Chapter 11 - Big Ideas Learning

11
11.1
11.2
11.3
11.4
Circumference and Area
Circumference and Arc Length
Areas of Circles and Sectors
Areas of Polygons and Composite Figures
Effects of Changing Dimensions
P t
((p. 628)
Posters
Basaltic Columns (p.
(p 621)
SEE the Big Idea
(p 618)
Table Top (p.
Population Density (p
(p. 607)
London Eye (p. 603)
Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems
arising in everyday life, society, and the workplace.
Maintaining Mathematical Proficiency
Finding Areas of Two-Dimensional Figures
Example 1
a.
(6.8.D)
Find the area of each figure.
b.
15.8 m
3.35 in.
8.4 m
1.4 in.
11.1 m
A = —12 h(b1 + b2)
=
1
—2 (8.4)(15.8
Write area formula.
+ 11.1)
A = bh
= (3.35)(1.4)
Substitute.
= 112.98
= 4.69
Simplify.
The area is 112.98 square meters.
The area is 4.69 square inches.
Find the area of the figure.
1.
2.
3.
36 cm
28.8 cm
3.9 ft
8.6 ft
5
12 8 yd
5
12 8 yd
21.6 cm
Finding a Missing Dimension
(A.5.A)
Example 2 A rectangle has a perimeter of 10 meters and a length of 3 meters. What is the
width of the rectangle?
P = 2ℓ + 2w
Write formula for perimeter of a rectangle.
10 = 2(3) + 2w
Substitute 10 for P and 3 for ℓ.
10 = 6 + 2w
Multiply 2 and 3.
4 = 2w
Subtract 6 from each side.
2=w
Divide each side by 2.
The width is 2 meters.
Find the missing dimension.
4. A rectangle has a perimeter of 28 inches and a width of 5 inches. What is the length of
the rectangle?
5. A triangle has an area of 12 square centimeters and a height of 12 centimeters. What is
the base of the triangle?
6. A rectangle has an area of 84 square feet and a width of 7 feet. What is the length of the
rectangle?
7. ABSTRACT REASONING How is the area formula for a parallelogram derived from the area
formula for a rectangle?
595
Mathematical
Thinking
Mathematically proficient students analyze mathematical relationships
to connect and communicate mathematical ideas. (G.1.F)
Solving a Simpler Form of a Problem
Core Concept
Composite Figures and Area
A composite figure is a figure that consists of triangles, squares, rectangles, and other
two-dimensional figures. To find the area of a composite figure, separate it into figures
with areas you know how to find. Then find the sum of the areas of those figures.
triangle
trapezoid
Finding the Area of a Composite Figure
Find the area of the composite figure.
10.2 cm
SOLUTION
The figure consists of a parallelogram and a rectangle.
Area of parallelogram
11.7 cm
Area of rectangle
A = bh
A =ℓw
23.1 cm
= (23.1)(10.2)
= (23.1)(11.7)
= 235.62
= 270.27
The area of the figure is 235.62 + 270.27 = 505.89 square centimeters.
Monitoring Progress
Find the area of the composite figure.
24 m
1.
2.
3.
6 in.
3.4 ft
18 in.
10.2 m
5.2 ft
8 in.
10.2 m
4.4 ft
6 in.
9.8 ft
596
Chapter 11
Circumference and Area
11.1
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
G.12.B
G.12.D
Circumference and Arc Length
Essential Question
How can you find the length of a circular arc?
Finding the Length of a Circular Arc
Work with a partner. Find the length of each red circular arc.
a. entire circle
b. one-fourth of a circle
5
y
5
3
1
−5
−3
3
1
A
−1
1
3
−3
B
1
−3
−5
−5
3
5x
d. five-eighths of a circle
y
y
4
4
2
2
−2
A
−1
−3
A
−4
−5
5x
c. one-third of a circle
C
y
C
A
B
2
4
−4
x
−2
−2
C
−4
B
2
4
x
−2
−4
Writing a Conjecture
ANALYZING
MATHEMATICAL
RELATIONSHIPS
To be proficient in math,
you need to notice if
calculations are repeated
and look both for general
methods and for shortcuts.
Work with a partner. The rider is attempting to stop with
the front tire of the motorcycle in the painted rectangular
box for a skills test. The front tire makes exactly
one-half additional revolution before stopping.
The diameter of the tire is 25 inches. Is the
front tire still in contact with the
painted box? Explain.
3 ft
Communicate Your Answer
3. How can you find the length of a circular arc?
4. A motorcycle tire has a diameter of 24 inches. Approximately how many inches
does the motorcycle travel when its front tire makes three-fourths of a revolution?
Section 11.1
Circumference and Arc Length
597
11.1 Lesson
What You Will Learn
Use the formula for circumference.
Core Vocabul
Vocabulary
larry
Use arc lengths to find measures.
circumference, p. 598
arc length, p. 599
radian, p. 601
Measure angles in radians.
Solve real-life problems.
Using the Formula for Circumference
Previous
circle
diameter
radius
The circumference of a circle is the distance around the circle. Consider a regular
polygon inscribed in a circle. As the number of sides increases, the polygon
approximates the circle and the ratio of the perimeter of the polygon to the diameter
of the circle approaches π ≈ 3.14159. . ..
For all circles, the ratio of the circumference C to the diameter d is the same. This
C
ratio is — = π. Solving for C yields the formula for the circumference of a circle,
d
C = πd. Because d = 2r, you can also write the formula as C = π(2r) = 2πr.
Core Concept
Circumference of a Circle
r
The circumference C of a circle is C = πd
or C = 2πr, where d is the diameter of the
circle and r is the radius of the circle.
d
C
C = π d = 2π r
Using the Formula for Circumference
USING PRECISE
MATHEMATICAL
LANGUAGE
Find each indicated measure.
a. circumference of a circle with a radius of 9 centimeters
You have sometimes used
3.14 to approximate the
value of π. Throughout this
book, you should use the
π key on a calculator, then
round to the hundredths
place unless instructed
otherwise.
b. radius of a circle with a circumference of 26 meters
SOLUTION
a. C = 2πr
⋅ ⋅
=2 π 9
= 18π
≈ 56.55
The circumference is about
56.55 centimeters.
Monitoring Progress
b.
C = 2πr
26 = 2πr
26
2π
—=r
4.14 ≈ r
The radius is about 4.14 meters.
Help in English and Spanish at BigIdeasMath.com
1. Find the circumference of a circle with a diameter of 5 inches.
2. Find the diameter of a circle with a circumference of 17 feet.
598
Chapter 11
Circumference and Area
Using Arc Lengths to Find Measures
An arc length is a portion of the circumference of a circle. You can use the measure of
the arc (in degrees) to find its length (in linear units).
Core Concept
Arc Length
In a circle, the ratio of the length of a given arc to the
circumference is equal to the ratio of the measure of the
arc to 360°.
Arc length of AB
m
AB
2πr
360°
A
P
r
—— = —, or
B
m
AB
Arc length of AB = — 2πr
360°
⋅
Using Arc Lengths to Find Measures
Find each indicated measure.
a. arc length of AB
8 cm
60°
P
c. m
RS
b. circumference of ⊙Z
A
4.19 in.
Z
40° Y
B
S
15.28 m
X
T
R
44 m
SOLUTION
60°
a. Arc length of AB = — 2π(8)
360°
≈ 8.38 cm
⋅
Arc length of m
RS
RS
c. —— = —
2πr
360°
Arc length of m
XY
XY
b. —— = —
C
360°
4.19
C
40°
360°
4.19
C
1
9
44
2π(15.28)
—=—
m
RS
360°
—=—
44
360° — = m
RS
2π(15.28)
⋅
—=—
37.71 in. = C
Monitoring Progress
165° ≈ m
RS
Help in English and Spanish at BigIdeasMath.com
Find the indicated measure.
3. arc length of PQ
4. circumference of ⊙N
61.26 m
Q
9 yd
75°
R
P
5. radius of ⊙G
E
G
270°
S
150°
N
L
Section 11.1
M
10.5 ft
Circumference and Arc Length
F
599
Solving Real-Life Problems
Using Circumference to Find Distance Traveled
The dimensions of a car tire are shown. To the
nearest foot, how far does the tire travel when
it makes 15 revolutions?
5.5 in.
SOLUTION
15 in.
Step 1 Find the diameter of the tire.
d = 15 + 2(5.5) = 26 in.
5.5 in.
Step 2 Find the circumference of the tire.
C = π d = π 26 = 26π in.
⋅
COMMON ERROR
Always pay attention to
units. In Example 3, you
need to convert units to
get a correct answer.
Step 3 Find the distance the tire travels in 15 revolutions. In one revolution, the tire
travels a distance equal to its circumference. In 15 revolutions, the tire travels
a distance equal to 15 times its circumference.
Distance
traveled
Number of
revolutions
=
⋅
Circumference
⋅
= 15 26π ≈ 1225.2 in.
Step 4 Use unit analysis. Change 1225.2 inches to feet.
