Sectors, Arcs, and
Regular Polygons
Sections 11.1,11.3,13.3
Geometry PAP
Name ____________________________
Period _______________
Teacher ____________
1|Page
5th Six Weeks 2015-2016 Geometry PAP
MONDAY
TUESDAY
WEDNESDAY
THURSDAY
FRIDAY
Feb 15
16
17
18
19
Quiz
Workday Arc Length
and Sectors
CW: Finish Arc length
& Sector Area
HW: Video Area
Regular Polygons
13.3 Area of Regular
Polygons day 1
Warm up 2
CW: Regular Polygons
Day 1
HW: Video Regular
Polygons day 2
13.3 Area of Regular
polygons 2 and 3
Warm up 3
CW: Regular Polygons 2
&3
HW: Complete
classwork
13.3 Area of Regular
polygons 2 and 3
Warm up 3
CW: Regular Polygons
2&3
HW: Complete
classwork
Review
Formula Quiz
CW: Review
EOC #3 Due
22
23
24
25
26
12.1 Exploring Solids
Test #13
Volume of Prisms
Volume of Prisms
Volume of Pyramids
HW: Study
EOC #4
29
March 1
2
3
4
Volume of Cylinders,
Cones, and Spheres
Workday
Surface Area and
Volume of Similar
Solids
Surface Area and
Volume of Similar
Solids
Review
EOC #4 Due
Formula Quiz
7
8
9
10
11
Surface Area of
Prisms
Test #14
EOC #5
Surface Area of
Pyramids
Surface Area of
Pyramids
Workday Surface area
of Prisms and
Pyramids
14
15
16
17
18
Spring Break
Spring Break
Spring Break
Spring Break
Spring Break
21
22
23
24
25
Surface area of
Spheres
Surface area of
Cylinders and Cones
Quiz
Formula Quiz
Mix of all figures
Quiz
Formula Quiz
Mix of all figures
28
29
30
31
April 1
Review Day
EOC #5 due
Test #15
EOC #6
9.2 Use Properties of
Matrices
9.2 Use Properties of
Matrices
9.1 Translate figures
and Vectors
School Holiday
2|Page
Worksheet 11.1&11.3 Arc length and Sector Area
Use the diagram to find the indicated measures.
1. Find the circumference.
2. Find the circumference.
3. Find the radius.
Find the indicated measure.
4. The exact radius of a circle with circumference 74 centimeters.
5. The exact circumference of a circle with radius 31.9 meters.
Find the exact area of the circle.
6.
7.
8.
Find the indicated measure.
9. The area of a circle is 128 square feet. What is the exact radius?
10. The area of a circle is 98 square inches. What is the exact diameter?
Find the exact arc length and area of each shaded sector.
11.
12.
13.
14.
120
1
4
6
90
1
4
6
45
4
4
15. In a circle with radius 6, mAB  60 . Make a sketch and find the exact area of the region
bounded by AB and AB .
16. A circle has area 160 cm2 . If a sector of the circle has area of 40 cm2 , find the measure of
the arc of the sector to the nearest degree.
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Sector AOB is described by giving mAOB and the radius of circle O . Make a sketch and find the
exact length of AB and the exact area of sector AOB .
mAOB
17.
18.
19.
20.
21.
30
45
120
240
180
Radius
12
4
3
3
1.5
mAOB
22.
23.
24.
25.
26.
270
192
.8
320
1
1
5
108
Radius
40
9
2
5 2
3 3
Arc setup
Arc Length
Sector
setup
Sector Area
Arc setup
Arc Length
Sector
setup
Sector Area
For the following problems you are given an arc length and an arc or angle measure. Find the indicated length.
27. Find exact radius.
28. Find Circumference.
29. Find exact diameter.
5 ft
16 m
80
30. Find exact radius.
50
31. Find exact radius.
44
39 m
80
15 in
ft
32. Find exact radius.
4 3 cm
150
165
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For the following problems you are given a sector area and an angle or arc measure. Find the exact radius for each.
30
A  15 m2
33.
34.
35.
A  12
A  30
ft2
72
in2
60
310
36.
37.
38.
312
A  123 ft2
164
108
A  20 cm2
A  60 km2
39. The area of sector AOB is 10 and mAOB  100 . Find the exact radius of circle O.
40. The area of sector AOB is
7
and mAOB  315 . Find the exact radius of circle O.
2
Find the exact perimeter of the region.
41.
42.
43.
Find the exact area of each shaded region. Point O marks the center of a circle.
44.
45.
46.
90
4 ft
60
O
O
47.
O
3 in
48.
6 cm
49.
5|Page
50. Posterboard Each of the following is cut from a 6 inch  12 inch piece of poster board.
Which waste the least? (Show your work for each)
60’
a. Two 6-inch diameter circles
b. Eight 3-inch diameter circles
c. Eighteen 2-inch diameter circles
325’
51. The diagram show some dimensions in a baseball
stadium. H represents home plate. What is the area
of the fair territory(shaded region) and the foul
territory(nonshaded region)?
325’
H
52. On a large machine, the centers of two pulleys are 16 feet
apart and the radius of each pulley is 24 inches. How long
a belt is needed to wrap around both pulleys?
