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 2

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

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

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

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 9

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

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 11

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) 12