⋅
1 ft
1225.2 in. — = 102.1 ft
12 in.
The tire travels approximately 102 feet.
Using Arc Length to Find Distances
The curves at the ends of the track shown are 180° arcs
of circles. The radius of the arc for a runner on the
red path shown is 36.8 meters. About how far does
this runner travel to go once around the track? Round
to the nearest tenth of a meter.
44.02 m
36.8 m
84.39 m
SOLUTION
The path of the runner on the red path is made of two straight sections and two
semicircles. To find the total distance, find the sum of the lengths of each part.
Distance
=
⋅
2 Length of each
straight section
(⋅ ⋅
⋅
2 Length of
each semicircle
+
= 2(84.39) + 2 —12 2π 36.8
)
≈ 400.0
The runner on the red path travels about 400.0 meters.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
6. A car tire has a diameter of 28 inches. How many revolutions does the tire make
while traveling 500 feet?
7. In Example 4, the radius of the arc for a runner on the blue path is 44.02 meters,
as shown in the diagram. About how far does this runner travel to go once around
the track? Round to the nearest tenth of a meter.
600
Chapter 11
Circumference and Area
Measuring Angles in Radians
Recall that in a circle, the ratio of the length of a given arc
to the circumference is equal to the ratio of the measure of
the arc to 360°. To see why, consider the diagram.
A circle of radius 1 has circumference 2π, so the arc
m
CD
length of CD is — 2π.
360°
A
C
r
⋅
1
D
B
Recall that all circles are similar and corresponding lengths
of similar figures are proportional. Because m
AB = m
CD ,
AB and CD are corresponding arcs. So, you can write the following proportion.
Arc length of AB
r
1
—— = —
Arc length of CD
Arc length of AB = r Arc length of CD
⋅
m
CD
Arc length of AB = r ⋅ — ⋅ 2π
360°
This form of the equation shows that the arc length associated with a central angle
mCD
is proportional to the radius of the circle. The constant of proportionality, — 2π,
360°
is defined to be the radian measure of the central angle associated with the arc.
⋅
In a circle of radius 1, the radian measure of a given central angle can be thought of
as the length of the arc associated with the angle. The radian measure of a complete
circle (360°) is exactly 2π radians, because the circumference of a circle of radius 1
is exactly 2π. You can use this fact to convert from degree measure to radian measure
and vice versa.
Core Concept
Converting between Degrees and Radians
Degrees to radians
Multiply degree measure by
2π radians
360°
π radians
180°
—, or —.
Radians to degrees
Multiply radian measure by
360°
2π radians
180°
π radians
—, or —.
Converting between Degree and Radian Measure
a. Convert 45° to radians.
3π
b. Convert — radians to degrees.
2
SOLUTION
π radians π
a. 45° — = — radian
180°
4
⋅
π
So, 45° = — radian.
4
Monitoring Progress
8. Convert 15° to radians.
Section 11.1
3π
180°
b. — radians — = 270°
2
π radians
⋅
3π
So, — radians = 270°.
2
Help in English and Spanish at BigIdeasMath.com
4π
3
9. Convert — radians to degrees.
Circumference and Arc Length
601
11.1 Exercises
Tutorial Help in English and Spanish at BigIdeasMath.com
Vocabulary and Core Concept Check
1. WRITING Describe the difference between an arc measure and an arc length.
2. WHICH ONE DOESN’T BELONG? Which phrase does not belong with the other three? Explain
your reasoning.
π times twice the radius
the distance around a circle
π times the diameter
the distance from the center to any point on the circle
Monitoring Progress and Modeling with Mathematics
12. ERROR ANALYSIS Describe and correct the error in
In Exercises 3–10, find the indicated measure.
(See Examples 1 and 2.)
finding the length of GH .
✗
3. circumference of a circle with a radius of 6 inches
4. diameter of a circle with a circumference of 63 feet
Arc length of GH
= mGH 2πr
= 75 2π(5)
= 750π cm
G
75°
H
C 5 cm
5. radius of a circle with a circumference of 28π
⋅
⋅
6. exact circumference of a circle with a diameter of
13. PROBLEM SOLVING A measuring wheel is used to
5 inches
7. arc length of AB
C
P
8 ft
calculate the length of a path. The diameter of the
wheel is 8 inches. The wheel makes 87 complete
revolutions along the length of the path. To the nearest
foot, how long is the path? (See Example 3.)
8. m
DE
D
A
45°
Q
8.73 in.
E
B
9. circumference of ⊙C
your bicycle is 32.5 centimeters. You ride 40 meters.
How many complete revolutions does the front
wheel make?
10. radius of ⊙R
L
F
76°
7.5 m
14. PROBLEM SOLVING The radius of the front wheel of
10 in.
C
38.95 cm
260° R
In Exercises 15–18, find the perimeter of the shaded
region. (See Example 4.)
15.
M
G
6
11. ERROR ANALYSIS Describe and correct the error in
13
finding the circumference of ⊙C.
✗
602
Chapter 11
9 in.
C
C = 2πr
= 2π(9)
=18π in.
Circumference and Area
16.
6
3
3
6
17.
18.
2
90°
5
90°
5
25. x2 + y2 = 16
6
120°
90°
90°
In Exercises 25 and 26, find the circumference of the
circle with the given equation. Write the circumference
in terms of π.
5
6
26. (x + 2)2 + (y − 3)2 = 9
27. USING STRUCTURE A semicircle has endpoints
In Exercises 19–22, convert the angle measure.
(See Example 5).
(−2, 5) and (2, 8). Find the arc length of the
semicircle.
28. REASONING EF is an arc on a circle with radius r.
19. Convert 70° to radians.
Let x° be the measure of EF . Describe the effect on
the length of EF if you (a) double the radius of the
circle, and (b) double the measure of EF .
20. Convert 300° to radians.
11π
12
21. Convert — radians to degrees.
29. MAKING AN ARGUMENT Your friend claims that it is
π
22. Convert — radian to degrees.
8
possible for two arcs with the same measure to have
different arc lengths. Is your friend correct? Explain
your reasoning.
23. PROBLEM SOLVING The London Eye is a Ferris
wheel in London, England, that travels at a speed of
0.26 meter per second. How many minutes does it
take the London Eye to complete one full revolution?
67.5 m
30. PROBLEM SOLVING Over 2000 years ago, the Greek
scholar Eratosthenes estimated Earth’s circumference
by assuming that the Sun’s rays were parallel. He
chose a day when the Sun shone straight down into
a well in the city of Syene. At noon, he measured the
angle the Sun’s rays made with a vertical stick in the
city of Alexandria. Eratosthenes assumed that the
distance from Syene to Alexandria was equal to about
575 miles. Explain how Eratosthenes was able to use
this information to estimate Earth’s circumference.
Then estimate Earth’s circumference.
t
ligh
m∠2 = 7.2°
Alexandria
sun
stick
t
ligh
sun
2
well
24. PROBLEM SOLVING You are planning to plant a
circular garden adjacent to one of the corners of a
building, as shown. You can use up to 38 feet of fence
to make a border around the garden. What radius can
the garden have? Choose all that apply. Explain
your reasoning.
1
1
Syene
center
of Earth
Not drawn to scale
31. ANALYZING RELATIONSHIPS In ⊙C, the ratio of the
length of PQ to the length of RS is 2 to 1. What is the
ratio of ∠PCQ to ∠RCS?
A 4 to 1
○
B 2 to 1
○
C 1 to 4
○
D 1 to 2
○
32. ANALYZING RELATIONSHIPS A 45° arc in ⊙C and a
A 7
○
B 8
○
C 9
○
D 10
○
30° arc in ⊙P have the same length. What is the ratio
of the radius r1 of ⊙C to the radius r2 of ⊙P? Explain
your reasoning.
Section 11.1
Circumference and Arc Length
603
33. PROBLEM SOLVING How many revolutions does the
38. MODELING WITH MATHEMATICS What is the
smaller gear complete during a single revolution of
the larger gear?
measure (in radians) of the angle formed by the hands
of a clock at each time? Explain your reasoning.
a. 1:30 p.m.
3
b. 3:15 p.m.
39. MATHEMATICAL CONNECTIONS The sum of the
7
circumferences of circles A, B, and C is 63π. Find AC.
x B
3x
34. USING STRUCTURE Find the circumference of each
5x
A
circle.
C
a. a circle circumscribed about a right triangle whose
legs are 12 inches and 16 inches long
b. a circle circumscribed about a square with a side
length of 6 centimeters
c. a circle inscribed in an equilateral triangle with a
side length of 9 inches
40. THOUGHT PROVOKING Is π a rational number?
35. REWRITING A FORMULA Write a formula in terms of
the measure θ (theta) of the central angle (in radians)
that can be used to find the length of an arc of a circle.
Then use this formula to find the length of an arc of a
circle with a radius of 4 inches and a central angle of
3π
— radians.
4
355
Compare the rational number — to π. Find a
113
different rational number that is even closer to π.
