Central Angles, Arc Length, and Sector Area
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1 CHAPTER 5 A Central Angles, Arc Length, and Sector Area c GOAL Identify central angles and determine arc length and sector area formed by a central angle. Yo will need a calclator a compass a protractor Learn abot the Math An angle whose vertex is the centre of a circle and whose sides pass throgh a pair of points on the circle is called a central angle. The following symbol, prononced theta, is sed to represent a central angle:. Each central angle forms an arc length between the points that it passes throgh on the circle. The arc length of a circle can be fond by first calclating what fraction of the circle is represented by the central angle. Then yo find the circmference of the circle. Finally, yo find the fraction of that circmference. For example, if a circle has a radis of 0.0 cm and a central angle of 60, to calclate the arc length for that angle we mst first determine what fraction of the circle is represented by the central angle 60. Since there are 360 in a fll circle, we can find the fraction of a circle by simplifying ; The sector represents of the circle. 6 central angle an angle whose vertex is the centre of a circle and whose sides pass throgh a pair of points on the circle arc length the length of a section of a circle s circmference To find the arc length, we now need to find the circmference of the circle. Circmference of a circle is eqal to 2pr. C 5 2pr 5 2p cm The circmference of this circle is abot 62.8 cm. To find the arc length, we need to find of 62.8 cm cm 6 3 The arc length formed by this central angle is abot 0.5 cm.
2 To make the process qicker, yo can refer to the formla sed to determine the arc length formed by a central angle: 5, where is the arc length, and pr r is the radis of the circle. A sector of a circle is a pie-shaped region bonded by an arc and an angle. If yo know the radis of a circle and a central angle, yo can determine the sector area for that angle by first calclating the fraction of the circle that the sector represents, then finding the area of the entire circle, and finally calclating that fraction of the total area. sector of a circle pie-shaped region bonded by an arc and an angle? For example, if we again se a circle with a radis of 0 cm and a central angle of 60, we can determine the sector area by first finding the fraction of the circle, which we calclated earlier to be 6 and then calclating the area of the entire circle. The area of a circle is eqal to pr 2. The area of this circle can be fond sing this formla. A 5 pr 2 5 p (0) cm 2 Now we need to find 6 of 34 cm 2 to determine the area of the sector formed by this central angle cm The area of the sector formed by central angle is abot 52.3 cm 2. The formla sed to determine the sector area for any central angle is A S 5 r 2, where A S is the area of the sector p and r is the radis of the circle. Denise needs to determine the arc length and sector area formed by a 40 central angle on a circle with a radis of 5 cm. How can Denise calclate the arc length and sector area for this central angle? A. Sketch a circle on a sheet of paper with a radis of 5 cm. B. Use a protractor to draw a 40 central angle on the circle. 2 Nelson Mathematics Secondary Year Two, Cycle One
3 C. Find the arc length between the two points where this angle passes throgh the circle, sing the formla pr D. Find the area for the sector of this circle formed by the central angle and the arc formed in step B, sing the formla A s pr 2 Reflecting. What is a central angle of a circle? 2. What is the formla for finding the arc length formed by the points where a central angle passes throgh the circle? 3. Why mst yo calclate the area of the entire circle in order to calclate the area of jst one sector? Work with the Math Example : Determining the arc length and sector area for a central angle of a circle Calclate the arc length and the area of a sector formed by a 30 central angle on a circle with a radis of 2 cm. Li Ming s Soltion First, I will calclate the arc length. To do this I mst first determine what fraction of the circle is represented by the central angle 30. Since there are in a fll circle, I can find the fraction of a circle by simplifying 360; The sector represents of the circle. 2 To find the arc length, I now need to find the circmference of the entire circle. Circmference of a circle is eqal to 2pr. C 5 2pr 5 2p cm 3
4 The circmference of this circle is abot 2.56 cm. To find the arc length, I need to find of 2.56 cm cm 2 3 The arc length formed by this central angle is abot.047 cm. Next, I will determine the area of the sector by calclating the area of the entire circle and then mltiplying by 2. The area of a circle is eqal to pr 2. The area of this circle then can be fond sing this formla. 2 A 5 pr 2 5 p (2) cm 2 Now I need to find of 2.56 cm 2, to determine the area of the sector formed by this central angle cm2 2 The area of the sector formed by this central angle is abot cm 2. A Checking 4. Calclate the arc length and the area of the sector formed by a 20 central angle on a circle with a radis of 5 cm. B Practising 5. Complete each of the following statements with a term that will make it a tre statement. a) A of a circle is an angle whose vertex is the centre of a circle and whose sides pass throgh a pair of points on the circle. b) A of a circle is a pie-shaped region bonded by an arc and an angle. c) is the length of a section of a circle s circmference. 6. Determine the area of the sector formed by each of the following central angles, on a circle with the given radis or diameter. a) central angle: 40 radis: 5 m b) central angle: 55 diameter: 28 cm c) central angle: 60 radis: 5.0 cm d) central angle: 90 diameter: 6.0 cm 4 Nelson Mathematics Secondary Year Two, Cycle One
5 7. Determine the arc length formed by each of the following central angles, on a circle with the given radis or diameter. a) central angle: 25 diameter: 22 m b) central angle: 0 radis: 25 cm c) central angle: 5 diameter: 66 mm d) central angle: 80 radis: 6.0 m 8. Determine the arc length and area for the sectors formed by each of the following central angles, on a circle with the given radis or diameter. a) central angle: 70 radis: 4 m b) central angle: 75 diameter: 20 cm c) central angle: 40 radis: 3 m d) central angle: 05 diameter: 8 cm C Extending 9. For a particlar circle, the central angle is x º. The corresponding arc length is, and the corresponding sector area is A S. For this circle, what wold be the arc length and sector area corresponding to a central angle of 2x º? 0. For a particlar circle with radis r, the arc length corresponding to a central angle of x º is and the corresponding sector area is A S. For a circle with radis 2r what wold be the arc length and sector area corresponding to a central angle of x º? 5
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