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Adrian wants to carpet two rectangular bedrooms. One bedroom measures 12 ft by 16 ft. The other measures 10 ft by 12 ft. How much carpet does Adrian need to buy?

1 What is the area of each room?

Area of rectangle = length × width

ft × ft = sq ft

ft × ft = sq ft

2 What is the total area?

Adrian needs to buy sq ft of carpet.

Example 1Berry bushes are being planted on a triangular piece of land to stop soil erosion. Each bush is planted in 1 sq yd of ground area. How many bushes are needed? (1 square represents 1 sq yd.)

SolutionWhat are the approximate dimensions of the triangle? A. Height: about yd Base: about yd

Area of triangle = B.

= sq yd

About berry bushes are needed.

Calculating Area of 2-D Shapes4 . 3

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1 square represents 1 sq. yd.

Match each shape name with a formula for its area.

i) 12 (base × height) triangle

ii) 12 (sum of parallel lengths) × height parallelogram

iii) base × height trapezoid

Try These

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height

base

height

height

base

base

b × h2

yd × yd2

parallel

two or more lines that are always the same distance apart

96 Copyright © 2011 by Nelson Education Ltd. Apprenticeship and Workplace 10

12 16 192

10 12 120

312

192 sq ft +120 sq ft = 312 sq ft

6.7e.g., 8.5

28.475

28

6.7 8.5

04C04.indd 96 4/21/10 9:51:40 AM

Example 2Awet is a sheet metal specialist. He cut a piece of sheet metal in the shape of a parallelogram. Then he cut out a circle so that a 5 inch pipe can fi t through. What area of the shape remains, to the nearest square inch?

SolutionArea of parallelogram = base × heightA. in. × in. = sq in.

Area of circle = B. π × r 2 = sq in.

Area of remaining shape = Area of parallelogram – Area of circleC. sq in. – sq in. = sq in.

The area of the remaining shape is about

Example 3Beverly bought a square rug for her home offi ce. The label on the

rug says it covers 24012 sq ft. What are the dimensions of the rug?

SolutionHow can you use the formula for area of a square to calculate A. the length of a side?

length of side = sArea of a square = s2

sq ft = s2

√ sq ft = s, so s = ft

What are the dimensions of the rug? (How precise will you be?)B.

How can you check your answer?C.

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9.8 in.

15.2 in.

5 in.

Keep the decimal places through the calculations and round at the end.

Hint

To calculate the inverse (opposite) of a square, press the square root key. For example, press 16 √ and then press x2 to check.

Tech Tip

REFLECTINGWhat are some other inverse operations?

Copyright © 2011 by Nelson Education Ltd. Chapter 4 Area Measurement 97

15.2 9.8 148.96

129.325...

π x (2.5 in.)2

19.634...

19.634...

148.96

129 sq in.

240.5

240.5 15.508...

e.g., Each side of the rug is about 15.5 ft long.

e.g., When I square my answer, I should get the original area.

It’s almost the same, except for the rounding.

04C04.indd 97 4/21/10 9:51:40 AM

PracticeCalculate the area of each rectangular photo.1.

medium size: 5˝ by 7˝ a)

poster size: 16˝ by 20˝ b)

small size: 3c) 12

˝ by 5˝

Calculate each area. If necessary, round to one decimal place.2.

a) c)

b) d)

The city of Winnipeg would like to build a new dog park. The 3. grassy area will be surrounded by a 2 m wide sidewalk. What is the area of sidewalk around the park?

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base = 1.2 m

height= 0.9 m

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10.0 m

8.6 m

height= 5.2 m

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radius = 514 in.

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base = 18.0 cm

height= 5.5 cm

40 m

80 m

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Use the formulas on page 91.

Hint


5 in. x 7 in. = 35 sq in.

16 in. x 20 in. = 320 sq in.

3.5 in. x 5 in. = 17.5 sq in.

Area of sidewalk = area of large rectangle – area of grass

= (80 m x 40 m) – (36 m x 76 m)

= 464 m2

The area of the sidewalk is 464 m2.

Area of parallelogram

= b x h

0.9 m x 1.2 m =· 1.1 m2

Area of trapezoid

= 12(a +b) x h

= 12(10.0 m +8.6 m) x 5.2 m

=· 48.4 m2

Area of circle = π x r 2

π x (5.25 in.)2

=· 86.6 sq in.

Area of triangle = 12(b x h)

12(18.0 cm x 5.5 cm)

= 49.5 cm2

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What is the area of each circle, rounded to one decimal place?4.

a) b)

What is the height of the parallelogram?5.

What are the dimensions of the surface of each baking pan?6. If necessary, round your answer to one decimal place.

a rectangular cookie sheet: a) area 198 sq in., width 12 in., length in.

a round pizza pan: area 730 cmb) 2, radius cm

Dylan has a fi eld in the shape of a trapezoid. He needs to 7. know its area in square metres before he buys fertilizer.

Label the dimensions of the fi eld in metres. a)

What is the area of the fi eld in square metres?b)

Each truckload of fertilizer costs $52.14 and covers c) 10 000 m2. How much will it cost to cover the fi eld?

Meredith built a rectangular display case for her collector 8. plates. How much of the case is visible behind the plates?

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Area =5300 mm2

base = 50 mm

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5˝ 20.5 cm

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5˝ 20.5 cm

The radius is half the diameter.