41. PROOF The circles in the diagram are concentric
— ≅ GH
—. Prove that have the
and FG
JK and NG
same length.
M
36. HOW DO YOU SEE IT?
L
Compare the circumference
of ⊙P to the length of DE .
Explain your reasoning.
N
F
D
C
P
E
G
H
K
J
37. MAKING AN ARGUMENT In the diagram, the measure
of the red shaded angle is 30°. The arc length a is 2.
Your classmate claims that it is possible to find the
circumference of the blue circle without finding the
radius of either circle. Is your classmate correct?
Explain your reasoning.
r
— is divided into four
42. REPEATED REASONING AB
congruent segments, and semicircles with radius r
are drawn.
A r
a. What is the sum of the four arc lengths?
2r
a
b. What would the sum of the arc lengths be if
— was divided into 8 congruent segments?
AB
16 congruent segments? n congruent segments?
Explain your reasoning.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Find the area of the polygon with the given vertices. (Section 1.4)
43. X(2, 4), Y(8, −1), Z(2, −1)
604
Chapter 11
Circumference and Area
B
44. L(−3, 1), M(4, 1), N(4, −5), P(−3, −5)
11.2
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
Areas of Circles and Sectors
Essential Question
How can you find the area of a sector of
a circle?
G.12.C
Finding the Area of a Sector of a Circle
Work with a partner. A sector of a circle is the region bounded by two radii of the
circle and their intercepted arc. Find the area of each shaded circle or sector of a circle.
a. entire circle
8
b. one-fourth of a circle
y
8
4
−8
4
−4
4
−8
8x
−4
4
−4
−4
−8
−8
c. seven-eighths of a circle
4
y
8x
d. two-thirds of a circle
y
y
4
−4
REASONING
To be proficient in math,
you need to explain to
yourself the meaning of a
problem and look for entry
points to its solution.
4x
−4
4
x
−4
Finding the Area of a Circular Sector
Work with a partner. A center pivot irrigation system consists of 400 meters of
sprinkler equipment that rotates around a central pivot point at a rate of once every
3 days to irrigate a circular region with a diameter of 800 meters. Find the area of the
sector that is irrigated by this system in one day.
Communicate Your Answer
3. How can you find the area of a sector of a circle?
4. In Exploration 2, find the area of the sector that is irrigated in 2 hours.
Section 11.2
Areas of Circles and Sectors
605
11.2 Lesson
What You Will Learn
Use the formula for the area of a circle.
Core Vocabul
Vocabulary
larry
Use the formula for population density.
population density, p. 607
sector of a circle, p. 608
Use areas of sectors.
Previous
circle
radius
diameter
intercepted arc
Find areas of sectors.
Using the Formula for the Area of a Circle
You can divide a circle into congruent sections and
rearrange the sections to form a figure that approximates a
parallelogram. Increasing the number of congruent sections
increases the figure’s resemblance to a parallelogram.
r
C = 2π r
r
The base of the parallelogram that the figure approaches
is half of the circumference, so b = —12 C = —12 (2πr) = πr.
The height is the radius, so h = r. So, the area of the
parallelogram is A = bh = (πr)(r) = πr2.
1
C
2
Core Concept
=πr
Area of a Circle
The area of a circle is
r
A = πr 2
where r is the radius of the circle.
Using the Formula for the Area of a Circle
Find each indicated measure.
a. area of a circle with a radius of 2.5 centimeters
b. diameter of a circle with an area of 113.1 square centimeters
SOLUTION
a. A = πr2
= π • (2.5)2
= 6.25π
≈ 19.63
Formula for area of a circle
Substitute 2.5 for r.
Simplify.
Use a calculator.
The area of the circle is about 19.63 square centimeters.
b.
A = πr2
113.1 = πr2
Formula for area of a circle
113.1
— = r2
π
6≈r
Divide each side by π.
Substitute 113.1 for A.
Find the positive square root of each side.
The radius is about 6 centimeters, so the diameter is about 12 centimeters.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
1. Find the area of a circle with a radius of 4.5 meters.
2. Find the radius of a circle with an area of 176.7 square feet.
606
Chapter 11
Circumference and Area
Using the Formula for Population Density
The population density of a city, county, or state is a measure of how many people
live within a given area.
number of people
Population density = ——
area of land
Population density is usually given in terms of square miles but can be expressed using
other units, such as city blocks.
Using the Formula for Population Density
a. About 430,000 people live in a 5-mile radius of a city’s town hall. Find the
population density in people per square mile.
b. A region with a 3-mile radius has a population density of about 6195 people
per square mile. Find the number of people who live in the region.
SOLUTION
a. Step 1
Find the area of the region.
⋅
A = πr2 = π 52 = 25π
The area of the region is 25π ≈ 78.54 square miles.
Step 2 Find the population density.
number of people
Population density = —— Formula for population density
area of land
430,000
=—
25π
Substitute.
≈ 5475
Use a calculator.
The population density is about 5475 people per square mile.
b. Step 1
Find the area of the region.
⋅
A = πr2 = π 32 = 9π
The area of the region is 9π ≈ 28.27 square miles.
Step 2 Let x represent the number of people who live in the region. Find the
value of x.
number of people
Population density = ——
area of land
x
6195 ≈ —
9π
175,159 ≈ x
Formula for population density
Substitute.
Multiply and use a calculator.
The number of people who live in the region is about 175,159.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
3. About 58,000 people live in a region with a 2-mile radius. Find the population
density in people per square mile.
4. A region with a 3-mile radius has a population density of about 1000 people
per square mile. Find the number of people who live in the region.
Section 11.2
Areas of Circles and Sectors
607
Finding Areas of Sectors
A sector of a circle is the region bounded by two radii of the circle and their
—, BP
—, and intercepted arc. In the diagram below, sector APB is bounded by AP
AB .
ANALYZING
MATHEMATICAL
RELATIONSHIPS
Core Concept
The area of a sector is a
fractional part of the area
of a circle. The area of a
sector formed by a 45° arc
1
45°
is —, or — of the area of
360°
8
the circle.
Area of a Sector
The ratio of the area of a sector of a circle to the
area of the whole circle (πr2) is equal to the ratio
of the measure of the intercepted arc to 360°.
Area of sector APB
πr
A
P
AB
m
360°
r
B
= —, or
——
2
AB
m
Area of sector APB = — πr2
360°
⋅
Finding Areas of Sectors
Find the areas of the sectors formed by ∠UTV.
U
S
T
SOLUTION
70°
8 in.
V
Step 1 Find the measures of the minor and major arcs.
Because m∠UTV = 70°, m
UV = 70° and m
USV = 360° − 70° = 290°.
Step 2 Find the areas of the small and large sectors.
m
UV
Area of small sector = — πr2
360°
⋅
Formula for area of a sector
⋅ ⋅
70°
= — π 82
360°
Substitute.
≈ 39.10
Use a calculator.
m
USV
Area of large sector = — πr2
360°
⋅
⋅ ⋅
Formula for area of a sector
290°
= — π 82
360°
Substitute.
≈ 161.97
Use a calculator.
The areas of the small and large sectors are about 39.10 square inches and
about 161.97 square inches, respectively.
Monitoring Progress
Find the indicated measure.
5. area of red sector
6. area of blue sector
Help in English and Spanish at BigIdeasMath.com
F
14 ft
120° D
E
608
Chapter 11
Circumference and Area
G
Using Areas of Sectors
Using the Area of a Sector
Find the area of ⊙V.
T
40° A = 35 m2
U
V
SOLUTION
m
TU
Area of sector TVU = — Area of ⊙V
360°
40°
35 = — Area of ⊙V
360°
315 = Area of ⊙V
⋅
⋅
Formula for area of a sector
Substitute.
Solve for area of ⊙V.
The area of ⊙V is 315 square meters.
Finding the Area of a Region
A rectangular wall has an entrance cut into it. You
want to paint the wall. To the nearest square foot,
what is the area of the region you need to paint?
10 ft
16 ft
36 ft
SOLUTION
COMMON ERROR
Use the radius (8 feet),
not the diameter (16 feet),
when you calculate the area
of the semicircle.
16 ft
The area you need to paint is the area of the rectangle minus the area of the entrance.
The entrance can be divided into a semicircle and a square.
Area of wall =
Area of rectangle
[
−
⋅ ⋅
(Area of semicircle + Area of square)
180°
= 36(26) − — (π 82) + 162
360°
= 936 − (32π + 256)
]
≈ 579.47
The area you need to paint is about 579 square feet.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
7. Find the area of ⊙H.
8. Find the area of the figure.
F
A = 214.37 cm2
7m
85° H
7m
G
9. If you know the area and radius of a sector of a circle, can you find the measure of
the intercepted arc? Explain.
Section 11.2
Areas of Circles and Sectors
609
11.2 Exercises
Tutorial Help in English and Spanish at BigIdeasMath.com
Vocabulary and Core Concept Check
1. VOCABULARY A(n) __________ of a circle is the region bounded by two radii of the circle and
their intercepted arc.
