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3 APPLY
ASSIGNMENT GUIDE
GUIDED PRACTICE
Vocabulary Check
✓
A
1. Identify the center of polygon ABCDE. J
BASIC
B
5.88
2. Identify the radius of the polygon. 5
Day 1: pp. 672–673 Exs. 9–29
Day 2: pp. 673–675 Exs. 30–44,
48–52, 54–64
K
3. Identify a central angle of the polygon. ™BJC
Concept Check
✓
Skill Check
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AVERAGE
Day 1: pp. 672–673 Exs. 9–29
Day 2: pp. 673–675 Exs. 30–52,
54–64
In Exercises 1–4, use the diagram shown.
ADVANCED
4. Identify a segment whose length is the apothem.
J
4.05
E
5
Æ
JK C
5. In a regular polygon, how do you find the
D
measure of each central angle? Divide 360° by the number of sides.
3
9 苶
4
6. What is the area of an equilateral triangle with 3 inch sides? }兹} ≈ 3.9 in.2
STOP SIGN The stop sign shown is a regular
octagon. Its perimeter is about 80 inches and its
height is about 24 inches.
Day 1: pp. 672–673 Exs. 9–29
Day 2: pp. 673–675 Exs. 30–64
BLOCK SCHEDULE
7. What is the measure of each central angle?
pp. 672–675 Exs. 9–52, 54–64
45°
8. Find the apothem, radius, and area of the stop sign.
about 12 in., about 13 in., about 480 in.2
EXERCISE LEVELS
Level A: Easier
9–15
Level B: More Difficult
16–52
Level C: Most Difficult
53
PRACTICE AND APPLICATIONS
FINDING AREA Find the area of the triangle.
STUDENT HELP
Extra Practice
to help you master
skills is on p. 823.
9.
10.
5
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 10,
18, 20, 22, 26, 28, 32, 38, 44. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 11
Resource Book, p. 40)
•
Transparency (p. 81)
11.
5
7兹5
11
5
25兹3苶
}} ≈ 10.8 sq. units
4
121兹3苶
}} ≈ 52.4 sq. units
4
7兹5
245兹3苶
}} ≈ 106.1 sq. units
4
MEASURES OF CENTRAL ANGLES Find the measure of a central angle of a
regular polygon with the given number of sides.
12. 9 sides
13. 12 sides
40°
14. 15 sides
30°
15. 180 sides
24°
2°
FINDING AREA Find the area of the inscribed regular polygon shown.
16. A
B
17.
STUDENT HELP
D
D
C
C
Example 1: Exs. 9–11, 17,
19, 25, 33
Example 2: Exs. 9–11, 17,
19, 25, 33
Example 3: Exs. 12–24, 26,
34
Example 4: Exs. 34, 45–49
G
20
E
C
10兹3
B
E
D
108兹3苶 ≈ 187.1 sq. units
600兹3苶 ≈ 1039.2 sq. units
PERIMETER AND AREA Find the perimeter and area of the regular polygon.
19.
20.
21.
10
30兹3苶 ≈ 52.0 units; 75兹3苶 ≈ 129.9 sq. units
672
B
F
12
128 sq. units
HOMEWORK HELP
A
6
4兹2
F
18.
A
E
8
672
7兹5
Chapter 11 Area of Polygons and Circles
4
16兹2苶 units; 32 sq. units
15
150 tan 36° ≈ 109.0 units;
1125 tan 36° ≈ 817.36 sq. units
PERIMETER AND AREA In Exercises 22–24, find the perimeter and area of
the regular polygon.
147 3苶
兹
22. 42 units; }}
≈
22.
23.
24.
2
127. 31 sq. units
23. 176 sin 22.5° ≈ 67.35 units;
9
968(sin 22.5°)(cos 22.5°) ≈
11
342.24 sq. units
7
24. 216 sin 15° ≈ 55.9 units;
972(sin 15°)(cos 15°) =
243 sq. units
25. AREA Find the area of an equilateral triangle that has a height of 15 inches.
75兹3苶 ≈ 129.9 in.2
27. True; let † be the measure
26.
A
REA Find the area of a regular dodecagon (or 12-gon) that has 4 inch sides.
of a central angle, n the
48(tan 75°) ≈ 179.14 in.2
number of sides, r the
LOGICAL REASONING Decide whether the statement is true or false.
radius, and P the perimeter. Explain your choice.
As n grows bigger † will
become smaller, so the
27. The area of a regular polygon of fixed radius r increases as the number of
apothem, which is given by
sides increases. See margin.
†
by r cos }} will get larger.