Hint

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20"

8 "12


e.g., 5300 mm2 = 50 mm x h

5300 mm2 50 mm = h

106 mm = h

16.5

15.2

e.g., Area visible = area of rectangle – area of two circles

= (20 in. x 8.5 in.) – (2 x π x (4.25 in.)2)

= 170 sq in. – 113.490… sq in.

= 56.509… sq in. About 57 sq in. is visible.

e.g., 390 000 m2 10 000 m2 = 39 truckloads

39 truckloads x $52.14/truckload = $2033.46

The fertilizer will cost $2033.46.

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0.8 km

0.5 km

0.6 km

500 m

600 m

800 mArea of a trapezoid = 1

2(a +b) x h

= 12(800 m +500 m) x 600 m

= 390 000 m2

The area of the field is 390 000 m2.

19.6 sq in. 330.1 cm2

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Evelyn made a model of a square metre using centimetre grid paper. How many grid squares are in the model?

1 What are the dimensions? 1 m = cm

1 m2 = 1 m × 1 m, or cm × cm

2 What is 1 m2 in square centimetres? cm2

Example 1Dylan’s fi eld is 390 000 m2 in area. What is the area in hectares?

Solution What is the relationship? 1 m = A. hm

1 m2 = 1 m × 1 m, or hm × hm = hm2

1 mB. 2 = 0.0001 hm2, so

390 000 m2 × hm2/m2 = hm2 or ha.

Example 2Phillip calculated the area of his driveway as 600 sq ft. How much will asphalt paving cost if the price is $12.25/sq yd?

Solution 1How many square feet are in 1 sq yd? A. 1 sq yd = 1 yd × 1 yd, or ft × ft = sq ft

What is 600 sq ft expressed in square yards? B. 600 sq ft ÷ sq ft/sq yd = sq yd

What is the cost of paving? C. sq yd × $ /sq yd =̇ $

Relating Area Units4 . 4i) 75 cm = m iii) 6 ft = yd

ii) 2 km = m iv) 2 ft = in.

Try These

In the metric system, land is often measured in hectares, using the symbol ha. 1 ha is the same area as 1 square hectometre. 1 ha = 1 hm2

1 hm = 100 m

Hint

When changing from a smaller unit to a larger unit, the number will decrease.

Hint


20.75

2000 24

100

100 100

10 000

0.01

0.01 0.01 0.0001

0.0001 39 39

3 3 9

9 66.666...

12.25 816.6766.666...

04C04.indd 100 4/21/10 9:51:43 AM

Solution 2What is the relationship? A. 1 sq yd = 1 yd × 1 yd, or ft × ft = sq ft

Set up an equation to relate the units. Then solve it.B.

=

sq ft × = × sq ft

= ?, so ? = sq yd

What is the cost of paving? C. sq yd × $ /sq yd

The cost to pave the driveway is $ .

PracticeExpress each area in the metric units given.1.

a vegetable garden, 23 ma) 2 cm2

a sports fi eld, 0.45 kmb) 2 m2 or ha

a postage stamp, 460 mmc) 2 cm2

a No Parking sign, 5500 cmd) 2 m2

Express each area in the imperial units given.2. a pond, 5 sq yd a) sq ft

a fl oor tile, 1 sq ft b) sq in.

a painting, 9 sq ft c) sq in.

a large envelope, about 175.4 sq in. about d) sq ft

Lia wrote: 1 sq ft = 1 ft × 1 ft, which is 3. 13 yd ×

13 yd =

19 sq yd.

How many square yards are in 18 sq ft?

sq ft × sq yd/sq ft = sq yd

Which lake has the greater surface area? 4. How many hectares greater is this?

REFLECTINGIf the dimensions of the driveway were given (e.g., 12 ft by 50 ft), how could you calculate the

area in square yards?

You can use Part b) to answer Parts c) and d).

Hint

1 sq yd sq ft1 sq yd

sq ft? sq yd

600 sq ft

? sq yd600 sq ft

LakeSurface

area

Great Bear Lake 31 153 km2

Great Slave Lake 27 200 km2


3 3 9

600 600

66.666...

12.25

816.67

230 000

450 000 45

4.6

0.55

45

144

1296

1.2

9

9

9600

66.666...

18 219

Great Bear Lake is larger, by 3953 km2. Its

surface area is greater by 395 300 ha.

04C04.indd 101 4/21/10 9:51:43 AM

You can use these charts to change from one system to another.

1 Complete the second chart below.

Metric (SI) to Imperial

1 cm2 =̇ 0.1550 sq in. 1 m2 =̇ 10.7639 sq ft 1 km2 =̇ 0.3861 sq mi

Imperial to Metric (SI)

1 in. =̇ cm, so

1 sq in. =̇ ( cm)2

=̇ cm2

1 ft =̇ m, so

1 sq ft =̇ ( m)2

=̇ m2

1 sq yd =̇ 0.8361 m2

1 sq mi =̇ 2.5900 km2

Example 1Antonio is designing a new auto body shop with a fl oor area of 40 m2. Each vehicle bay needs to be 38 sq ft. How many vehicle bays can there be?

SolutionHow many square feet are in 40 mA. 2?

=

m2 × = × m2

? =̇ sq ft

How many bays are possible?B. Total area area of each bay:

sq ft 38 sq ft/bay =̇ bays

Expressing Area in Different Systems4 . 5Use the charts inside the back cover.