2. WRITING The arc measure of a sector in a given circle is doubled. Will the area of the sector
also be doubled? Explain your reasoning.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 –10, find the indicated measure.
(See Example 1.)
3. area of ⊙C
4. area of ⊙C
In Exercises 15–18, find the areas of the sectors formed
by ∠DFE. (See Example 3.)
E
15.
10 in.
C
60°
E
C
0.4 cm
A
D
17.
5. area of a circle with a radius of 5 inches
6. area of a circle with a diameter of 16 feet
G
F 256°
14 cm
D
F
G
20 in.
16.
D
18.
137°
F 28 m
G
E
D
F
75°
4 ft
G
E
7. radius of a circle with an area of 89 square feet
8. radius of a circle with an area of 380 square inches
19. ERROR ANALYSIS Describe and correct the error in
finding the area of the circle.
9. diameter of a circle with an area of 12.6 square inches
10. diameter of a circle with an area of 676π square
centimeters
✗
A = π(12)2
C
In Exercises 11–14, find the indicated measure.
(See Example 2.)
12 ft
= 144π
≈ 452.39 ft2
11. About 210,000 people live in a region with a 12-mile
radius. Find the population density in people per
square mile.
12. About 650,000 people live in a region with a 6-mile
radius. Find the population density in people per
square mile.
13. A region with a 4-mile radius has a population density
of about 6366 people per square mile. Find the
number of people who live in the region.
14. About 79,000 people live in a circular region with a
population density of about 513 people per square
mile. Find the radius of the region.
610
Chapter 11
Circumference and Area
20. ERROR ANALYSIS Describe and correct the error in
finding the area of sector XZY when the area of ⊙Z
is 255 square feet.
✗
X
Let n be the area of
sector XZY.
W
Z
115°
Y
n
115
=
—
360 —
255
n ≈ 162.35 ft2
In Exercises 21 and 22, the area of the shaded sector is
shown. Find the indicated measure. (See Example 4.)
21. area of ⊙M
30. MAKING AN ARGUMENT Your friend claims that if
the radius of a circle is doubled, then its area doubles.
Is your friend correct? Explain your reasoning.
31. MODELING WITH MATHEMATICS The diagram shows
A = 56.87 cm2
K
50°
the area of a lawn covered by a water sprinkler.
M
J
L
22. radius of ⊙M
J
A = 12.36 m2
M
15 ft
89°
L
145°
K
In Exercises 23 –28, find the area of the shaded region.
(See Example 5.)
23.
24.
6m
a. What is the area of the lawn that is covered by
the sprinkler?
b. The water pressure is weakened so that the radius
is 12 feet. What is the area of the lawn that will
be covered?
20 in.
32. MODELING WITH MATHEMATICS The diagram shows
a projected beam of light from a lighthouse.
24 m
20 in.
25.
1 ft
26.
180°
245°
18 mi
8 cm
5 in.
27.
lighthouse
28.
3m
4m
a. What is the area of water that can be covered by
the light from the lighthouse?
b. What is the area of land that can be covered by the
light from the lighthouse?
29. PROBLEM SOLVING The diagram shows the shape of
a putting green at a miniature golf course. One part of
the green is a sector of a circle. Find the area of the
putting green.
(3x − 2) ft
5x ft
33. ANALYZING RELATIONSHIPS Look back at the
Perimeters of Similar Polygons Theorem (Theorem
8.1) and the Areas of Similar Polygons Theorem
(Theorem 8.2) in Section 8.1. How would you
rewrite these theorems to apply to circles? Explain
your reasoning.
34. ANALYZING RELATIONSHIPS A square is inscribed in
(2x + 1) ft
a circle. The same square is also circumscribed about
a smaller circle. Draw a diagram that represents this
situation. Then find the ratio of the area of the larger
circle to the area of the smaller circle.
Section 11.2
Areas of Circles and Sectors
611
35. CONSTRUCTION The table shows how students get
38. THOUGHT PROVOKING You know that the area of
a circle is πr2. Find the formula for the area of an
ellipse, shown below.
to school.
Method
Percent of
students
bus
65%
walk
25%
other
10%
b
a
a. Explain why a circle graph is appropriate for
the data.
b
a
39. MULTIPLE REPRESENTATIONS Consider a circle with
a radius of 3 inches.
b. You will represent each method by a sector of a
circle graph. Find the central angle to use for each
sector. Then construct the graph using a radius of
2 inches.
a. Complete the table, where x is the measure of the
arc and y is the area of the corresponding sector.
Round your answers to the nearest tenth.
c. Find the area of each sector in your graph.
x
36. HOW DO YOU SEE IT? The outermost edges of
30°
60°
90°
120°
150°
180°
y
the pattern shown form a square. If you know the
dimensions of the outer square, is it possible to
compute the total colored area? Explain.
b. Graph the data in the table.
c. Is the relationship between x and y linear? Explain.
d. If parts (a) –(c) were repeated using a circle with
a radius of 5 inches, would the areas in the table
change? Would your answer to part (c) change?
Explain your reasoning.
40. CRITICAL THINKING Find
C
the area between the three
congruent tangent circles.
The radius of each circle
is 6 inches.
37. ABSTRACT REASONING A circular pizza with a
12-inch diameter is enough for you and 2 friends. You
want to buy pizzas for yourself and 7 friends. A
10-inch diameter pizza with one topping costs $6.99
and a 14-inch diameter pizza with one topping costs
$12.99. How many 10-inch and 14-inch pizzas should
you buy in each situation? Explain.
A
41. PROOF Semicircles with diameters equal to three
sides of a right triangle are drawn, as shown. Prove
that the sum of the areas of the two shaded crescents
equals the area of the triangle.
a. You want to spend as little money as possible.
b. You want to have three pizzas, each with a different
topping, and spend as little money as possible.
c. You want to have as much of the thick outer crust
as possible.
Maintaining Mathematical Proficiency
Find the area of the figure.
42.
6 in.
18 in.
612
Chapter 11
Reviewing what you learned in previous grades and lessons
(Skills Review Handbook)
43.
4 ft
7 ft
10 ft
Circumference and Area
B
44.
45.
13 in.
3 ft
9 in.
5 ft
11.1–11.2
What Did You Learn?
Core Vocabulary
circumference, p. 598
arc length, p. 599
radian, p. 601
population density, p. 607
sector of a circle, p. 608
Core Concepts
Section 11.1
Circumference of a Circle, p. 598
Arc Length, p. 599
Converting between Degrees and Radians, p. 601
Section 11.2
Area of a Circle, p. 606
Population Density, p. 607
Area of a Sector, p. 608
Mathematical Thinking
1.
In Exercise 13 on page 602, why does it matter how many revolutions the wheel makes?
2.
Your friend is confused with Exercise 19 on page 610. What question(s) could you ask
your friend to help them figure it out?
3.
In Exercise 40 on page 612, explain how you started solving the problem and why you
started that way.
Study Skills
Kinesthetic Learners
Incorporate physical activity.
• Act out a word problem as much as possible. Use props
when you can.
• Solve a word problem on a large whiteboard. The physical
action of writing is more kinesthetic when the writing is
larger and you can move around while doing it.
• Make a review card.
613
11.1–11.2
Quiz
5
1. Find the circumference of a circle with a radius of 7 —8 inches. (Section 11.1)
2. Find the radius of a circle with a circumference of 30 meters. (Section 11.1)
Find the indicated measure. (Section 11.1)
3. m
EF
4. arc length of QS
13.7 m
E
Q
F
5. circumference of ⊙N
L 8 in.
M
48°
S
4 cm 83°
R
7m
G
N
5π
9
6. Convert 26° to radians and — radians to degrees. (Section 11.1)
Find the indicated measure. (Section 11.2)
7. area of a circle with a diameter of 10 yards
8. radius of a circle with an area of 38.5 square kilometers
Use the figure to find the indicated measure. (Section 11.2)
H
9. area of red sector
100°
K
J 12 yd
10. area of blue sector
L
11. Find the area of ⊙C. (Section 11.2)
A = 50 mm2
A
B
80°
C
56 ft
12 ft
12. You are using one of your school’s colors to paint around the
shaded region of the basketball court shown. Find the perimeter
of the shaded region. (Section 11.1)
3m
13. The two white congruent circles just fit into the blue circle.
What is the area of the blue region? (Section 11.2)
14. About 750,000 people live in a region with a 10-mile radius. (Section 11.2)
a. Find the population density in people per square mile.
b. The same number of people live in a region with a 20-mile radius. Is this population
density one-half of the population density you found in part (a)? Explain.
614
Chapter 11
Circumference and Area
Areas of Polygons and
Composite Figures
11.3
Essential Question
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
How can you find the area of a regular
polygon?
The center of a regular polygon is the center
of its circumscribed circle.
G.11.A
G.11.B
apothem CP
P
The distance from the center to any side of a
regular polygon is called the apothem of a
regular polygon.
