1.1
Area
Ski Area
40 m
T-bar
Lift
15 ft
At the summer ski camps on Horstman Glacier, you can learn to freeride,
ski moguls, and race.
Suppose that you were working at a summer ski racing camp on
Horstman Glacier. You have been asked to salt the ski area, which helps
the athletes to ski faster. How many bags of salt do you think you will
need? It is a very long chair lift ride from the bottom of the mountain to
the glacier, so you do not want to make more than one trip!
Investigate
Determining Area
Tools
1. Look at the ski lanes in the photograph, which begin between the
ruler
T-bar lift and the giant rock. What measures would you need to
know in order to estimate the total area of the ski area?
2. What would you need to know about each bag of salt?
3. What other information would be useful? Explain.
4. Reflect Suggest a method to determine the amount of salt needed to
cover the ski area shown in the photograph.
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Example 1
Area of a Composite Figure
Annette is the owner of a dance club. The Fire Marshall needs to
know the area of the club’s dance floor, illustrated below, to the
nearest square metre.
Mosh Pit
13 m
8m
Main Dance Floor
10 m
composite figure
• a figure made up of
two or more simple
geometric shapes
semi-circle
• a half-circle
component areas
• areas of simple shapes
that combine to form a
composite figure
Technology Tip
Scientific calculators
usually follow the
order of operations but
the actual keystrokes
may differ. If these
keystrokes do not
match your calculator,
refer to the instruction
manual for your
calculator.
Solution
The dance floor is a composite figure . Determine the area
of each component and then add them to find the total area.
Main Dance Floor
The main dance floor is a rectangle.
Calculate the area of the rectangle.
A=l×w
= 10 × 8
= 80
The area of the main dance floor is 80 m2.
8m
Mosh Pit
The mosh pit is a semi-circle with diameter
10 m. The radius of the semi-circle
is 10 ÷ 2, or 5 m.
Calculate the area of the semi-circle.
1 πr2
A= _
2
1 π(5)2
The radius is 5 m. Substitute r = 5.
=_
2
π
×
÷
39.3
5 x
2 =
The area of the mosh pit is approximately 39.3 m2.
10 m
5m
10 m
2
Add the component areas to determine the total area of the dance floor.
Total area = Area of main dance floor + Area of mosh pit
80 + 39.3
119.3
The total area of the dance floor is approximately 119 m2.
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Example 2
Net Area
Nazir is installing interlocking brick between the two triangular
gardens in his backyard as shown.
15 m
11 m
24 m
Determine the area to be covered with interlocking brick.
Solution
Calculate the total area of the backyard.
The dimensions of the backyard are 15 m by 24 m.
A=l×w
= 24 × 15
= 360
The total area of the backyard is 360 m2.
Calculate the area of the two gardens.
11 m
1b × h
A = 2 ×_
2
1 (11) × (11)
Substitute and simplify.
=2×_
2
=121
The area of the two gardens is 121 m2.
net area
• area found by
subtracting one or
more areas from a total
area
Subtract the area of the gardens from the total area to determine
the net area . This is the area to be covered with interlocking brick.
Net area = Total area - Area of gardens
= 360 - 121
= 239
Therefore, the area to be covered with interlocking brick is 239 m2.
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Example 3
Cost to Paint a Wall
Two coats of paint are to be applied to this wall. Determine the number
Receipt
of cans of paint needed, and the total cost.
Paint $43.17
Tax
$7.03
Total $50.20
19 ft
12 ft
36 in.
36 in.
25 ft
Solution
Determine the total area of the wall, including the windows.
The wall is a composite figure made up of a rectangle and a triangle. The
dimensions of the rectangle are 12 ft by 25 ft. The base of the triangle is
25 ft. Determine the height of the triangle.
19 ft - 12 ft = 7 ft
7 ft
+
12 ft
25 ft
25 ft
1 bh
ARectangle + ATriangle = lw + _
2
1 (25)(7)
= (25)(12) + _
2
= 300 + 87.5
= 387.5
The total area of the wall is 387.5 ft2.
Determine the area to be painted.
Subtract the area of the windows from the total area. The windows are
measured in inches and the wall is measured in feet. Convert the window
measures to feet.
36 ÷ 12 = 3
The windows have side lengths of 3 ft.
ATotal - 2AWindows = 387.5 - 2s2
= 387.5 - 2(3)2
= 387.5 - 18
= 369.5
The area to be painted is 369.5 ft2.