60’
24 in.
16 ft
53. The large sprocket on the pedals of a bicycle has 50 teeth and the small sprocket on the wheel
has 20 teeth. When the pedals make two complete revolutions, how many revolutions does
the wheel make?
3 in
54. Four posts with 3-in. radii are bound together with a wire. Find the
length of the shortest wire to the nearest hundredth that will go
around them.
In 55 – 57, find the exact area of the segment bounded by the given
arc and chord. Use the diagram given to help.
A
B
55. Segment bounded by AB and chord AB , radius 6 in., m AB = 90.
56. Segment bounded by AB and chord AB , radius 6 in., m AB = 120.
57. Segment bounded by AB and chord AB , radius 6 in., m AB = 60.
58. Find the exact perimeter and area of the figure.
15 in
8 in
6|Page
59. A bicycle rolls 100 feet, if the wheel has a diameter of 36 inches, how many complete
revolutions did the wheel make?
60. Thread A spool of thread contains 150
revolutions of thread. The diameter of
the spool is 3 centimeters. Find the length
of the thread to the nearest centimeter.
61. Pendulum Find the distance traveled in one
back and forth swing by the weight of a 16-inch
pendulum that swings through a 70 angle.
Challenge Problem
62. A cow is tied by a 25 meter rope to the corner of a barn shown.
A fence keeps the cow out of the garden. Find the total grazing
area. The cow cannot go around the buildings. Round answer
nearest thousands.
Garden
15 m
10 m
barn
20 m
10 m
25 m
7|Page
Guided Practice Regular Polygons
A regular polygon is both equilateral and equiangular. Any regular
polygon can be inscribed in a circle. Therefore, many of the terms
associated with circles are also used with regular polygons.
The center of a regular polygon is the center of the circumscribed circle.
The radius of a regular polygon is the distance from the center to a vertex.
AP and AR are radii.
T
A central angle is an angle formed by two radii drawn to consecutive
vertices. PAQ and SAR are central angles.
The measure of a central angle of a regular polygon with n sides
is
360
n
S
60
A
U
R
. For example, the measure of each central angle in regular
hexagon PQRSTU is
360
6
 60 .
P
The apothem of a regular polygon is the distance from the center to a
side. AB is an apothem. AB is the perpendicular bisector of side PQ .
B
Q
Note: PB  BQ
PAB  BAQ
If you know the apothem and perimeter of a regular polygon, the following
theorem allows you to find the area of the polygon.
The area of a regular polygon is equal to half the
product of the apothem and the perimeter.
A
1
Pa
2
For each regular polygon:
A) draw a radius AP and apothem AB
B) calculate the measure of the central angle PAQ
C) calculate the measure of PAB ( Write this measurement on the diagram)
D) calculate the length of the radius, apothem, and side of the polygon.
(use many have to use 30-60-90 triangles or 45-45-90 triangles or Trig. Formulas)
E) Calculate the area of the polygon using A 
1
Pa
2
F) Neatly show all formulas and work!
8|Page
1) Regular Hexagon. PQ  4 cm
S
T
B) mPAQ = _____________
C) mPAB = _____________
A
U
R
D) radius = _______________
apothem = ____________
E) Area = _______________
Q
P
2) Square. PQ  10 in
R
S
B) mPAQ = _____________
C) mPAB = _____________
A
D) radius = _______________
apothem = ____________
E) Area = _______________
Q
P
R
3) Regular Triangle. PQ  12 3 yd
B) mPAQ = _____________
C) mPAB = _____________
A
D) radius = _______________
apothem = ____________
E) Area = _______________
Q
P
S
T
4) Regular Hexagon. radius  4 3 km
B) mPAQ = _____________
C) mPAB = _____________
A
U
R
D) radius = _______________
apothem = ____________
E) Area = _______________
P
Q
9|Page
S
5) Regular Pentagon. radius  10 ft
B) mPAQ = _____________
T
R
A
C) mPAB = _____________
D) radius = _______________
apothem = ____________
E) Area = _______________
Q
P
T
U
6) Regular polygon. radius  6 ft
B) mPAQ = _____________
S
V
C) mPAB = _____________
A
D) radius = _______________
apothem = ____________
R
W
E) Area = _______________
Q
P
7) Regular polygon. radius  10 mm
U
V
T
B) mPAQ = _____________
C) mPAB = _____________
W
S
A
D) radius = _______________
apothem = ____________
E) Area = _______________
R
X
P
Q
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Regular Polygons Day 1
Name _______________________ Pd _____
Find the missing lengths and the area of the regular polygon to the nearest tenths, unless Special Right Triangle.
1)
2)