C
center
Finding the Area of a Regular Polygon
Work with a partner. Use dynamic geometry software to construct each regular
polygon with side lengths of 4, as shown. Find the apothem and use it to find the
area of the polygon. Describe the steps that you used.
a.
b.
7
4
6
C
5
3
E
3
2
1
−3
−1
0
c.
1
B
0
−2
1
2
A
3
−5
−4
−3
−1
F
0
1
2
3
4
5
E
10
D
7
B
0
−2
d.
8
E
C
4
2
A
D
9
8
6
G
6
4
F
5
C
4
3
H
2
−5
−4
−3
2
1
−1
A
B
0
−2
C
3
1
A
D
7
5
0
1
2
3
4
5
−6
−5
−4
−3
B
0
−2
−1
0
1
2
3
4
5
6
Writing a Formula for Area
REASONING
To be proficient in math,
you need to know and
flexibly use different
properties of operations
and objects.
Work with a partner. Generalize the steps you used in Exploration 1 to develop a
formula for the area of a regular polygon.
Communicate Your Answer
3. How can you find the area of a regular polygon?
4. Regular pentagon ABCDE has side lengths of 6 meters and an apothem of
approximately 4.13 meters. Find the area of ABCDE.
Section 11.3
Areas of Polygons and Composite Figures
615
11.3 Lesson
What You Will Learn
Find areas of rhombuses and kites.
Find angle measures and areas of regular polygons.
Core Vocabul
Vocabulary
larry
Find areas of composite figures.
center of a regular polygon,
p. 617
radius of a regular polygon,
p. 617
apothem of a regular polygon,
p. 617
central angle of a regular
polygon, p. 617
Finding Areas of Rhombuses and Kites
You can divide a rhombus or kite with diagonals d1 and d2 into two congruent triangles
with base d1, height —12 d2, and area —12 d1 —12 d2 = —14 d1d2. So, the area of a rhombus or kite
is 2 —14 d1d2 = —12 d1d2.
(
( )
)
1
1
A = 4 d1d2
Previous
rhombus
kite
A = 4 d1d2
1
d
2 2
d2
1
d
2 2
d2
d1
d1
Core Concept
Area of a Rhombus or Kite
The area of a rhombus or kite with diagonals d1 and d2 is —12 d1d2.
d2
d2
d1
d1
Finding the Area of a Rhombus or Kite
Find the area of each rhombus or kite.
a.
b.
8m
7 cm
6m
10 cm
SOLUTION
a. A = —12 d1d2
b. A = —12 d1d2
= —12 (6)(8)
= —12 (10)(7)
= 24
= 35
So, the area is 24 square meters.
Monitoring Progress
So, the area is
35 square centimeters.
Help in English and Spanish at BigIdeasMath.com
1. Find the area of a rhombus with diagonals d1 = 4 feet and d2 = 5 feet.
2. Find the area of a kite with diagonals d1 = 12 inches and d2 = 9 inches.
616
Chapter 11
Circumference and Area
Finding Angle Measures and Areas of Regular Polygons
M
center
P
apothem
Q
PQ
N
radius
PN
∠MPN is a central angle.
The diagram shows a regular polygon inscribed in a circle. The center of a regular
polygon and the radius of a regular polygon are the center and the radius of its
circumscribed circle.
The distance from the center to any side of a regular polygon is called the apothem
of a regular polygon. The apothem is the height to the base of an isosceles triangle
that has two radii as legs. The word “apothem” refers to a segment as well as a length.
For a given regular polygon, think of an apothem as a segment and the apothem as
a length.
A central angle of a regular polygon is an angle formed by two radii drawn to
consecutive vertices of the polygon. To find the measure of each central angle,
divide 360° by the number of sides.
Finding Angle Measures in a Regular Polygon
In the diagram, ABCDE is a regular pentagon inscribed
in ⊙F. Find each angle measure.
ANALYZING
MATHEMATICAL
RELATIONSHIPS
a. m∠AFB
— is an altitude of an
FG
isosceles triangle, so
it is also a median and
angle bisector of the
isosceles triangle.
b. m∠AFG
Q
D
F
360°
A
a. ∠AFB is a central angle, so m∠AFB = — = 72°.
5
— is an apothem, which makes it an altitude of isosceles △AFB.
b. FG
E
— bisects ∠AFB and m∠AFG = —1m∠AFB = 36°.
So, FG
2
c. By the Triangle Sum Theorem (Theorem 5.1), the sum of the angle measures of
right △GAF is 180°. So, m∠GAF = 180° − 90° − 36° = 54°.
Y
Help in English and Spanish at BigIdeasMath.com
In the diagram, WXYZ is a square inscribed in ⊙P.
3. Identify the center, a radius, an apothem, and a central angle of the polygon.
P
W
B
G
SOLUTION
Monitoring Progress
X
c. m∠GAF
C
Z
4. Find m∠XPY, m∠XPQ, and m∠PXQ.
You can find the area of any regular n-gon by dividing it into congruent triangles.
⋅
A = Area of one triangle Number of triangles
READING
In this book, a point
shown inside a regular
polygon marks the center
of the circle that can be
circumscribed about
the polygon.
( ⋅ ⋅ ) ⋅n
= —12 s a
Base of triangle is s and height of
triangle is a. Number of triangles is n.
⋅ ⋅ ⋅
= —a ⋅ P
Commutative and Associative
Properties of Multiplication
There are n congruent sides of
length s, so perimeter P is n s.
= —12 a (n s)
1
2
a
s
⋅
Core Concept
Area of a Regular Polygon
The area of a regular n-gon with side length s is one-half
the product of the apothem a and the perimeter P.
⋅
A = —12 aP, or A = —12 a ns
Section 11.3
a
Areas of Polygons and Composite Figures
s
617
Finding the Area of a Regular Polygon
K
4
L
4
M
J
A regular nonagon is inscribed in a circle with a radius of 4 units. Find the area of
the nonagon.
SOLUTION
360°
— bisects the central angle,
The measure of central ∠JLK is —
, or 40°. Apothem LM
9
so m∠KLM is 20°. To find the lengths of the legs, use trigonometric ratios for
right △KLM.
MK
sin 20° = —
LK
MK
sin 20° = —
4
4 sin 20° = MK
LM
cos 20° = —
LK
LM
cos 20° = —
4
4 cos 20° = LM
L
20°
4
4
J
M
K
The regular nonagon has side length s = 2(MK) = 2(4 sin 20°) = 8 sin 20°, and
apothem a = LM = 4 cos 20°.
⋅
⋅
So, the area is A = —12 a ns = —12 (4 cos 20°) (9)(8 sin 20°) ≈ 46.3 square units.
Finding the Area of a Regular Polygon
You are decorating the top
of a table by covering it
with small ceramic tiles.
The tabletop is a regular
octagon with 15-inch sides
and a radius of about
19.6 inches. What is the
area you are covering?
R
15 in.
19.6 in.
P
SOLUTION
Q
R
Step 1 Find the perimeter P of the tabletop. An octagon
has 8 sides, so P = 8(15) = 120 inches.
19.6 in.
Step 2 Find the apothem a. The apothem is height RS of △PQR.
— bisects QP
—.
Because △PQR is isosceles, altitude RS
P
So, QS = —12 (QP) = —12 (15) = 7.5 inches.
Q
S
7.5 in.
To find RS, use the Pythagorean Theorem (Theorem 9.1) for △RQS.
——
—
a = RS = √ 19.62 − 7.52 = √ 327.91 ≈ 18.108
Step 3 Find the area A of the tabletop.
—
A = —12 aP = —12 ( √ 327.91 )(120) ≈ 1086.5
The area you are covering with tiles is about 1086.5 square inches.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Find the area of the regular polygon.
5.
6.
7
6.5
618
Chapter 11
8
Circumference and Area
Finding Areas of Composite Figures
Finding Areas of Composite Figures
Find the area of each composite figure. Round your answers to the nearest hundredth,
if necessary.
a.
b.
1.6 in.
7 cm
c.
1 in.
4 cm
7 cm
1.5 in.
0.8 in.
7 ft
60°
6 ft
3 ft
2 in.
4 ft
SOLUTION
a. The composite figure consists of a regular hexagon and a trapezoid. The
length of one side of the hexagon is 2(0.8) = 1.6 inches. The apothem is
——
—
√(1.6)2 − (0.8)2 = √1.92 ≈ 1.386 inches.
Area of composite figure = Area of regular hexagon + Area of trapezoid
1 —
1
= — ( √1.92 )(6)(1.6) + —(1.5)(1 + 2)
2
2
≈ 8.90
The area is about 8.90 square inches.
b. The composite figure consists of a sector and a triangle.
Area of composite figure = Area of sector + Area of triangle
⋅ ⋅
60°
1
= — π 72 + —(7)(4)
360°
2
≈ 39.66
The area is about 39.66 square centimeters.
c. The composite figure consists of a kite and a parallelogram.
Area of composite figure = Area of kite + Area of parallelogram
1
= —(7)(6) + 3(4)
2
= 33
The area is 33 square feet.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
7. Find the area of the composite figure.
3 ft
Round your answer to the nearest hundredth,
if necessary.
