Arc Length and Area of a Sector

TrigonometryMATH 103

S. Rook

Overview

• Section 3.4 in the textbook:– Arc length– Area of a sector

2

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

• Like many formulas we can often solve for one variable in terms of the others

• Thus, we get a formula for arc length: 4

rs

r

s

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)

b) θ = 315°, r = 5 inches

5

cm 12,3

r

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?

6

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

7

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:

becomes which is the formula for Area of a Sector

9r

r

r

A

22 2

2

1rA

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)

b) θ = 15°, r = 10 m

10

m 3,5

2 r

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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