HintRound to the nearest tenth of a unit.i) 175 cm =̇ in. iii) 32 ft =̇ cm

ii) 12 km =̇ mi iv) 15 ft =̇ m

Try These

A. You can multiply 40 m2 by 10.7639 sq ft/m2 to get the number of square feet, or you can solve an equation.

Hint

sq ft1 m2

? sq ft m2

sq ft1 m2

? sq ft m2

REFLECTINGHow do you decide

when to round your answer up or

down?


68.9 975.4

7.5 4.6

0.092.54 0.30

6.4516

2.54 0.30

10.7639

4040

430.556

430.556 11

40

10.7639

40

04C04.indd 102 4/21/10 9:51:44 AM

Example 2 The area of a CFL football fi eld is 7150 sq yd. What is the area of the fi eld in acres and in hectares?

What is the area in acres? 1 acre = A. sq yd

=

? = , or acres

The area is about acres.

What is the area in hectares? 1 acre B. =̇ ha

acres × ha/acre = ha

The area is about ha.

PracticeComplete. If necessary, round to one decimal place.1.

area of a property for sale, 3 ha a) acres

area of the glass fl oor in the b) Calgary Tower, 144 sq ft m2

area of a 100 acre woodlot c) ha

area of Canada, 9 984 670 kmd) 2 sq mi

area of 8e) 12˝ by 11˝ paper sq in. or cm2

A hockey rink has a fl oor area of 17 000 sq ft. The rental price 2. of the rink is $0.25/m2 for 2 h. What will it cost the alumni hockey team to rent the rink for 2 h?

The fl oor area of Bailey’s living room is 26 sq yd. She wants to 3. buy hardwood fl ooring that is advertised at $23.40/m2. How much would it cost to buy that hardwood for her living room?

In the imperial system, land is measured in acres.1 acre = 4840 sq yd1 acre =̇ 0.4047 ha

Hint

REFLECTINGHow can you solve

an equation to complete this relationship?

1 ha = ? acres

1 ha =̇ 2.4711 acresHint

sq yd1 acre

7150 sq yd? acres

7150 sq yd × 1 acre sq yd


4840

1.477...

1.5

0.4047

1.4772... 0.4047 0.597...

0.6

7.4

13.0

40.5

3 855 081.1

93.5 603.2

e.g., 17 000 sq ft x 0.09 m2/sq ft = 1530 m2

$0.25/m2 x 1530 m2 = $382.50

The rental price is $382.50.

4840

4840

e.g., 26 sq yd x 0.8361 m2/sq yd = 21.7386 m2

$23.40/m2 x 21.7386 m2 =· $508.68

It would cost $508.68.

04C04.indd 103 4/21/10 9:51:45 AM

a) 1. Name a referent for 1 cm2.

b) Estimate the area of your calculator.

c) Measure the length and width of your calculator, in inches. Calculate the area to the nearest square inch.

a) 2. Name a referent for 1 sq ft.

b) Estimate the area of this arrow painted on a wall.

(1 square represents 1 sq ft.)

Two theatre stages each have an area of about 315 sq yd. 3.

About how long is a side of the square stage?a)

About how long is the radius of the circular stage?b)

In 2006, the area of Kelowna was 211.69 km4. 2, and Chilliwack was 260.19 km2. How much larger was Chilliwack? Express your answer in square miles, to the nearest tenth.

Express the area of each opening, in square feet. 5. a lacrosse net, 6 ft by 6 ft a) sq ft

a hockey net, 72 in. by 48 in. b) sq ft

a soccer net, 8 yd by 8 ft c) sq ft

You will need• a ruler (imperial)

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stage area= 315 sq yd

stage area= 315 sq yd

side = ?

radius = ?

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Mid-Chapter


e.g., area of your fingernail

e.g., about 20 cm2

e.g., 2.5 in. x 4.75 in. = 11.875 sq in.

The area is about 12 sq in.

e.g., Area of a square = s 2

315 sq yd = s 2, so s = √315, or about 17.7

The length of a side is about 17.7 yd.

e.g., Area of a circle = π x r 2

315 sq yd =· 3.14 x r 2, so r 2 = 315 3.14, or 100

r = √100, or 10 The radius is about 10 yd.

e.g., area of a floor tile

e.g., about 8 sq ft

36

24

192

e.g., 260.19 – 211.69 = 48.5 Chilliwack was 48.5 km2 larger.

1 km2 =· 0.3861 sq mi, so

48.5 km2 x 0.3861 sq mi/km2 =· 18.7 sq mi

04C04.indd 104 4/21/10 9:51:45 AM

Jordana is making face pins shaped like regular polygons. What is the area of each pin?

1 How can you divide each shape into congruent triangles?

2 What is the area of each triangle within each shape?

In square:

base = cm

height = cm

Area of triangle

=

= cm2

In regular hexagon:

base = cm

height = cm

Area of triangle

=

= cm2

In regular octagon:

base = cm

height = cm

Area of triangle

=

= cm2

3 What is the area of each regular polygon?

Area of square

= × (area of triangle)

= cm2

= cm2

Area of regular hexagon


= cm2

= cm2

Area of regular octagon


= cm2

= cm2

Areas of Regular Polygons4 . 7If you fold a square like this, you can fi nd the centre. What do you know about the four triangles?