45°
6 ft
7 ft
Section 11.3
Areas of Polygons and Composite Figures
619
11.3 Exercises
Tutorial Help in English and Spanish at BigIdeasMath.com
Vocabulary and Core Concept Check
1. WRITING Explain how to find the measure of a central angle of a regular polygon.
2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
B
A
Find the radius of ⊙F.
Find the apothem of polygon ABCDE.
Find AF.
8
5.5
G
6.8
E
Find the radius of polygon ABCDE.
C
F
D
Monitoring Progress and Modeling with Mathematics
In Exercises 3–6, find the area of the kite or rhombus.
(See Example 1.)
3.
4.
6
19
2
38
5.
6.
7
5
15. m∠GJH
16. m∠GJK
K
17. m∠KGJ
18. m∠EJH
C
J
D
F
E
In Exercises 19–24, find the area of the regular polygon.
(See Examples 3 and 4.)
5
5
H
B
A
G
10
6
In Exercises 15–18, find the given
angle measure for regular octagon
ABCDEFGH. (See Example 2.)
6
7
19.
20.
12
In Exercises 7–10, use the diagram.
7. Identify the center of
J
N
polygon JKLMN.
8. Identify a central angle
of polygon JKLMN.
5.88
4.05
Q
5
21.
22.
K
P
10
2 3
6.84
7
2.77
2.5
M
L
9. What is the radius of
polygon JKLMN?
23. an octagon with a radius of 11 units
10. What is the apothem of polygon JKLMN?
In Exercises 11–14, find the measure of a central angle
of a regular polygon with the given number of sides.
Round answers to the nearest tenth of a degree,
if necessary.
11. 10 sides
12. 18 sides
13. 24 sides
14. 7 sides
24. a pentagon with an apothem of 5 units
25. ERROR ANALYSIS Describe and correct the error in
finding the area of the kite.
✗
3.6
2
5.4
3 2 5
A = —12 (3.6)(5.4)
= 9.72
So, the area of the kite is 9.72 square units.
620
Chapter 11
Circumference and Area
26. ERROR ANALYSIS Describe and correct the error in
finding the area of the regular hexagon.
✗
—
s = √ 152 − 132 ≈ 7.5
32. MODELING WITH MATHEMATICS A watch has a
circular surface on a background that is a regular
octagon. Find the area of the octagon. Then find the
area of the silver border around the circular face.
⋅
A = —12a ns
15
13
1 cm
0.2 cm
≈ —12(13)(6)(7.5)
= 292.5
So, the area of the hexagon is about
292.5 square units.
CRITICAL THINKING In Exercises 33 –35, tell whether the
In Exercises 27–30, find the area of the composite
figure. (See Example 5.)
statement is true or false. Explain your reasoning.
33. The area of a regular n-gon of a fixed radius r
increases as n increases.
27.
34. The apothem of a regular polygon is always less than
4 ft
9 ft
the radius.
6 ft
35. The radius of a regular polygon is always less than the
side length.
28.
36. REASONING Predict which figure has the greatest
74°
10 m
29.
area and which has the least area. Explain your
reasoning. Check by finding the area of each figure.
7 in.
9 in.
A
○
B
○
13 in.
15 in.
9 in.
30.
6 cm
4 cm
10 cm
C
○
6.6 cm
15 in.
5.1 cm
18 in.
MATHEMATICAL CONNECTIONS In Exercises 37 and 38,
31. MODELING WITH MATHEMATICS Basaltic columns
are geological formations that result from rapidly
cooling lava. Giant’s Causeway in Ireland contains
many hexagonal basaltic columns. Suppose the top
of one of the columns is in the shape of a regular
hexagon with a radius of 8 inches. Find the area of the
top of the column to the nearest square inch.
write and solve an equation to find the indicated
length(s). Round decimal answers to the nearest tenth.
37. The area of a kite is 324 square inches. One diagonal
is twice as long as the other diagonal. Find the length
of each diagonal.
38. One diagonal of a rhombus is four times the length
of the other diagonal. The area of the rhombus is
98 square feet. Find the length of each diagonal.
39. USING EQUATIONS Find the area of a regular
pentagon inscribed in a circle whose equation is given
by (x − 4)2 + (y + 2)2 = 25.
Section 11.3
Areas of Polygons and Composite Figures
621
40. HOW DO YOU SEE IT? Explain how to find the area
46. CRITICAL THINKING The area of a dodecagon, or
of the regular hexagon by dividing the hexagon into
equilateral triangles.
12-gon, is 140 square inches. Find the apothem of
the polygon.
U
V
Z
47. REWRITING A FORMULA Rewrite the formula for
the area of a rhombus for the special case of a square
with side length s. Show that this is the same as the
formula for the area of a square, A = s 2.
W
Y
X
48. REWRITING A FORMULA Use the formula for the
41. REASONING The perimeter of a regular nonagon,
or 9-gon, is 18 inches. Is this enough information to
find the area? If so, find the area and explain your
reasoning. If not, explain why not.
42. MAKING AN ARGUMENT Your friend claims that it is
possible to find the area of any rhombus if you only
know the perimeter of the rhombus. Is your friend
correct? Explain your reasoning.
area of a regular polygon to show that the area of an
equilateral
triangle can be found by using the formula
—
A= —14 s2√ 3 , where s is the side length.
49. COMPARING METHODS Find the area of regular
pentagon ABCDE by using the formula A = —12 aP, or
A = —12 a • ns. Then find the area by adding the areas
of smaller polygons. Check that both methods yield
the same area. Which method do you prefer? Explain
your reasoning.
A
43. PROOF Prove that the area of any quadrilateral with
1
—2 d1d2,
perpendicular diagonals is A =
are the lengths of the diagonals.
5
where d1 and d2
B
E
P
Q
D
P
R
T
S
d1
C
50. THOUGHT PROVOKING The area of a regular n-gon is
given by A = —12 aP. As n approaches infinity, what does
the n-gon approach? What does P approach? What
does a approach? What can you conclude from your
three answers? Explain your reasoning.
d2
44. USING STRUCTURE In the figure, an equilateral
triangle lies inside a square inside a regular pentagon
inside a regular hexagon. Find the approximate area of
the entire shaded region to the nearest whole number.
51. USING STRUCTURE Two regular polygons both have
8
45. CRITICAL THINKING The area of a regular pentagon
is 72 square centimeters. Find the length of one side.
Maintaining Mathematical Proficiency
Find the perimeter and the area of the figure.
52.
n sides. One of the polygons is inscribed in, and the
other is circumscribed about, a circle of radius r. Find
the area between the two polygons in terms of n and r.
Reviewing what you learned in previous grades and lessons
(Skills Review Handbook)
53.
54.
9 cm
15 yd
5m
4 cm
12 m
622
Chapter 11
Circumference and Area
8 yd
10 yd
11.4
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
Effects of Changing Dimensions
Essential Question
How does changing one or more dimensions
of a rectangle affect its perimeter and area?
G.10.B
Changing One Dimension
Work with a partner.
a. Fold a piece of paper in half twice so that there are
four layers.
b. Draw a rectangle on the paper. Then use scissors to
cut through the four layers so that you cut out four
congruent rectangles.
MAKING
MATHEMATICAL
ARGUMENTS
c. Place two rectangles side-by-side along either
the length or the width so that you form a figure
with double the length or double the width of a
single rectangle.
d. Compare the perimeter and the area of the figure
formed by the two rectangles to the perimeter and the
area of a single rectangle.
To be proficient in math, you
need to make conjectures and
build a logical progression of e. Make a conjecture about how doubling the length
or the width of a rectangle affects the perimeter
statements to explore the truth
and the area.
of your conjectures.
f. Make a conjecture about how multiplying the length or
the width of a rectangle by a positive number k affects
the perimeter and the area.
Changing Dimensions Proportionally
Work with a partner. Use the rectangles from Exploration 1.
a. Arrange the four rectangles so that you form a
rectangle with double the length and double the
width of a single rectangle.
b. Compare the perimeter and the area of the figure
formed by the four rectangles to the perimeter and
the area of a single rectangle.
c. Make a conjecture about how doubling the length
and the width of a rectangle affects the perimeter
and the area.
d. Make a conjecture about how multiplying the length
and the width of a rectangle by a positive number k
affects the perimeter and the area.
Communicate Your Answer
3. How does changing one or more dimensions of a rectangle affect its
perimeter and area?
Section 11.4
Effects of Changing Dimensions
623
11.4 Lesson
What You Will Learn
Describe the effects of non-proportional dimension changes.
Describe the effects of proportional dimension changes.
Core Vocabul
Vocabulary
larry
Changing Dimensions Non-Proportionally
Previous
perimeter
area
similar figures
When you change one or more dimensions of a figure, you also change the perimeter
and the area of the figure.
