Arc Length and Area of a Sector 1

Arc Length
and
Area of a Sector
Feb 22­4:25 PM
1
Recall:
s
O
Solve for s.
r
Feb 27­10:06 PM
2
Definition (Arc Length):
If θ (in radians) is a central angle in a circle
with radius r, then the length of the arc cut off
by θ is given by
(θ in radians)
s
O
r
Feb 27­10:11 PM
3
Example:
Give the length of the arc cut off by a central
angle of 2 radians in a circle of radius 4.3
inches.
Feb 27­10:18 PM
4
Example:
Below is a model of George Ferris's wheel.
Recall that the diameter of the wheel is 250
feet, and θ is the central angle formed as a
rider travels from his or her initial position P0
to position P1. Find the distance traveled by
the rider if
and if
Feb 27­10:21 PM
5
Example:
The minute hand of a clock is 1.2 centimeters
long. To the nearest tenth of a centimeter,
how far does the tip of the minute hand move
in 20 minutes?
First find θ, then use
Feb 27­10:35 PM
6
Note:
If we are working with relatively small central
angles in circles with large radii, we can use
the length of the intercepted arc to
approximate the length of the associated
chord.
Feb 28­8:59 PM
7
Example:
The figure below shows a central angle of 1
degree in a circle of radius 1,800 feet, along
with the arc and chord cut off by 1 degree.
B
Chord AB
1800 ft
s
Arc AB
A
Later we will learn a way to calculate the exact
length of chord AB which is 31.4155. If we
carried out the number above, it would be
31.4159. It seems reasonable then to
approximate the length of the chord AB with
the length of arc AB.
Feb 28­9:10 PM
8
Example:
A person standing on the earth notices that a
747 Jumbo Jet flying overhead subtends an
angle of
. If the length of the jet is 230
feet, find its altitude to the nearest thousand
feet.
jet
A
B
230 ft
r
r
Feb 27­10:38 PM
9
Find the area of the sector.
5
Mar 2­10:12 AM
10
Area of a Sector:
s
Area of sector
Central angle θ
Area of circle
One full rotation
r
Feb 28­9:50 PM
11
Definition (Area of Sector):
If θ (in radians) is a central angle in a circle
with radius r, then the area of the sector
formed by angle θ is given by
(θ in radians)
s
r
Feb 28­9:33 PM
12
Example:
Find the area of the sector formed by a central
angle of 1.4 radians in a circle of radius 2.1
meters.
Feb 28­9:59 PM
13
Example:
If the sector formed by a central angle of
has an area of
cm2, find the radius of the
circle.
Feb 28­10:04 PM
14
Example:
A lawn sprinkler located at the corner of a
yard is set to rotate through
and project
water out 30 feet. To the nearest square foot,
what area of lawn is watered by the sprinkler?
Feb 28­10:06 PM
15
3.4 Page 139: 3, 7, 11, 15, 21, 27, 29, 37,
41, 45, 49, 55
Feb 2­11:17 PM
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Feb 28­10:25 PM
17
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