16
8.5
a
a
Measure of central angle (  ): _________
Measure of central angle (  ): _________
Length of apothem( a ): _______________
Length of apothem( a ): _______________
Length of a side ( s ): _________________
Length of a radius ( r ): _________________
Perimeter: _________________________
Perimeter: _________________________
Area setup:
1
( _________) (___________)
2
Area setup:
1
( _________) (___________)
2
Area: _______________________________
Area: _______________________________
3)
4)
12


s
8
a
Measure of central angle (  ): _________
Measure of central angle (  ): _________
Length of apothem( a ): _______________
Length of apothem( a ): _______________
Length of a radius ( r ): _________________
Length of a side ( s ): _________________
Perimeter: _________________________
Perimeter: _________________________
Area setup:
1
( _________) (___________)
2
Area: _______________________________
Area setup:
1
( _________) (___________)
2
Area: _______________________________
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5)
6)


s
s
12
7.5
Measure of central angle (  ): _________
Measure of central angle (  ): _________
Length of radius ( r ): _______________
Length of apothem( a ): _______________
Length of a side ( s ): _________________
Length of a side ( s ): _________________
Perimeter: _________________________
Perimeter: _________________________
Area setup:
1
( _________) (___________)
2
Area setup:
1
( _________) (___________)
2
Area: _______________________________
Area: _______________________________
7)
8)