2
28. The apothem of a regular polygon is always less than the radius. See margin.
The perimeter of the
polygon, which is given
29. The radius of a regular polygon is always less than the side length.
†
False; for example, the radius of a regular hexagon is equal to the side length.
by n 2rsin }} will grow
2
larger, too. Although the AREA In Exercises 30–32, find the area of the regular polygon. The area of
factor involving the sine the portion shaded in red is given. Round answers to the nearest tenth.
will get smaller, the
30. Area = 16兹3
苶
31. Area = 4 tan 67.5°
32. Area = tan 54°
increase in n more than
makes up for it.
Consequently, the area,
1
q
which is given by }}aP
q
2
will increase.
q
28. True; the apothem is a leg
length in a right ¤ of
32 tan 67.5° ≈ 77.3
5 tan 54° ≈ 6.9
96兹3苶 ≈ 166.3
which the radius is the
33. USING THE AREA FORMULAS Show that the area of a regular hexagon is
length of the hypotenuse.
冉
冊
six times the area of an equilateral triangle with the same side length.
冉
冉
TEACHING TIPS
EXERCISES 16–21 Students
may come up with slightly different
answers to these problems. To get
the most accurate answer possible,
have students carry trigonometric
expressions or radical expressions
through until the very last step, then
round, as is shown in Example 3 on
page 671.
COMMON ERROR
EXERCISES 22–24 Students
might use the central angle when
using trigonometry to find the
apothem and side length. Point out
that the acute angle in the triangle
formed by a radius and a segment
whose length is the apothem is half
the measure of the central angle of
the polygon. Also, students might
forget to double the length of the
leg of this triangle to get the side
length. This error would result in an
area that is half the actual area of
the polygon.
冊冊
33. Let s = the length of a side
1
1
of the hexagon and of the
Hint: Show that for a hexagon with side lengths s, ᎏᎏ aP = 6 • ᎏᎏ兹3苶s 2 .
2
4
equilateral triangle. The
apothem of the hexagon
34.
BASALTIC COLUMNS Suppose the top of one of the columns along the
1
is }}兹3苶s and the perimeter
Giant’s Causeway (see p. 659) is in the shape of a regular hexagon with a
2
of the hexagon is 6s. The
diameter of 18 inches. What is its apothem? 4.5兹3苶 ≈ 7.8 in.
area of the hexagon, then, is
1
1 1
A = }}aP = }} }}兹3苶s • 6s,
CONSTRUCTION In Exercises 35–39, use a straightedge and a
2
2 2
3
or }}兹3苶s2. The area of an compass to construct a regular hexagon and an equilateral triangle.
2
35–37. Check drawings.
Æ
equilateral triangle with 35. Draw AB with a length of 1 inch. Open the
side length s is
compass to 1 inch and draw a circle with that radius.
A
B
1
A = ᎏᎏ兹3苶s2. Six of these
4
36. Using the same compass setting, mark off equal
equilateral triangles
parts along the circle.
together (forming the
hexagon), then, would have 37. Connect the six points where the compass marks
s2 3苶
3s2兹3苶
and circle intersect to draw a regular hexagon.
area 6 • }兹} = }}
.
冉
4
冊
2
The two results are the
same.
3 3苶
兹
38. What is the area of the hexagon? }2} ≈ 2.60 in.2
39.
Writing
Explain how you could use this
construction to construct an equilateral triangle.
Draw segments connecting 3 of the compass marks, skipping every second mark.
11.2 Areas of Regular Polygons
673
673
STUDENT HELP
INT
STUDENT HELP NOTES
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell for
help with construction in
Exs. 40–44.
Homework Help Students can
find help for Exs. 40–44 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
CONSTRUCTION In Exercises 40–44, use a straightedge and a compass
to construct a regular pentagon as shown in the diagrams below.
40–44. Check drawings.
A
CAREER NOTE
EXERCISES 45 AND 46
Additional information about
astronomers is available at
www.mcdougallittell.com.
q
B
A
Exs. 40, 41
q
B
q
A
Ex. 42
Exs. 43, 44
Æ
40. Draw a circle with center Q. Draw a diameter AB. Construct the perpendicular
Æ
bisector of AB and label its intersection with the circle as point C.
ENGLISH LEARNERS
EXERCISES 45–46
Understanding terminology in word
problems can be difficult for English
learners. For Exercises 45–46,
students may have difficulty understanding what a primary mirror is,
and thus have trouble interpeting
the photograph.
Æ
41. Construct point D, the midpoint of QB.
42. Place the compass point at D. Open the compass to the length DC and draw
Æ
Æ
an arc from C so it intersects AB at a point, E. Draw CE.
43. Open the compass to the length CE. Starting at C, mark off equal parts along
the circle.
44. Connect the five points where the compass marks and circle intersect to draw
a regular pentagon. What is the area of your pentagon?
FOCUS ON
CAREERS
Check results.
TELESCOPES In Exercises 45 and 46, use the following information.