Try TheseYou will need• a ruler

regular polygon

a closed fi gure with all sides equal and all angles equal

Area of a triangle

= 12

(b × h)

Hint

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congruent

two or more polygons with the same size and shape

REFLECTINGWhat pattern do

you see in the way these areas are

calculated?


e.g., The triangles have equal areas.

The triangles have the same size and shape.

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1.3 cm

2.6 cm 1.6 cm1.8 cm 1.6 cm

1.7 cm

Join opposite corners.

2.61.3

1.69

1.81.6

1.44

1.61.7

1.36

44 x 1.69

6.76

66 x 1.44

8.64

88 x 1.36

10.88

2.6 cm x 1.3 cm2

1.8 cm x 1.6 cm2

1.6 cm x 1.7 cm2

04C04.indd 106 4/21/10 9:51:48 AM

Example 1What is the area of this regular pentagon in square inches? Measure each dimension to the nearest quarter inch.

SolutionWhat is the area of each triangle in the regular pentagon? A.

Base = in. Height = in.

Area of triangle: 12 in. × in. = sq in.

What is the total area? B. Area of regular pentagon: × sq in. = sq in.

Example 2How can you calculate the area of other regular polygons?

SolutionWhat is the area of this sandbox, which has the shape of a A. regular heptagon? Round your answer to one decimal place.

Area of triangle: ( m × m) ÷ 2 = m2

Area of regular heptagon: × m2 = m2

The area of the sandbox is about m2.

Look at the area calculations you did for the square and B. regular pentagon, hexagon, heptagon, and octagon. Complete a general formula in words:

Area of a regular polygon =

Trish says that the following formula works. C. “Area of a regular polygon = (number of sides in the regular polygon × length of a side × distance to the centre from the middle of each side) all divided by 2.” Do you agree? Explain your thinking.

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base = 1.3 m

height =1.4 m

penta means 5hexa means 6hepta means 7octa means 8nona means 9deca means 10

Hint


5

1.3 1.4

7

0.91

6.4

6.370.91

the number of congruent triangles

in the polygon times the area of one triangle

Yes, e.g., because the distance from the centre to the middle

of each side is the height of the triangle and the length of a

side is the base of the triangle

34

12

34

12

316

1516

316

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PracticeA helicopter landing pad, on top of a tall office building, has 1. the shape of a regular octagon. How big is this landing area?

Area of triangle: sq ft

Area of regular octagon: × sq ft = sq ft

The area of the helicopter pad is sq ft.

A $1 coin looks like a regular polygon with 11 sides. It is 2. called a hendecagon. What is the area of the coin?

Length of a side = 7.9 mm

Distance from centre to middle of a side =̇ 13.3 mm

Area of regular polygon:

× ( mm × mm) ÷ 2 =̇ mm2

A banquet hall has the shape of a regular heptagon. Each 3. side measures 7.87 m, and the distance to the centre of the room is 8.17 m.

What is the floor area of the room, to the nearest square a) metre?

The floor area required for each table is 9 mb) 2. How many tables can fit in the room?

Lionel wants to stain the floor of his new gazebo that is 4. shaped like a regular hexagon as shown. The label on a can of stain says that 1 L covers about 9.3 m2. Is 1 L of stain enough for two coats?

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7 ft

10 ft

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2.8 m

1.6 m


(7 x 10) 2 = 35

8 35 280

11 7.9 13.3 578

280

e.g., height: 2.8 m 2 = 1.4 m

Area of triangle: 12(1.6 m x 1.4 m) = 1.12 m2

Area of regular hexagon: 6 x 1.12 m2 = 6.72 m2

The area of the floor is about 7 m2. For 2 coats, you need

enough stain to cover about 14 m2, so 1 L is not enough.

e.g., Area of triangle: 12 (7.87 m x 8.17 m) = 32.148… m2

Area of heptagon: 7 x 32.148… =· 225 m2

2259

= 25

There can be up to 25 tables in the room.

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The Giant’s Causeway is an area in Northern Ireland of nearly 5. 40 000 interlocking blocks that were formed from a volcanic eruption. Most of the blocks have the shape of a regular hexagon. The area of the top of one block is 585 cm2, and the distance from each side to the centre is 13 cm. What is the length of each side?

Dante wants to create a new school crest. He will use two 6. identical regular pentagons—one green, on top of one black. He will cut a square out of the green pentagon so that the black pentagon shows through.

Each pentagon can be cut from a 2 m by 2 m square piece a) of felt fabric. What area of felt will Dante buy?

What is the area of the black pentagon? b)

Area of regular pentagon:

How much green felt is wasted? c)

The area of a regular polygon is 180 sq in. The length of a 7. side is 5 in., and the shortest distance from a side to the centre is 8 in. How many sides does the polygon have?

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1.5 m

1.0 m

40 cm

Area = # of congruent triangles x area of each

180 sq in. = ? x 12(5 in. x 8 in.)

180 sq in. = ? x 20

180 sq in. 20 = ?

? = 9, so the shape is a regular nonagon.


Area of 1 big square of felt: 2 x 2 = 4 m2.

He needs to buy 2 pieces, so the total is 8 m2.

585 cm2 = 6 x 12(s cm x 13 cm)

585 cm2 3 = (s cm x 13 cm)

195 cm2 = s cm x 13 cm

195 13 = s

The length of each side is 15 cm.