Changing One Dimension
Describe how the change affects the perimeter and the area of the figure.
b. multiplying the length by —13
a. doubling the height
5 ft
4 cm
12 ft
3 cm
SOLUTION
ANALYZING
MATHEMATICAL
RELATIONSHIPS
a.
Notice that when one
dimension is multiplied
by k, the area is multiplied
by k.
Aoriginal = —12 bh
Anew =
=
1
—2 b(kh)
k —12bh
(
)
Before change
After change
Dimensions
b = 3 cm, h = 4 cm
b = 3 cm, h = 8 cm
Perimeter
P = sum of side lengths
—
= 3 + 4 + √ 32 + 42
= 12 cm
P = sum of side lengths
—
= 3 + 8 + √ 32 + 82
—
= 11 + √ 73 cm
Area
A = —12 bh = —12 (3)(4) = 6 cm2
A = —12 bh = —12 (3)(8) = 12 cm2
Doubling the height increases the perimeter by
—
—
11 + √73 − 12 = √ 73 − 1 ≈ 7.54 centimeters and increases the area
12
by a factor of —
= 2.
6
= kAoriginal
b.
Before change
After change
Dimensions
ℓ= 12 ft, w = 5 ft
ℓ= 4 ft, w = 5 ft
Perimeter
P = 2ℓ + 2w
= 2(12) + 2(5)
= 34 ft
P = 2ℓ + 2w
= 2(4) + 2(5)
= 18 ft
Area
A =ℓw = 12(5) = 60 ft2 A =ℓw = 4(5) = 20 ft2
Multiplying the length by —13 decreases the perimeter by 34 − 18 = 16 feet and
20
decreases the area by a factor of —
= —13 .
60
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
1. Describe how multiplying the width by —1 affects the
perimeter and the area of the rectangle.
2
8 ft
10 ft
624
Chapter 11
Circumference and Area
Changing Dimensions Non-Proportionally
Describe how the change affects the perimeter and the area of the figure.
b. multiplying the base by —12
and doubling the height
a. doubling the length
and tripling the width
6m
ANALYZING
MATHEMATICAL
RELATIONSHIPS
Notice that when one
dimension is multiplied by
j and another dimension is
multiplied by k, the area
is multiplied by jk.
8 in.
7m
6 in.
SOLUTION
a.
Before change
Dimensions
ℓ= 7 m, w = 6 m
ℓ= 14 m, w = 18 m
Perimeter
P = 2ℓ + 2w
= 2(7) + 2(6)
= 26 m
P = 2ℓ + 2w
= 2(14) + 2(18)
= 64 m
Area
A =ℓw = 7(6) = 42 m2
A =ℓw = 14(18) = 252 m2
Aoriginal = bh
Anew = (jb)(kh)
After change
= jk(bh)
Doubling the length and tripling the width increases the perimeter by
252
64 − 26 = 38 meters and increases the area by a factor of —
= 6.
42
= jkAoriginal
b.
Before change
After change
Dimensions
b = 6 in., h = 8 in.
b = 3 in., h = 16 in.
Perimeter
P = sum of side lengths
—
= 6 + 8 + √ 62 + 82
= 24 in.
P = sum of side lengths
—
= 3 + 16 + √ 32 + 162
—
= 19 + √265 in.
Area
A = —12 bh = —12 (6)(8) = 24 in.2
A = —12 bh = —12 (3)(16) = 24 in.2
Multiplying the base by —12 and doubling the height increases the perimeter
—
by √ 265 − 5 ≈ 11.28 inches and does not change the area.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Describe how the change affects the perimeter and the area of the figure.
2. multiplying the length by 4
and the width by —1
2
7m
3. doubling the base and
tripling the height
5 ft
16 m
Section 11.4
12 ft
Effects of Changing Dimensions
625
Changing Dimensions Proportionally
When you change all the dimensions of a figure proportionally, the resulting figure is
similar to the original figure.
Core Concept
Changing Dimensions Proportionally
When you multiply all the linear dimensions of a figure by a positive number k,
the perimeter and the area change as shown.
Before multiplying
all dimensions by k
After multiplying
all dimensions by k
Perimeter
P
kP
Area
A
k 2A
Changing Dimensions Proportionally
Describe how doubling all the linear dimensions affects the perimeter and the area
of the parallelogram.
4m
5m
10 m
SOLUTION
Before change
After change
Dimensions
b = 10 m, h = 4 m
b = 20 m, h = 8 m
Perimeter
P = sum of side lengths
= 2(10) + 2(5)
= 30 m
kP = 2(30)
= 60 m
Area
A = bh = 10(4) = 40 m2 k 2A = 22(40) = 4(40) = 160 m2
Doubling all the linear dimensions of the parallelogram doubles the perimeter
and increases the area by a factor of 4.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
1
4. Describe how multiplying all the linear dimensions by —2 affects the perimeter
and the area of the triangle.
12 ft
14 ft
626
Chapter 11
Circumference and Area
11.4 Exercises
Tutorial Help in English and Spanish at BigIdeasMath.com
Vocabulary and Core Concept Check
1. VOCABULARY What is the difference between changing the linear dimensions of a figure
non-proportionally and proportionally?
2. COMPLETE THE SENTENCE When you change all the dimensions of a figure proportionally,
the resulting figure is _________ to the original figure.
Monitoring Progress and Modeling with Mathematics
7. doubling all the linear dimensions
In Exercises 3–10, describe how the change
affects the perimeter and the area of the figure.
(See Examples 1–3.)
8 yd
3. doubling the base
10 yd
18 yd
9 ft
1
8. multiplying all the linear dimensions by —2
12 ft
13 m
1
4. multiplying the width by —4
10 m
24 m
16 in.
9. tripling all the linear dimensions
20 in.
8 ft
5. multiplying the length by 4 and tripling the width
9 in.
14 in.
1
6. tripling the base and multiplying the height by —2
1
10. multiplying all the linear dimensions by —5
10 m
24 cm
7 cm
Section 11.4
Effects of Changing Dimensions
627
11. ERROR ANALYSIS Describe and correct the error in
14. HOW DO YOU SEE IT? Describe the relationship
between the areas of ABCD and AEFD.
finding the perimeter and the area of the rectangle
when the width is multiplied by —12 .
✗
D
F
C
A
E
B
2 cm
8 cm
Poriginal = 20 cm
Aoriginal =
16 cm2
Pnew = —12(20) = 10 cm
Anew =
15. MODELING WITH MATHEMATICS Your patio is 6 feet
( )
1 2
(16) = 4 cm2
—
2
long and 4 feet wide. Explain two different ways you
can change the dimensions of your patio to double
its area.
12. ERROR ANALYSIS Describe and correct the error in
16. ANALYZING RELATIONSHIPS You multiply the height
finding the perimeter and the area of the triangle when
all the linear dimensions are doubled.
✗
of a triangle by a positive number k, where k ≠ 1.
Describe how you can change the base so that the
area of the triangle is the same as the area of the
original triangle.
8 in.
17. ANALYZING RELATIONSHIPS Describe how the
15 in.
Poriginal = 40 in.
Pnew = 2(40) = 80 in.
Aoriginal = 60 in.2
Anew = 2(60) = 120 in.2
change affects the circumference and the area of
the circle.
a. doubling the radius
b. multiplying the radius by
9 ft
1
—3
c. squaring the radius
13. MAKING AN ARGUMENT You and your friend are
making posters for a school dance. Your friend claims
that doubling the length and the width of the posters
will double their areas. Is your friend correct? Explain
your reasoning.
18. THOUGHT PROVOKING The perimeter of rectangle
EFGH is k times the perimeter of rectangle ABCD
and the area of EFGH is k2 times the area of ABCD.
Can you be certain that ABCD and EFGH are always
similar? Explain.
Maintaining Mathematical Proficiency
D
C
A
B
Perimeter = P
Area = A
H
G
E
F
Perimeter = kP
Area = k2A
Reviewing what you learned in previous grades and lessons
Determine whether the figure has line symmetry, rotational symmetry, both, or neither. If the figure
has line symmetry, determine the number of lines of symmetry. If the figure has rotational symmetry,
describe any rotations that map the figure onto itself. (Section 4.2 and Section 4.3)
19.
628
20.
Chapter 11
Circumference and Area
21.
22.
11.3–11.4
What Did You Learn?
Core Vocabulary
center of a regular polygon, p. 617
radius of a regular polygon, p. 617
apothem of a regular polygon, p. 617
central angle of a regular polygon, p. 617
Core Concepts
Section 11.3
Area of a Rhombus or Kite, p. 616
Area of a Regular Polygon, p. 617
Area of a Composite Figure, p. 619
Section 11.4
Changing Dimensions Non-Proportionally, p. 624
Changing Dimensions Proportionally, p. 626
Mathematical Thinking
1.
In Exercise 50 on page 622, what conjecture did you make about the shape the n-gon
approaches? What logical progression led you to determine whether your conjecture
was correct?
2.
What was the first step in the process you used to solve Exercise 15 on page 628?
Why did you begin with this step?