16

6
a
Measure of central angle (  ): _________
Measure of central angle (  ): _________
Length of apothem( a ): _______________
Length of apothem( a ): _______________
Length of a radius ( r ): _________________
Length of a side ( s ): _________________
Perimeter: _________________________
Perimeter: _________________________
Area setup:
1
( _________) (___________)
2
Area: _______________________________
Area setup:
1
( _________) (___________)
2
Area: _______________________________
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Regular Polygons 11.6 Day 2
Name ____________________________
1) Find the area of a regular hexagon with apothem 3 3
2) Find the area of regular hexagon with radius of 8.
3) If the area of a regular hexagon is 36 3 cm2, find its apothem and the length of its side.
4) If the apothem of a regular hexagon is 5 m, find its perimeter and area.
5) An equilateral triangle and a regular hexagon are both inscribed in a circle with
radius 6. Find the area of each polygon.
6) A side of a regular hexagon is s units long. Find the area of the hexagon in terms of s.
7) A side of a regular hexagon is twice the square root of its apothem. Find the length of
the apothem and the side. Hint: put the apothem in terms of the side.
8) The apothem of a regular pentagon is 13 cm. Find the area of the pentagon to the
nearest hundredth.
9) A regular hexagon and a square are circumscribed about a circle with radius 10. How
much more area does the square have than the hexagon?
10) A flower garden is made with a white decorative brick border as shown
by the unshaded part of this figure. The inner square planting area is
formed by connecting the midpoints of the 12 foot sides of the out regular
octagon. Find the area of the brick border.
11)* A regular octagon has a radius of 11.5 inches. What is the length of its apothem? Round
your answer to the nearest hundredth?
12)* A regular decagon has a diameter of 35 meters. What is the length of its apothem? Round
your answer to the nearest hundredth.
13)* What is the area of an equilateral triangle with radius of 15 centimeters? Give an exact answer.
14)* What is the area of a regular hexagon with radius 8.5 inches? Give an exact answer.
15)* What is the area of a square with diagonal 6.3 centimeters? Given an exact answer.
16)* What is the side length of a regular hexagon with area 100 square centimeters? Give an exact answer.
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Regular Polygons Day 3
Write an equation for the area of the following regular polygons.
1)
Name __________________________
2)
 4x
4x
 2x
 1
 2x
3)
 6
 3
4)
6x
10x
1

 4 x  6


 4
2x
5)
6)
3x
x
 3x  2
7)
12x
 8
8)
12x
 20  4x 
10  2x 
9)
 7
6x 2
10)
7x
6x
 6  2x 
1

 2 x  8


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Find the area of each shaded region. Assume that all polygons are regular. Given an exact answer.
11)
12)
5m
12 m
16 m
6m
8m
13)
14)
9m
8m
12 m
3.2 m
4m
Review
15) Find the sum of the measures of the interior angles of 12 sided polygon.
16) For a 20 sided polygon, Find the measure of an interior angle and an exterior angle for each polygon.
17) Find the area of the shaded region.
16 m
10 m
8m
4m
7x 2 y 8
18) Find the area.
19) Find the area.
12a 3b 5
4x 2 y 3
6a 3b 6
20)  4x  2  3x  4 
5x 2 y 8
21) 5x  10  x  5
22)  5x  5 
2
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Review Chapter 11.4-11.6
Name _________________________
Find the indicated measure. Round to the nearest hundredth.
1)
2)
3) Find length of WX
Find the area of the shaded region.
4)
5)
6)
Find the area of the shaded region.
4)
5)
6)
8 in.
7) A bike tire has a diameter of about 26 inches. You ride a straight distance of
about 75 feet. About how many revolutions does the tire make along this distance.
8) Find the shaded area.
9) Write an expression for the area
10) Same as #9
9m
4m
6x  4
1

 4 x  6


6xy 2
3xy
4xy
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Find the area of the following regular polygons.
11)
12)
13)
14 ft
12cm
14)
9m
15)
2
2
yd
5
6 yd
16) The following is a diagram of a statue in a garden. The statue and garden need to be
fenced. How many feet of fencing is needed for the statue and the garden? What is
the area of the garden to the nearest foot?
17) A regular hexagon has an area of 150 3 ft 2 and a apothem of 5 3 ft . What is the length
of one side and the radius?
18) A regular octagon has an area of 212 cm2 and a apothem of 8 cm . What is the length of one side?
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