UNIT 3 CIRCLES AND VOLUME Lesson 4: Finding Arc Lengths and Areas of Sectors Instruction
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1 Prerequiite Skill Thi leon require the ue of the following kill: finding the circumference of a circle undertanding cale factor in imilar hape uing ratio and proportion Introduction All circle are imilar; thu, o are the arc intercepting congruent angle in circle. A central angle i an angle with it vertex at the center of a circle. We have meaured an arc in term of the central angle that it intercept, but we can alo meaure the length of an arc. Arc length, the ditance between the endpoint of an arc, i proportional to the radiu of the circle according to the central angle that the arc intercept. The contant of proportionality i the radian meaure of the angle. You already know how to meaure angle in degree. Radian meaure i another way to meaure angle. An angle meaure given in degree include a degree ymbol. An angle meaure given in radian doe not. Key Concept Arc length i the ditance between the endpoint of an arc, and i commonly written a mab. The radian meaure of a central angle i the ratio of the length of the arc intercepted by the angle to the radiu of the circle. B C r A The definition of radian meaure lead u to a formula for the radian meaure of a central angle θ in term of the intercepted arc length,, and the radiu of the circle, r: θ r. U3-171
2 When the intercepted arc i equal in length to the radiu of the circle, the central angle meaure 1 radian. 1 radian C B arc length radiu A Recall that the circumference, or the ditance around a circle, i given by C 2πr or C πd, where C repreent circumference, r repreent radiu, and d repreent the circle diameter. Since the ratio of the arc length of the entire circle to the radiu of the circle i 2 π r 2π, there are 2π radian in a full circle. r We know that a circle contain 360º or 2π radian. We can convert between radian meaure and degree meaure by implifying thi ratio to get π radian 180º. To convert between radian meaure and degree meaure, et up a proportion. radian meaure degree meaure π 180º To find the arc length when the central angle i given in radian, ue the formula for radian meaure to olve for. To find arc length when the central angle i given in degree, we determine the fraction of the circle that we want to find uing the meaure of the angle. Set up a proportion with the circumference, C. degree meaure C 360º Common Error/Miconception confuing arc meaure with arc length forgetting to check the mode on the graphing calculator inconitently etting up ratio incorrectly implifying ratio involving π U3-172
3 Guided Practice Example 1 Convert 40º to radian. 1. Set up a proportion. radian meaure degree meaure π 180º x 40º π 180º 2. Multiply both ide by π to olve for x. x 40 π 2π radian Example 2 Convert 3 π radian to degree Set up a proportion. radian meaure degree meaure π 180º 3π 4 x π 180º 3 x 4 180º 2. Multiply both ide by 180 to olve for x. x 3 ( 180 ) 135º 4 U3-173
4 Example 3 A circle ha a radiu of 4 unit. Find the radian meaure of a central angle that intercept an arc of length 10.8 unit unit 4 unit 1. Subtitute the known value into the formula for radian meaure. θ arclength radiu r Simplify. θ 2.7 radian The radian meaure i 2.7 radian. U3-174
5 Example 4 A circle ha a radiu of 3.8 unit. Find the length of an arc intercepted by a central angle meauring 2.1 radian unit 1. Subtitute the known value into the formula for radian meaure. θ arclength radiu r Multiply both ide by 3.8 to olve for arc length unit The arc length i 7.98 unit. U3-175
6 Example 5 A circle ha a diameter of 20 feet. Find the length of an arc intercepted by a central angle meauring 36º. 36 d 20 feet 1. Find the circumference of the circle. circumference π diameter C πd 20π feet 2. Set up a proportion. arclength circumference C degree meaure 360º 20π degree meaure 360º 3. Multiply both ide by 20π to find the arc length π π 2π feet 6.28 feet The length of the arc i approximately 6.28 feet. U3-176
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