Transcript Angles, Degrees, and Special Triangles

Arc Length and Area of a Sector
Trigonometry
MATH 103
S. Rook
Overview
• Section 3.4 in the textbook:
– Arc length
– Area of a sector
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Arc Length
Arc Length
• Recall that in Section 3.2
we derived a formula
relating the central angle
θ (in radians), radius r, and
the arc of length s cut off
by θ   rs
• Like many formulas we can often solve for one
variable in terms of the others
• Thus, we get a formula for arc length: s  r
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Arc Length (Example)
Ex 1: θ is a central angle in a circle of radius r.
Find the length of arc s cut off by θ:
a)  

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, r  12 cm
b) θ = 315°, r = 5 inches
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Arc Length (Example)
Ex 2: The minute hand of a circular clock is 8.4
inches long. How far does the tip of the
minute hand travel in 10 minutes?
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Arc Length (Example)
Ex 3: θ is a central angle in a circle that cuts off
arc length s. Find the radius r of the circle:
θ = 150°, s = 5 km
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Area of a Sector
Area of a Sector
• Sometimes we wish to
know the area of the
sector of a circle with
central angle θ in radians
and radius r
– Let A be the area of this sector
• Using a part to whole proportion with area
and arc length:
1 2
A
r
A  r  which is the

becomes
2
2
r
2r
formula for Area of a Sector
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Area of a Sector (Example)
Ex 4: Find the area of the sector formed by
central angle θ in a circle of radius r if:
a)
2

,r  3m
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b) θ = 15°, r = 10 m
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Area of a Sector (Example)
Ex 5: An automobile windshield wiper 6 inches
long rotates through an angle of 45°. If the
rubber part of the blade covers only the last 4
inches of the wiper, approximate the area of
the windshield cleaned by the windshield
wiper
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Summary
• After studying these slides, you should be able
to:
– Calculate arc length
– Calculate the area of a sector
• Additional Practice
– See the list of suggested problems for 3.4
• Next lesson
– Velocities (Section 3.5)
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