387.5 ft2
-2×
3 ft
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Determine the total cost.
Since two coats of paint must be applied, double the area to be painted.
2 × 369.5 = 739
Technology Tip
For Web sites that will
convert units between
the metric and imperial
systems, go to the
Ryerson McGraw-Hill
Web site and follow the
links to Foundations for
College Mathematics 12,
Section 1.1, Example 3.
Therefore, there must be enough paint to cover 739 ft2. According to the
label, each can of paint covers approximately 45 m2. Convert the net area
of the wall from square feet to square metres.
Since 1 ft = 0.3048 m, 1 ft2 = (0.3048)2 m2.
739 × 0.30482 = 68.7
Each can of paint will cover 45 m2. Divide the total area by 45.
68.7 ÷ 45 1.5267
Two cans of paint will be needed to complete the job.
Find the total cost of the paint.
2 × 50.20 = 100.40
The total cost of the paint is $100.40.
Key Concepts
• To apply an area formula, all measures must be in the same units.
• The area of a composite shape can be found by adding the areas of its
component shapes.
• To find the net area of a shape, subtract the unneeded component
areas from the total area.
Discuss the Concepts
D1. Explain the steps you would follow to find the area of this tabletop.
80 cm
60 cm
1.8 m
D2. How is the area of a composite shape related to the areas of its
components? Provide an example to illustrate your answer.
D3. a) What is meant by net area?
b) Provide a sketch to represent a real-world situation where you
would need to calculate the net area of a figure.
c) Describe the steps you would follow to calculate the net area of
your sketch.
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Practise
A
1. a) Copy each composite shape. Then, draw lines to divide the shape
into component shapes and name each component shape.
i)
ii)
b) Calculate the total area of each composite shape in part a)
using components.
2. Pick one of the shapes in question 1. Calculate the total area
using net area.
3. a) Explain how the steps you used in questions 1 and 2 are different.
b) Which method do you prefer? Explain your reasoning.
Apply
B
4. Julio works for the Parks Department. One of his responsibilities is
to maintain a grassed playing field surrounded by a running track.
Determine the area of the playing
field to the nearest tenth of a
32 m
square metre.
100 m
5. Courtney has a summer job working for the Ministry of Transportation.
She has to paint 10 square
signs like the one shown.
Each sign needs two coats
of paint for each colour.
50 cm
a) How much yellow paint
30 cm
does Courtney need?
b) How much black paint
does she need?
30 cm
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6. a) Describe two different methods of finding the area of this shape.
8 cm
5 cm
10 cm
16 cm
b) Use one method to determine the area and the other to check
your answer.
7. a) What is the net area of this shape, to the nearest hundredth
of a square inch?
1.5 in.
b) Could you have found the area by considering the area of
composite figures? Explain.
c) Explain the main difference between the two methods.
8. Consider this swimming pool.
10 m
300 cm
m
15 m
a) Calculate the total area
using components.
b) Calculate the total area
using net area.
c) Compare your answers
to parts a) and b).
Are they the same?
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Achievement Check
9. a) Give the missing dimensions for Martin’s garden, shown below.
Explain how you calculated these measures.
12 ft
16 ft
b) Determine the total area of Marvin’s garden, to two decimal places.
Math
Connect
1 ft2 = 0.09 m2
c) A bag of topsoil costs $2.99 and will cover 3 m2. How much will it
cost Marvin to cover his garden with a layer of topsoil?
10. A roof of a barn has a pentagonal gable with a base length of 14 ft and
a height of 8 ft. The radius of the circular window is 2 ft.
Use The Geometer’s Sketchpad® to estimate the area of the gable,
less the window, by following these steps.
a) Open The Geometer’s Sketchpad® and begin a new sketch. From
the Graph menu, choose Show Grid.
b) Construct and measure the window.
• From the Graph menu, choose Plot Points … .
Plot points at (8, 4) and (10, 4).
• Select the left point and then the right point. From the
Construct menu, choose Circle By Center+Point.
• From the Construct menu, choose Circle Interior. Use the
Display menu to change the colour to green.
• From the Measure menu, choose Area.
c) Construct and measure the gable.
• Plot points at (1, 1), (15, 1), (12, 7), (8, 9), and (4, 7).
• Click to select the points in order. From the Construct menu,
choose Polygon Interior. Use the Display menu to change the
colour to yellow.
