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 16 Feb 2810:25 PM 17

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