Waste from the big piece is 4 – 3.75 = 0.25 m2.

Area of little square cut out is 0.4 x 0.4 = 0.16 m2.

0.25 m2 +0.16 m2 = 0.41 m2

The total amount of green felt wasted is 0.41 m2.

5 x 12(1.5 x 1) = 3.75 m2

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Katie designed an irregular four-sided patch of fabric to sew on her jeans. She divided the quadrilateral into triangles to make it easier to calculate the area. Then she drew the height of each triangle using a right triangle to make sure that the height was perpendicular to the base. What is the area of the patch?

1 What are the measurements of the two triangles?

Triangle Base Height Area

A cm cm cm2

B cm cm cm2

2 What is the area of the patch, to the nearest tenth?

Area of patch = area of triangle A + area of triangle B

= cm2 + cm2

=∙ cm2

Example Does Katie’s method work with a fi ve-sided patch?

SolutionDraw an irregular polygon with fi ve sides on paper.A.

Draw as few triangles as possible to divide the shape.B.

Follow Katie’s steps above. Does the method work?C.

Area of Irregular Polygons4 . 8You will need• a ruler (imperial

and SI)• a right triangle

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Try These

i) (18) + 40 = ii) (3.1 × 3.4) ÷ 2 = 12

Measure on the diagram to the nearest millimetre.

Hint

REFLECTINGCan an irregular polygon always be divided into triangles? Try it.


5.2749

5.31 14.455

19.8

The method works. You can calculate the total area by

adding the areas of all the triangles.

5.9

5.9

1.8

4.9

5.31

14.455

04C04.indd 110 4/21/10 9:51:50 AM

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1.50 km

1.35

km

0.80 km

0.80 km

base X

F

G

E

base Y

PracticeWhat is the area of this irregular polygon?1.

Show your measurements on the diagram. a)

Area of C: b)

Area of D:

Area of the polygon: about sq in.

What should the selling price of this land be if 2. the going rate is $100 000/km2?

Yolly, a real estate agent, divided the polygon a) into three triangles. Then he measured:

base X = 0.88 km base Y = 1.14 km

height E = 0.64 km height G = 0.74 km

height F = 1.06 km

Complete the chart.

Triangle Base Height Area of triangle

G 1.14 km 0.74 km 1.14 km × 0.74 km ÷ 2 = 0.4218 km2

E km km km × km ÷ 2 = km2

F km km km × km ÷ 2 = km2

What is the approximate total area? b)

What should the price be? c)

Which person below made a mistake? Explain.3. • Keir divided the shape and calculated:

• Sherry divided the shape a different way and calculated:

Area of triangle 1: (2 cm x 3.5 cm) 2 = 3.5 cm2

Area of triangle 2: (6.4 cm x 4.2 cm) 2 = 13.44 cm2

Total area: 3.5 cm2 +13.44 cm2 = 16.94 cm2

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6.4 cm 4.2 cm2

1

3.5 cm

1 square represents 1 cm2Area of rectangle: 2 cm x 3.5 cm = 7 cm2

Area of remaining triangle: (5 cm x 3.5 cm) 2 = 8.75 cm2

Total area: 7 cm2 +8.75 cm2 = 15.75 cm2


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D38

1 ˝

58

1 ˝

6

81 ˝

78

1 ˝

7

81 ˝

2˝

e.g.,

3

0.88

0.88

0.64

1.06

0.2816

0.88 1.06 0.4664

0.88 0.64

( 118 in. x 15

8 in.) 2 =· 1.29 sq in.

( 138 in. x 14

8 in.) 2 =· 1.42 sq in.

e.g., The area is about 1.2 km2.

1.2 km2 x $100 000/km2 = $120 000

Keir made the mistake. e.g., He used 4.2 as the height of

triangle 2, but it’s a side length, not a height.

04C04.indd 111 4/21/10 9:51:51 AM

Danielle wants to use paving stones to pave a rectangular loading area outside her store’s entrance. Paving stones are about $8.50/sq ft. How much should she expect to pay?

1 How can you divide the shape into areas you can calculate?

2 What is the area of each part?

Area A: sq ft

Area B: sq ft

Area C: sq ft

3 What is the total area of the loading area?

sq ft + sq ft + sq ft = sq ft

4 What is the cost for paving stones?

Example 1Liam is a window installer. A Norman window has this shape. What is the area of the window?

Solution

Total area = area of square + area of semicircle

Area of square: ( m)2 = m2

Area of semicircle = ( ) ÷ 2

( ) ÷ 2 =̇ m2

The area of the window is m2 + m2 = m2

Area of Composite Shapes4 .9Is the area of this shape more than or less than 48 sq ft? Explain your thinking.

Try These

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12 ft

REFLECTINGCould the shape

have been divided differently? Would this still give you

the correct answer?

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3 m

3 mREFLECTINGWhy divide by 2?


Less; it's like a rectangle

with area 48' with part removed.

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4 ft

2 ft

12 ft B

A

C

(See diagram for example.)

4 ft x 2 ft = 8

8 ft x 2 ft = 16

4 ft x 2 ft = 8

8 16 8 32

32 sq ft x $8.50/sq ft = $272 The cost is $272.