Performance Task
sk
Window Design
n
The art department at your school has decided to replace a broken
window with an art project. Each color of glass has a different
price. The principal asks your class to calculate the cost. Can the
school afford the window?
To explore the answer to this question and more,
go to BigIdeasMath.com.
629
62
9
11
Chapter Review
11.1
Circumference and Arc Length (pp. 597–604)
The arc length of QR is 6.54 feet. Find the radius of ⊙P.
m
Arc length of QR
QR
—— = —
2πr
360°
75°
6.54
—=—
2πr 360°
Q
Formula for arc length
P
75°
6.54 ft
Substitute.
6.54(360) = 75(2πr)
R
Cross Products Property
5.00 ≈ r
Solve for r.
The radius of ⊙P is about 5 feet.
Find the indicated measure.
1. diameter of ⊙P
3. arc length of AB
2. circumference of ⊙F
C = 94.24 ft
A
P
F
B
115°
13 in.
C
G
5.5 cm
H
35°
4. A mountain bike tire has a diameter of 26 inches. To the nearest foot, how far does the
tire travel when it makes 32 revolutions?
11.2
Areas of Circles and Sectors (pp. 605–612)
Find the area of sector ADB.
m
AB
Area of sector ADB = — πr2
360°
80°
= — π 102
360°
≈ 69.81
⋅
⋅ ⋅
A
10 m
Formula for area of a sector
80°
D
Substitute.
B
Use a calculator.
The area of the small sector is about 69.81 square meters.
Find the area of the blue shaded region.
5.
T
W
6.
7.
R
9 in.
240°
4 in.
V
U
Chapter 11
A = 27.93 ft2
50°
Q
6 in.
630
S
Circumference and Area
T
11.3
Areas of Polygons and Composite Figures (pp. 615–622)
A regular hexagon is inscribed in ⊙H. Find (a) m∠EHG, and (b) the area of the hexagon.
A
B
H
F
C
16
G
E
D
360°
— bisects ∠FHE.
a. ∠FHE is a central angle, so m∠FHE = — = 60°. Apothem GH
6
So, m∠EHG = 30°.
⋅
⋅
—
—
1
b. Because △EHG is a 30°-60°-90° triangle, GE = — HE = 8 and GH = √ 3 GE = 8√ 3 .
2
—
So, s = 2(GE) = 16 and a = GH = 8√3 .
⋅
—
1
1
The area is A = — a ns = — ( 8√ 3 )(6)(16) ≈ 665.1 square units.
2
2
Find the area of the kite or rhombus.
9.
8.
10.
6
13
8
7
3
8
6
20
12
7
Find the area of the regular polygon.
11.
12.
13.
8.8
7.6
5.2
3.3
4
Find the area of the composite figure.
14.
15.
19 in.
7 in.
14 in.
27 in.
16.
223°
2 ft
5m
7 ft
4m
9 in.
8 ft
8m
17. A platter is in the shape of a regular octagon with an apothem of 6 inches. Find the area of
the platter.
Chapter 11
Chapter Review
631
11.4
Effects of Changing Dimensions (pp. 623–628)
Describe how the change affects the perimeter and the area of the figure.
a. multiplying the base by 2 and multiplying the height by 5
3 in.
4 in.
Before change
After change
Dimensions
b = 4 in., h = 3 in.
b = 8 in., h = 15 in.
P = sum of side lengths
—
= 3 + 4 + √ 32 + 42
P = sum of side lengths
Perimeter
—
= 8 + 15 + √ 82 + 152
= 40 in.
= 12 in.
Area
A =ℓw = —12 (3)(4) = 6 in.2
A =ℓw = —12(8)(15) = 60 in.2
Multiplying the base by 3 and multiplying the height by 5 increases the
60
perimeter by 40 − 12 = 28 inches and increases the area by a factor of —
= 10.
6
b. multiplying all the linear dimensions by —12
6 ft
18 ft
Before change
After change
Dimensions
ℓ = 18 ft, w = 6 ft
ℓ = 9 ft, w = 3 ft
Perimeter
P = 2ℓ + 2w = 2(18) + 2(6) = 48 ft kP = —12 (48) = 24 ft
Area
A =ℓw = 18(6) = 108 ft2
()
2
k 2A = —12 (108) = —14 (108) = 27 ft2
1
Multiplying all the linear dimensions by —12 decreases the perimeter by a factor of —2 and
1
decreases the area by a factor of —4.
Describe how the change affects the perimeter and the area of the figure.
18. multiplying the base by 7
19. multiplying the length by
1
—3 and the width by 6
20. multiplying all the
linear dimensions by 5
5 in.
5m
12 cm
5 cm
632
Chapter 11
Circumference and Area
2 in.
9m
3 in.
11
Chapter Test
Find the area of the composite figure.
1.
2.
111°
6 yd
3.
17 mm
13 mm
5.2 cm
8 mm
9 cm
8 yd
15 mm
2.8 cm
4.8 cm
12 yd
Find the indicated measure.
5. m
GH
4. circumference of ⊙F
6. area of shaded sector
64 in.
210°
F
D
J
27 ft
E
Q
T
G
S
35 ft
105°
8 in.
H
R
7. One diagonal of a rhombus is three times as long as the other diagonal. The area of the
rhombus is 108 square inches. Find the length of each diagonal.
Find the area of the regular polygon.
8. a hexagon with an apothem of 9 centimeters
9. a nonagon (9-gon) with a radius of 1 meter
Describe how the change affects the perimeter and the area of the rectangle.
10. multiplying the width by 3
15 cm
1
11. multiplying the length by —5 and the width by 4
7 cm
12. The area of a circular pond is about 138,656 square feet. You are going to walk
around the entire edge of the pond. About how far will you walk? Round your
answer to the nearest foot.
13. You want to make two wooden trivets, a large one and a small one. Both trivets
will be shaped like regular pentagons. The perimeter of the small trivet is
15 inches, and the perimeter of the large trivet is 25 inches. Find the area of
the small trivet. Then use this area to find the area of the large trivet.
14. In general, a cardboard fan with a greater area does a better job of moving air
and cooling you. The fan shown is a sector of a cardboard circle. Another fan
has a radius of 6 centimeters and an intercepted arc of 150°. Which fan does a
better job of cooling you?
120°
Chapter 11
Chapter Test
9 cm
633
11
Standards Assessment
1. About 6200 people live in one-fourth of a region with a 5-mile radius. What is the most
reasonable estimate for the population density in people per square mile? (TEKS G.12.C)
A 80 people per square mile
○
B 250 people per square mile
○
C 315 people per square mile
○
D 425 people per square mile
○
—
2. The point ( 1, √3 ) lies on the circle centered at the origin and containing which
point? (TEKS G.12.E)
F (3, 4)
○
G (4, 0)
○
H (0, 2)
○
J (5, 3)
○
1
3. All the linear dimensions of a rectangle are multiplied by —4 . Which of the following
statements describes the area of the new rectangle? (TEKS G.10.B)
1
A The area of the new rectangle is —
○
64 times the area of the original rectangle.
1
B The area of the new rectangle is —
○
16 times the area of the original rectangle.
1
C The area of the new rectangle is —8 times the area of the original rectangle.
○
1
D The area of the new rectangle is —4 times the area of the original rectangle.
○
—
4. Rectangle ABCD is shown. What is the length of AB? (TEKS G.9.B)
A
B
15 in.
45 in.
D
E
4 in.
C
—
F 14 in.
○
G √270 in.
○
H 18 in.
○
J 26 in.
○
5. What is the equation of the line passing through the point (2, 5) that is parallel to the
line x + —12 y = −1? (TEKS G.2.C)
634
A y = −2x + 9
○
B y = 2x + 1
○
1
C y = —2 x + 4
○
1
D y = −—2 x + 6
○
Chapter 11
Circumference and Area
6. The figure shows two regular hexagons with center C and apothems a and b.
Each vertex of the smaller hexagon is a midpoint of a side—of the larger hexagon.
What is the total area of the shaded regions when a = 8√ 3 centimeters and
b = 12 centimeters? (TEKS G.11.A)
b
C
a
—
—
F 24√3 cm2
○
G 96√3 cm2
○
—
—
H 288√3 cm2
○
J 672√3 cm2
○
—
7. In the diagram, ⃖
⃗
RS is tangent to ⊙P at Q and PQ is a radius of ⊙P. What must be true
—
⃖
⃗
about RS and PQ? (TEKS G.12.A)
R
Q
P
S
1
A PQ = —2RS
○
B PQ = RS
○
—
C PQ is tangent to ⊙P.
○
—
RS
D PQ ⊥ ⃖
⃗
○
8. GRIDDED ANSWER What is the length of an arc of a circle with a radius of 3 inches
π
and a central angle of — radians? Round to the nearest hundredth. (TEKS G.12.D)
3
9. What is the approximate area of the composite figure? (TEKS G.11.B)
F 110 ft2
○
G 120 ft2
○
H 130 ft2
○
J 140 ft2
○
8 ft
Chapter 11
Standards Assessment
635

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