short Side of a Right Angled Triangle (S 2 + M2 = L2)
5m
x
7m
S =
M=
L=
x
11 m
S =
M=
L=
8m
17 m
y
14 m
S =
M=
L=
Finding the Middle Side of a Right Angled Triangle (S2 + M2 = L2)
3m
x
6m
S =
M=
L=
S =
M=
L=
18 m
8m
4m
9m
x
S =
M=
L=
y
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Pythagorean Theorem Applications
1. How wide is the ramp in the following picture?
10 m
1.2 m
Width
=?
2. TV’s are measured based on their “diagonal” screen distance. What size is this TV? (draw
the diagonal first)
20
inches
36
inches
3. How high does the ladder in the following diagram reach up the wall?
6m
h
2m
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Pythagorean Theorem Worksheet #2
Round your answers to 1 decimal place.
1. Find the value of each variable.
a) x 2  122  52
b)
x 2  7 2  82
d) 92  x 2  42
e) 112  82  x 2
c) x 2  10 2  92
f) 122  x 2  102
2. Find the missing measure in each triangle (assume each triangle is a right triangle)
a)
b)
x
c)
14
x
6
5
6
x
8
9
d)
15
x
11
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3. Devon was asked to use Pythagorean Theorem to find the missing side in the following
triangle. Explain (in detail) what the problem is with this question?
6
8
x
4. Find the length of the ramp in the following diagram.
1.8m
16.4m
5. Three hills in a ski resort are shown in the diagram below. Calculate the height of each.
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Sketch a diagram to help you answer questions 6-8.
6. A ladder leans against a wall. The foot of the ladder is 1.9m from the base of the wall
and the ladder reaches 4.5m up the wall. How long is the ladder?
7. A ladder that is 7.0m long is placed against a wall so that the base of the ladder is
2.1m from the base of the wall. How far up the wall does the ladder reach?
8. As a short cut to school, Joelle cuts across the diagonal of a rectangular field. If the
field is 160.0m long and 120.0m wide, how much walking does Joelle save by cutting across
the field instead of walking around it?
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Volume Worksheet
9 cm
10 cm
Open
Topped
8 cm
15 cm
12 cm
18 cm
20 cm
V=
11cm
V=
12 m
6 m
10 m
10 m
V=
r = 10 cm
V=
h = 10 cm
d = 8 cm
7 cm
11 cm
9 cm
4 cm
V=
V=
5cm
4 cm
6 cm
V=
10
cm
V=
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7 cm
8 cm
Open
Topped
7cm
13 cm
10 cm
17 cm
V=
9cm
V=
17 cm
10 m
9 m
15 m
6 m
V=
d = 16 cm
V=
h = 7 cm
d = 12 cm
6cm
8 cm
12
cm
3 cm
V=
V=
4cm
3 cm
8 cm
5 cm
V=
V=
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Relationships in Measurement
Comparing the Volumes of Cylinders and Cones
If you have a cylinder with a base that has a diameter of 8cm and a height of 12cm and
compare that with a cone that has a base with a diameter of 8cm and a height of 12cm,
which one will hold the most?
vs.
Volume of the Cylinder
Volume of the Cone
What is the relationship between the volume of the cone and the volume of the cylinder?
(hint: look at the formulas for both volumes)
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Comparing the Volumes of Rectangular Prisms and Pyramids
If you have a rectangular prism with a square base with sides 7cm long and a height of 9cm
and compare that with a square based pyramid with the same base and height, which of the
two shapes will have the smallest volume?
vs.
Volume of the Rectangular Prism
Volume of the Square Based Pyramid
What is the relationship between the volume of the prism and the volume of the pyramid?
(hint: look at the formulas for both volumes)
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Relationships in Measurement Worksheet
1. If the volume of a square-based rectangular prism is 45cm3, then what is the volume of
the square-based prism with the same base and height dimensions? Explain your answer.
2. If the volume of a cone with a base of 10cm and a height of 8cm is approximately
628cm3, what is the volume of a cylinder with the same base and height dimensions?