3 9

π x (1.5 m)2

9 3.5 12.5

3.5

π x radius2

04C04.indd 112 4/21/10 9:51:53 AM

Example 2Angie is re-shingling a roof with two sides. Each side is rectangular with a skylight in the shape of a regular hexagon as shown. One bundle of shingles covers 2.25 m2. How many bundles of shingles are needed?

SolutionWhat is the area to be shingled? A. Area of shingles = area of rectangles – area of skylights

Area of rectangles: ft × ft × 2 = sq ft

Area of skylights (2 regular hexagons)

= [ × area of triangles] × 2

= [ × ( )] × 2

= sq ft

Area of shingles: sq ft – sq ft = sq ft

How many bundles of shingles are needed? B. 1 bundle covers 2.25 m2, and 1 m2 =̇ 10.76 sq ft, so

m2 × sq ft/m2 covers about sq ft

sq ft ÷ sq ft =̇

About bundles of shingles are needed.

PracticeMarlene charges $2.15 per square metre to 1. install sod. How much should she estimate for sodding the interior of this athletic track?

REFLECTINGWhy multiply by 2?

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25 ft

312 ft

4 ft

HintYou can express the height of each triangle (3 1

2 ft) as a decimal.

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60 m

0.15 km

60 m

0.15 km

Watch for different units.

Hint


40 25 2000

6

6 4 ft x 3.5 ft 2

84

2000 84 1916

2.25 10.76 24.2

1916

79

24.2 79

Area of rectangle: 150 m x 60 m = 9000 m2

Area of circle: π x (30 m)2 = 2827.4333… m2

Total area: 9000 m2 +2827 m2 = 11 827 m2

11 827 m2 x $2.15/m2 = $25 428.05

Marlene should estimate about $25 500.

04C04.indd 113 4/21/10 9:51:53 AM

Calculate the area of each composite shape.2.

a)

Area of parallelogram:

Area of triangle:

Total area:

c)

All corners are right b) angles.

d) Divide the diagram in Part b) differently. Calculate the area again.

Maria would like to paint 3. both sides of her steps in a brighter colour. One side is shown in the diagram. All corners are square.

How much area needs to be painted?a)

The smallest can of paint covers 10 sq ft. If Maria decides b) to paint two coats, how many cans will she need?

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5 m6 m

10 m

4 m>

>

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8 in.4 in.

11 in.

19 in.

>>

>>

>

>


Area of trapezoid: 12

(4 m +10 m) x 5 m

= 35 m2

Area of semicircle:

3.14(2 m)2 2 =· 6.28 m2

Total: 35 m2 – 6.28 m2

= 28.72 m2

11 in. x 8 in. = 88 sq in.

104 sq in.

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1.9km

2.1 km

3.0 km

4.3 km

4.5 kmB

C

A

Rectangle A: 13.50 km2

Rectangle B: 19.32 km2

Rectangle C: 3.99 km2

Total: 36.81 km2

Rectangle D: 9.03 km2

Rectangle E: 7.98 km2

Rectangle F: 19.80 km2

Total: 36.81 km2

6 sq ft x 2 = 12 sq ft

24 sq ft are needed for 2 coats, so 3 cans of paint are

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2.1 km

3.0 km

4.3 km

4.5 km

D

E

F

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1 ft

3 ft

B

= 16 sq in.4 in. x 8 in.2

04C04.indd 114 4/21/10 9:51:54 AM

Tyler installed new siding on the front of a 4. house. How much area did he cover, to the nearest tenth?

In a city, condominium fees are often 5. based on the floor area of the condo. Linda chose this floor plan. If the monthly condo fees are $0.31/sq ft, how much should Linda expect to pay each month?

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15 ft

3 ft

16 ft

34

4 ft

6

18 ft12

Each window measures 2 ft by 2 ft.12

Diameter of circular window = 1 ft.12

ft

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26 ft

5 ft

8 ft

10 ft

8 ft

32 ft

28 ft

28 ft12

14

Area of rectangle – area of windows &door

18.5 ft x 15 ft = 277.5 sq ft

4(2.5 ft x 2 ft) = 20 sq ft

3 ft x 6.75 ft = 20.25 sq ft

277.5 sq ft – 20 sq ft – 20.25 sq ft

= 237.25 sq ft

Area of triangle – area of circle

= 12(16 ft x 4 ft) – π x (1.5 ft 2)2

32 sq ft – 1.7671 sq ft =· 30.23 sq ft

Total area = 237.25 +30.23

= 267.48 sq ft

Tyler covered 267.5 sq ft.

Area of rectangle:

40 ft x 28 ft = 1120 sq ft

Area of regular octagon:

8 x 12(8 ft x 10 ft) = 320 sq ft

Area of trapezoid: 12 (26

14 ft +32 ft) x 5 1

2 ft

= 0.5(26.25 ft +32 ft) x 5.5 ft

=· 160 sq ft

Total area: 1120 sq ft +320 sq ft +160 sq ft

= 1600 sq ft

Cost each month: 1600 sq ft x $0.31/sq ft = $496

04C04.indd 115 4/21/10 9:51:54 AM

Surface Area of 3-D Objects4 .10

Imagine cutting apart boxes so that you can put them in a recycling bin. How much cardboard was used to make this triangular prism? (Do not include any fl aps.)

1 What faces are in the net?

5 faces: congruent triangles

(for the ends) and congruent

rectangles (for the sides)

2 What is the surface area of the prism?

Surface area

= (area of triangle) × 2 + (area of rectangle) ×

Try TheseMatch each package shape with its net.You will need

• a paper cone• scissors• a ruler

chocolate

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134 in.