• From the Measure menu, choose Area.
d) Determine the net area.
• From the Measure menu, choose Calculate.
• Click on the Polygon area measure. Then click –.
• Click on the Circle area measure. Then click OK.
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11. a) Create a problem similar to the one in question 10.
b) Solve the problem.
c) Trade problems with a classmate. Solve each other’s problems
and check your solutions.
12. a) Find a wall in your classroom, school, or home that could use
a fresh coat of paint. Choose a wall that has features such as
windows, doors, or chalkboards.
b) Sketch the wall and identify any areas, such as windows, that
would not be painted.
c) Use a measuring tape, metre stick, or other measurement tools
to measure the required dimensions of the wall.
d) Determine the net area of the wall to be painted in square metres.
e) Research the cost of paint. Pick a colour that you like and
determine the cost of applying two coats of paint to the wall.
Chapter Problem
Reasoning and Proving
Representing
Selecting Tools
13. The top sections of the ski area of Horstman Glacier are indicated by
the dark red rectangle in the photograph. To improve summer skiing
conditions, this entire area must be covered with salt.
Problem Solving
Connecting
Reflecting
Communicating
T-bar
40 m Lift
Ski Area
15 ft
Use this information:
• The 7 T-bar support poles are spaced approximately 40 m apart.
• The giant slalom gate lane is approximately 15 ft wide.
• There are ten ski lanes.
• Each 20-kg bag of salt will cover 400 m2 of snow. The entire ski
area must be salted twice per day.
a) Estimate the number of bags of salt needed per day.
b) What is the estimated total mass of salt needed per day? Would it be
reasonable to carry this up on a chair lift? Explain why or why not.
c) Describe how you solved this problem and discuss any
assumptions that you made.
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Extend
C
14. A shed has walls that are 12 ft in length, 8 ft in width, and 3 m in
height. The roof is in the shape of a rectangular-based pyramid whose
height reaches 1 m above the walls at its highest point. The doorway
measures 1.5 m by 2.5 m. The two congruent windows each have an
area of 1.4 m2.
1m
3m
1.5 m
2.5 m
8 ft
12 ft
The entire shed, excluding the windows and doorway, needs to be
covered in plastic sheeting to make it waterproof. Determine the
surface area that needs to be covered. Include diagrams to support
your solution. Discuss any assumptions you must make.
15. a) Find a shed, garage, or other small building in your neighbour.
Measure its dimensions.
b) Draw a diagram of the building. Include all features such as
windows and doors. Add the measurements to your diagram.
c) Suppose the building is to be panelled with siding. Determine how
much siding is required. Indicate the width of the siding and list
the considerations you made when calculating this value.
d) A one-litre can of paint will cover 10 m2. How many cans of
paint will you need to paint the shed with two coats? List any
restrictions you placed on your estimate.
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Use Technology
Unit Conversions Using the TI-Nspire™ CAS
Graphing Calculator
Tools
TI-Nspire™ CAS graphing
calculator
Part 1: Converting units using the TI-Nspire™ graphing calculator.
Follow these steps to convert 5 in. to centimetres.
1. Turn on the TI-Nspire™ CAS and open a new Calculator page.
Technology Tip
2. Enter 5.
Press k to access the
catalogue.
3. Press k. Use the arrow keys to scroll to
If you need help
opening pages and
documents, refer to the
Technology Appendix
on page 498.
.
Press ·. Scroll down to Length, and press x.
A list of length units will appear.
4. Scroll down to _in, and press ·.
5. Press k. Scroll up to the Conversion Operator, (¢).
Press ·.
6. Press k. Scroll down to _cm. Press ·.
7. Press ·. The answer 12.7 cm will be displayed.
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Part 2: Shortcuts
1. Type the desired units, preceded by the underscore.
The underscore is accessed by pressing /_.
To enter 2.5 ft, press 2.5 /_ FT.
2. You can access the conversion operator by pressing /k.
A matrix of selections will appear. Use the cursor keys to select
the conversion operation, and press ·.
3. Type _m, and press ·. The answer of 0.762 m will be displayed.
You can browse the catalogue to determine which units the TI-Nspire™
CAS can convert. You can also convert other kinds of measurements,
such as area and volume. However, you must choose two units that
belong in the same category. For example, you cannot convert
centimetres to square inches.
Perform several other length conversions of your choice. Try several
within the imperial system, some within the metric system, and some
that convert between the two.
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