3. Find the volume of the cone that has the same base and height as the following cylinder.
(round final answer to 1 decimal place)
11 cm
9 cm
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Volume Problems
1. Sugar cubes come in boxes containing 2 layers
of cubes. Each layer forms a 12 by 6 rectangle and
each cube has sides that are 1cm in length. What
is the volume of the box that will hold all of the
cubes?
2. Golf balls are sold in packages that hold 3 balls and are shaped like rectangular prisms.
The diameter of a golf ball is 4cm.
a) What is the volume of the box that holds the golf balls?
b) What is the volume of each golf ball?
c) How much wasted space (extra volume) is there in the box of 3 golf balls?
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3. A tub of ice cream in the shape of a rectangular prism measures 21 cm x 15 cm x 12 cm.
a) What is the volume of the ice cream container?
b) If a scoop of ice cream (think spherical shapes!) has a diameter of 6cm, how many scoops
of ice cream could you serve from the container?
4. A can of concentrated orange juice is shown in
the image to the right.
a) What is the volume of the orange juice container?
b) If the manufacturer of the container wanted to change the shape so that it would hold
the same amount of orange juice concentrate but only be 8.5cm tall instead of 11.6cm,
what would the new dimensions be?
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5. A can of tennis balls is shown in the diagram to the right.
a) What would the volume of the canister be?
b) What is the volume of each tennis ball inside of the can?
c) How much wasted space would there be inside the can?
d) What would the volume be of a rectangular prism shaped box that would hold the same 3
tennis balls?
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More Applications of Volume
1.
The accompanying diagram shows the side view of a pool. If the pool is 7 m wide,
calculate the volume of water it can hold in cubic metres.
15 m
1m
3.5 m
6m
4m
2. A room is in the shape of a rectangular prism. Its dimensions are 7.5m, 6.0 m, and
3.0m.
a) Using a ruler, draw a labeled diagram of this room.
b) What volume of air (in cubic metres) can this room hold? How much would it cost to
operate air cleaning unit that costs $0.67 per cubic meter each year?
c) What is the area of the walls in this room? If one can of paint covers 40 m2, how
many cans of paint are needed to paint this room?
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3. A silo has a cylindrical base and a hemi-spherical roof (see diagrams below). Answer
the following questions for each silo below. What is the maximum volume of grain
each silo could hold?
i)
ii)
10 m
14 m
6m
4m
4. A spherical planet has a diameter of 6500 km. What amount of space does this
planet occupy?
5. A dump truck pours out a load of gravel, creating a pile that is in the shape of a
cone. If the diameter and the height of the gravel pile are 7.5 m and 4.4 m
respectively, what is the volume of gravel that the dump truck delivered?
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6. An Egyptian pyramid, which has a square base, stands 180 m from the ground at its
highest point. Each side of the square base measures 220 m.
a) Draw a diagram of this pyramid.
b) What is the volume of this pyramid?
c) Use The Pythagorean Theorem to calculate the slant height of the pyramid.
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Measurement Review
1. For each of the following triangles, find x.
a)
b)
x
16
x
7
cm
m
15
m
12
cm
2. Find the area of the shaded region.
4 cm
9 cm
3. Consider the track below.
90m
200 m
30 m
a) Find the perimeter of the track.
b)
Find the area inside the track.
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4. Find the volume of the following triangular prism
7m
12 m
10 m
8m
5. Frozen ice cream treats are sold in cone-shaped containers. The containers are 12 cm
high and have a 5-cm diameter. Indicate the formula you would use to determine the
volume of one ice cream treat container and find the volume. What would the volume be
if the container was shaped like a cylinder instead of a cone?
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6. Mrs. Coffin wants to fence a rectangular garden in her backyard. She has 36 metres of
fencing. Sketch and label the largest garden that Mrs. Coffin can enclose. Explain your
reasoning.
7. If Mrs. Coffin had fenced in an area of 65m2 and used the least amount of fencing
possible, what are the dimensions of the area and how much fencing did she use?
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