2 in.

6 in.

face

a 2-D shape that forms a fl at surface of a 3-D object

surface area

the sum of all the areas of the faces of a 3-D object


= 2 x 12 (2 in. x 1.75 in.) +3 x (6 in. x 2 in.)

= 3.5 sq in. +36 sq in.

The amount of cardboard is the surface area, 39.5 sq in.

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spaghetti tea juice

3

2

3

04C04.indd 116 4/21/10 9:51:56 AM

ExampleHow much cardboard is needed to make this cone?

SolutionWhat is the area of the curved surface?A. Area of large circle:

Area of curved surface = 14 area of large circle

What is the area of the small circle? B.

What is the total area? C.

The area for the part of the large circle can be calculated using this formula: r

s × π s2 = π rs, where r is the radius of thesmall circle, and s is the radius of the large circle. s is also the slant height of the cone.

What is the area of the curved surface, using the formula D. πrs?

PracticeCalculate the surface area of each 3-D object.1. a) b)

slant height

the distance from the top to the base, at a right angle, along a slanted side of a pyramid or cone

REFLECTINGIf you made the cone taller, how

would the areas of the circles change?

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9.0 m

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54 in.

37 in.

• When the base of a pyramid is a regular polygon, you can use the slant height as the height of each triangular face.

• Surface area of a closed cone = π r 2 + π rs, where r = radius and s = slant height

Hints

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s = 8 in.

slantheights = 8 in.

r = 2 in.

r = 2 in.


π(8 in.)2 = 201.061… sq in.

0.25 x 201.061… sq in. = 50.265… sq in.

π(2 in.)2 = 12.566… sq in.

50.27 sq in. +12.57 sq in. = 62.8 sq in.

It’s the same, 50.265… sq in.

(9.0 m)2 = 81 m2

4[(9.0 m x 14.4 m) 2]

= 259.2 m2

Total: 340.2 m2

π(37 in.)2 =· 4300 sq in.

π(37 in.)(54 in.)

=· 6277 sq in.

Total: 10 578 sq in.

04C04.indd 117 4/21/10 9:51:57 AM

Which 3-D object requires more 2. stained glass to make, the hexagonal prism at the left or the pentagonal pyramid at the right? Show how you know.

Hexagonal prism:

Area of regular hexagon

Area of rectangle

Pentagonal pyramid:

Area of regular pentagon

Area of triangle

Surface area of hexagonal prism:

Surface area of pentagonal pyramid:

The surface area of a Rubik’s cube is 54 sq in. 3. What are the dimensions of each face? Explain your thinking.

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20 cm

17 cm

10 cm

16 cm

24 c

m

16 cm 11 cm

24 cm

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17 cm

10 cm

16 cm

24 c

m

16 cm 11 cm

24 cm


e.g., The net of a cube has 6 faces.

54 sq in. 6 faces = 9 sq in. for each face

Each face is 3 in. by 3 in.

= 20 cm x 10 cm

= 200 cm2

The hexagonal prism has the greater surface area, so it will take

more stained glass to make.

= 6 x 12 (20 cm x 17 cm)

= 1020 cm2

= 5 x 12 (16 cm x 11 cm)

= 440 cm2

= 192 cm2

= 12 (16 cm x 24 cm)

2(area of regular hexagon)

+6(area of rectangle)

= 2(1020 cm2) +6(200 cm2)

= 3240 cm2

(area of regular pentagon)

+5(area of triangle)

= (440 cm2) +5(192 cm2)

= 1400 cm2

04C04.indd 118 4/21/10 9:51:58 AM

Trish wrapped two gifts with dimensions as shown. 4. Which gift required more wrapping paper? Show how you know.

Surface area of rectangular prism:

Surface area of cylinder:

Area of curved side (rectangle) = circumference of circle × height

× cm × cm =̇ cm2

Surface area of cylinder = area of circular ends + area of curved side

Use the formula below to calculate the surface area of the 5. cylinder in Question 4. 2πr 2 + 2πrh, where r = radius and h = height

Manny wants to make an enclosed tent with these 6. dimensions. The two sides and the bottom are rectangular. The ends are triangular.

How much canvas fabric would he have to buy?a)

If canvas costs $4.89/mb) 2, how much will the fabric cost to make the tent?

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20 cm

10 cm

15 cm

30 cm

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3 yd

5 yd3 yd12

3 yd12


2 x 20 cm x 10 cm = 400 cm2

2 x 30 cm x 10 cm = 600 cm2

2 x 30 cm x 20 cm = 1200 cm2

Total: 2200 cm2

Area of circular ends: 2 x π x (7.5 cm)2 =· 353.43 cm2

353.43 cm2 +1413.72 cm2 =· 1767.15 cm2

The rectangular gift required more wrapping paper.

2 x π x (7.5)2 +2 x π x 7.5 x 30 =· 1767.15

π 15 30 1413.72

e.g., 1 sq yd =· 0.836 m2

63 sq yd x 0.836 m2/yd = 52.668 m2

52.668 m2 x $4.89/m2 =· $257.55

e.g., Area of bottom: 3.5 yd x 5 yd = 17.5 sq yd

Area of triangle: 12 (3 yd x 3.5 yd) x 2 = 10.5 sq yd

Area of sides: 3.5 yd x 5 yd x 2 = 35 sq yd

Surface area: 63 sq yd

04C04.indd 119 4/21/10 9:51:58 AM

Almira is tiling the walls around a bathtub. Small bathroom 1. tiles are 1 sq in., and 144 tiles are placed on each sheet of mesh. (There’s a space between the tiles for the grout to fi ll.)

