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Lesson 11-6
Arc Lengths and Areas of Sectors
(page 452)
Essential Question
How can you calculate the
area of any figure?
Arc Lengths & Areas of Sectors
Two different numbers that
describe the size of an arc are:
(1)
the measure of the arc, ie.
mAB .
A
B
O
(2)
the arc length, which is the length of
a piece of the circumference.
The arc length is a fraction of the
whole circumference.
x
If mAB = xº, then the length of AB =
2p r.
360
xº
A
x
=
p d.
B
360
O
The length of the arc equals the fraction times
the circumference of the circle.
A sector of a circle is a region bounded
by two radii and an arc of the circle.
xº
A
B
O
The area of a sector is a fraction of
the area of the whole circle.
Area of a Sector
x 2
If mAB = xº, then the area of sector AOB =
pr .
360
xº
A
B
O
The area of a sector equals the
fraction times the area of the circle.
#1 In ⨀O with radius 6 and m∠AOB = 150º,
find the lengths of
150º
B
10º
A
150º
r=6
D
O
C
AB and ACD .
mAB
2p r
length of AB =
360
150
=
2p (6)
360
5
= (12p )
12
= 5p u
#1 In ⨀O with radius 6 and m∠AOB = 150º,
find the lengths of
150º
B
10º
A
150º
r=6
D
O
C
200º
mACD = 360º-160º= 200º
AB and ACD .
mACD
2p r
length of ACD =
360
200
=
2p (6)
360
5
= (12p )
9
20p
=
u
3
#2 Find the area of the shaded sector in the
circle with radius equal to 12.
40º
r = 12
m
2
=
pr
sector 360
40
2
=
p (12)
360
1
= (144p )
9
A
= 16p u
2
#3 Find the area of the shaded region in the
circle with radius equal to 6.
120º
Please note:
This is also
referred to as the
Area of a Segment.
#3 Find the area of the shaded region in the
circle with radius equal to 6.
120º
WARNING?
We do not
have a
formula for
this figure!
#3 Find the area of the shaded region in the
circle with radius equal to 6.
120º
#3 Find the area of the shaded region in the
circle with radius equal to 6.
120º
General approach:
Ashaded region = Abig region - Asmall region
For this problem:
Ashaded region = Asector - Atriangle
#3 Find the area of the shaded region in the
circle with radius equal to 6.
120º
m
2
=
pr
sector 360
120
2
=
p (6)
360
1
= (36p )
3
A
r=6
Ashaded region = Asector - Atriangle
= 12p u
2
#3 Find the area of the shaded region in the
circle with radius equal to 6.
120º
r=6
60º
60º 30º
h=3
r=6
Ashaded region = Asector - Atriangle
#3 Find the area of the shaded region in the
circle with radius equal to 6.
120º
A
3 3
r=6
60º
3 3
60º 30º
h=3
r=6
\ b= 6 3
Ashaded region = Asector - Atriangle
1
= bh
2
1
= 6 3 (3)
2
(
)
=9 3u
2
#3 Find the area of the shaded region in the
circle with radius equal to 6.
120º
3 3
r=6
60º
3 3
60º 30º
h=3
r=6
\ b= 6 3
Ashaded region = Asector - Atriangle
(
)
= 12p - 9 3 u
» 22.11 u
2
2
#4 Find the area of the shaded region in the
circle with radius equal to 2 inches.
WARNING?
300º
We do not
have a
formula for
this figure!
#4 Find the area of the shaded region in the
circle with radius equal to 2 inches.
300º
#4 Find the area of the shaded region in the
circle with radius equal to 2 inches.
General approach:
Ashaded region = Aregion + Aregion
300º
For this problem:
Ashaded region = Asector + Atriangle
#4 Find the area of the shaded region in the
circle with radius equal to 2 inches.
m
2
=
pr
sector
360
300
2
=
p (2)
360
5
= (4p )
6
A
r = 2”
300º
Ashaded region = Asector + Atriangle
10p 2
=
in
3
#4 Find the area of the shaded region in the
circle with radius equal to 2 inches.
1
h= 3
2”
60º
30º 30º
1
60º
r = 2”
\ b= 2
300º
Ashaded region = Asector + Atriangle
#4 Find the area of the shaded region in the
circle with radius equal to 2 inches.
1
h= 3
2”
60º
30º 30º
1
60º
r = 2”
\ b= 2
300º
Ashaded region = Asector + Atriangle
A
1
= bh
2
1
= (2)
2
( 3)
= 3 in
2
#4 Find the area of the shaded region in the
circle with radius equal to 2 inches.
1
h= 3
2”
60º
30º 30º
\ b= 2
300º
1
60º
r = 2”
Ashaded region = Asector + Atriangle
æ10p
æ 2
=æ
+ 3 æ in
æ 3
æ
» 12.20in
2
Chapter 11 Project Assignment
Circle Graph Activity
This is due _________________!
Please make this VERY nice!
How can you calculate the
area of any figure?
Assignment
Written Exercises on pages 453 & 454
REQUIRED: 1 to 19 odd numbers
UPDATE YOUR STUDENT AID CARD!
How can you calculate the
area of any figure?