The Hobby-Eberly Telescope in Fort Davis,
Texas, is the largest optical telescope in
North America. The primary mirror for the
telescope consists of 91 smaller mirrors
forming a hexagon shape. Each of the
smaller mirror parts is itself a hexagon with
side length 0.5 meter.
45. What is the apothem of one of the
smaller mirrors? }1}兹3苶 ≈ 0.43 m
4
46. Find the perimeter and area of one of
the smaller mirrors. 3.0 m, about 0.65 m 2
RE
• Practice Levels A, B, and C
(Chapter 11 Resource Book,
p. 30)
• Reteaching with Practice
(Chapter 11 Resource Book,
p. 33)
•
See Lesson 11.2 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
674
ASTRONOMERS
use physics and
mathematics to study the
universe, including the sun,
moon, planets, stars, and
galaxies.
INT
For Lesson 11.2:
FE
L
AL I
ADDITIONAL PRACTICE
AND RETEACHING
B
NE
ER T
CAREER LINK
www.mcdougallittell.com
TILING In Exercises 47–49, use the following information.
You are tiling a bathroom floor with tiles that are regular hexagons, as shown.
Each tile has 6 inch sides. You want to choose different colors so that no two
adjacent tiles are the same color.
47. What is the minimum number of colors that
you can use? 3 colors
48. What is the area of each tile? 54兹3苶 ≈ 93.5 in.2
49. The floor that you are tiling is rectangular. Its
width is 6 feet and its length is 8 feet. At least
how many tiles of each color will you need?
about 25 tiles
674
Chapter 11 Area of Polygons and Circles
Test
Preparation
QUANTITATIVE COMPARISON In Exercises 50–52, choose the statement
that is true about the given quantities.
¡
B
¡
C
¡
D
¡
A
DAILY HOMEWORK QUIZ
The quantity in column B is greater.
The relationship cannot be determined from the given information.
Column B
r
B
1
1
s
P
N
1
9
9
81兹3苶
} ≈ 35.1 square units
4
q
1
50.
m™APB
m™MQN
B
51.
Apothem r
Apothem s
A
52.
Perimeter of octagon with center P
Perimeter of heptagon with center Q
A
A
A
B
14
5
of ABCDE by using two methods. First, use the
1
2
2. Find the measure of the
central angle of a regular
polygon with 10 sides. 36°
3. Find the area of the inscribed
regular polygon shown.
P
53. USING DIFFERENT METHODS Find the area
1
2
B
E
formula A = ᎏᎏaP, or A = ᎏᎏ a • ns. Second, add
www.mcdougallittell.com
9
M
A
EXTRA CHALLENGE
Transparency Available
1. Find the area of the triangle.
The two quantities are equal.
Column A
★ Challenge
4 ASSESS
The quantity in column A is greater.
C
P
the areas of the smaller polygons. Check that
both methods yield the same area.
E
D
392 square units
4. Find the area of a regular
hexagon that has 4-inch sides.
C
D
1
1
1
1
}}aP ≈ }}(3.44)(25) ≈ 43; }}a • ns ≈ }}(3.44)(5)(5) ≈ 43;
2
2
2
2
area of †ABE + area of EDCB ≈ 11.9 + 31.1 ≈ 43
7 2
24兹3苶 ≈ 41.6
MIXED REVIEW
EXTRA CHALLENGE NOTE
SOLVING PROPORTIONS Solve the proportion. (Review 8.1 for 11.3)
x
11 11
54. ᎏᎏ = ᎏᎏ }2}
6
12
12
13
56. ᎏᎏ = ᎏᎏ º91
x+7
x
20
15
55. ᎏᎏ = ᎏᎏ 3
4
x
USING SIMILAR POLYGONS In the diagram shown,
¤ABC ~ ¤DEF. Use the figures to determine
whether the statement is true. (Review 8.3 for 11.3)
AC
DF
DF
EF + DE + DF
58. ᎏᎏ = ᎏᎏ true 59. ᎏ = ᎏᎏ
BC
EF
AC
BC + AB + AC
Æ
Æ
º33
x
x+6
57. ᎏᎏ = ᎏᎏ
11
9
A
D
4
3
true
B
60. ™B £ ™E true 61. BC £ EF false
E
C
F
FINDING SEGMENT LENGTHS Find the value of x. (Review 10.5)
62. 6
63. 7
x
7
9
x
12
14
8
10
64. 12
8
4
x
11.2 Areas of Regular Polygons
675
Challenge problems for
Lesson 11.2 are available in
blackline format in the Chapter 11
Resource Book, p. 37 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. OPEN ENDED Draw a regular
polygon and show how to find its
area. Check student’s work.
2. WRITING Explain why you
might use the Pythagorean
Theorem when finding the area
of a regular polygon.
Sample answer: If you know the
radius and side length of the
regular polygon, you can use the
Pythagorean Theorem with the
radius and half the side length to
find the apothem.
675