Estimate the area of one mesh sheet in square feet. a) about

How many sheets does Almira need to cover the walls? b)

Express each area in the units given. If necessary, round your 2. answer to the nearest unit.

the fi eld inside an Olympic track, 1 ha: a) m2

the fenced area around a working oil rig, 11 000 sq ft: b) sq yd

a fi nished jigsaw puzzle, 16 in. by 20 in.: c) sq in. or cm2

the square base of the world’s tallest building (in 2009),d) the Burj Khalifa, 3 595 100 sq ft:

m2 or ha

What is the radius of the circle, to the nearest tenth? 3.

Corbett is a diamond cutter. He cut a diamond so that the 4. face is a regular hexagon. Is the area of the face about 1 cm2? Explain your thinking.

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6' 6

"

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Area of circle= 324 sq in.

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5 mm

4.3 mm

Use the charts and formulas inside the back cover.

Hint

Chapter


Area: 2(6.5 ft x 3.5 ft) +6.5 ft x 6 ft = 84.5 sq ft

She needs about 85 sheets.

1 sq ft

10 000

1222

320 2065

323 559 32

e.g., 324 sq in. = πr 2

324 sq in. π = r 2

√(324 π) = r

The radius is about 10.2 in.

e.g., Area of regular hexagon:

6 x 12 (0.5 cm x 0.215 cm) =· 0.3 cm2

The face of the diamond is less than half a square centimetre.

04C04.indd 120 4/21/10 9:51:59 AM

On paper, draw a polygon that has an irregular shape. Explain 5. how you can determine its area. (Don’t do the calculations.)

a)6. Rayza is building a deck. What is the area of the deck?

b) Rayza wants to put one coat of stain on the deck. A can of stain covers about 15 m2 and costs $27.99. About how much will she pay for the cans of stain?

What is the surface area, to one decimal place? 7.

an ice cream cone, top opena)

a gas storage tank, top closedb)

The shaded area on the map shows the oil 8. sands in Alberta. (1 square represents 2500 sq mi.)

Estimate the area of the oil sands.

about sq mi

about km2

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4 m

9 m

6 m

7 m

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Edmonton

ALBERTA

SA

SK

ATCH

EWA

N

GrandePrairie

FortMcMurray

FortMacKay

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9.4 cm

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6 yd

7 yd12


69 m2 15 m2/can = 4.6 cans

5 cans x $28 = $140

She will pay about $140.

50 000

129 500

1. Divide it into triangles.

2. Measure the base and height of each triangle and

calculate Area of triangle = 12 (base x height).

3. Add all the areas.

Area of trapezoid: 12 (4 m +7 m) x 6 m = 33 m2

Area of rectangle: 36 m2

Total area is 69 m2.

Surface area of open cone:

π(4.7 cm)(16.6 cm) =· 245.1 cm2

Circumference: π(12 yd) = 37.699… yd

Surface area: 2 x π(6 yd)2 +(7.5 yd x 37.699… yd)

=· 508.9 sq yd

e.g.,

04C04.indd 121 4/21/10 9:52:00 AM

Complete. If necessary, round the area to the nearest unit.1.

a sheet of drywall, 4 ft by 8 ft: a) sq ft or m2

the square base of the Great Pyramid of Giza in Egypt, b) 230.37 m each side: about m2 or ha

What is the area of this 2. irregular-shaped piece of plastic tarp, rounded to the nearest square foot?

What is the surface area of the metal can?3.

Jen will paint two coats of paint on the outside of her dog’s 4. house. What is the surface area that she will paint?

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5 ' 5'

8'

8' 2'

1 2

34

A

B

Use the charts and formulas inside the back cover.

Hint

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5 cm

20 cm

Chapter

Copyright © 2011 by Nelson Education Ltd. Apprenticeship and Workplace 10122

32 3

53 070 5

e.g., Area of trapezoid:12 (2 ft +8 ft) x 5 ft = 25 sq ft

Area of triangle A: 12 (2.5 ft x 8 ft) = 10 sq ft

Area of triangle B: 12 (5.75 ft x 5.75 ft) =· 16.53 sq ft

Total: 25 sq ft +10 sq ft +16.53 sq ft = 51.53 sq ft

The area of the tarp is about 52 sq ft.

e.g., Area of circles: π(5 cm)2 x 2 = 157.079… cm2

Area of curved part: π(10 cm) (20 cm) =· 628.32

Surface area of can: 628.32 cm2 +157.08 cm2 =· 785.4 cm2

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25"

20"36" 10"

20"

540 sq in.540 sq in.

100 sq in.

300 sq in.100 sq in.

39 sq in.

10"

36"

e.g., Surface area =· 1741 sq in., or 12.1 sq ft

Jen will paint two coats, so she will paint an area of

3482 sq in. or 24.2 sq ft.

04C04.indd 122 4/21/10 9:52:00 AM

try these - itismath.comitismath.com/wp-content/uploads/2015/02/ch4-answers.pdftry these aw10...

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