11.1 Circumference and Arc Length

Transcription

1 . ircumference and rc Length Essential Question How can you find the length of a circular arc? Finding the Length of a ircular rc Work with a partner. Find the length of each red circular arc. a. entire circle b. one-fourth of a circle y y x x c. one-third of a circle d. five-eighths of a circle 4 y 4 y 4 4 x 4 4 x 4 4 Using rc Length LOOKING FOR REGULRITY IN REETED RESONING To be proficient in math, you need to notice if calculations are repeated and look both for general methods and for shortcuts. Work with a partner. The rider is attempting to stop with the front tire of the motorcycle in the painted rectangular box for a skills test. The front tire makes exactly one-half additional revolution before stopping. The diameter of the tire is inches. Is the front tire still in contact with the painted box? Explain. ommunicate Your nswer. How can you find the length of a circular arc? 4. motorcycle tire has a diameter of 4 inches. pproximately how many inches does the motorcycle travel when its front tire makes three-fourths of a revolution? ft Section. ircumference and rc Length 69

2 . Lesson What You Will Learn ore Vocabulary circumference, p. 640 arc length, p. 64 radian, p. 64 revious circle diameter radius Use the formula for circumference. Use arc lengths to find measures. Solve real-life problems. Measure angles in radians. Using the Formula for ircumference The circumference of a circle is the distance around the circle. onsider a regular polygon inscribed in a circle. s the number of sides increases, the polygon approximates the circle and the ratio of the perimeter of the polygon to the diameter of the circle approaches π For all circles, the ratio of the circumference to the diameter d is the same. This ratio is = π. Solving for yields the formula for the circumference of a circle, d = πd. ecause d = r, you can also write the formula as = π(r) = πr. ore oncept ircumference of a ircle The circumference of a circle is = πd or = πr, where d is the diameter of the circle and r is the radius of the circle. d r = πd = πr Using the Formula for ircumference TTENDING TO REISION You have sometimes used.4 to approximate the value of π. Throughout this chapter, you should use the π key on a calculator, then round to the hundredths place unless instructed otherwise. Find each indicated measure. a. circumference of a circle with a radius of 9 centimeters b. radius of a circle with a circumference of 6 meters a. = πr = π 9 = 8π 6. The circumference is about 6. centimeters. b. = πr 6 = πr 6 π = r 4.4 r The radius is about 4.4 meters. Monitoring rogress 640 hapter ircumference, rea, and Volume Help in English and Spanish at igideasmath.com. Find the circumference of a circle with a diameter of inches.. Find the diameter of a circle with a circumference of 7 feet.

3 Using rc Lengths to Find Measures n arc length is a portion of the circumference of a circle. You can use the measure of the arc (in degrees) to find its length (in linear units). ore oncept rc Length In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 60. rc length of = m πr 60, or rc length of = m 60 πr r Using rc Lengths to Find Measures Find each indicated measure. a. arc length of b. circumference of Z c. m RS 8 cm 60 Z 40 X 4.9 in. Y.8 m S T R 44 m a. rc length of = 60 π(8) 8.8 cm b. rc length of XY = m XY c. rc length of RS = m RS 60 πr = π(.8) = m RS = π(.8) = m RS 7.7 in. = 6 m RS Monitoring rogress Help in English and Spanish at igideasmath.com Find the indicated measure.. arc length of Q 4. circumference of N. radius of G Q 9 yd 7 R S L 70 N 6.6 m M E G 0 0. ft F Section. ircumference and rc Length 64

4 Solving Real-Life roblems Using ircumference to Find Distance Traveled The dimensions of a car tire are shown. To the nearest foot, how far does the tire travel when it makes revolutions?. in. OMMON ERROR lways pay attention to units. In Example, you need to convert units to get a correct answer. Step Find the diameter of the tire. d = + (.) = 6 in.. in. Step Find the circumference of the tire. = πd = π 6 = 6π in. Step Find the distance the tire travels in revolutions. In one revolution, the tire travels a distance equal to its circumference. In revolutions, the tire travels a distance equal to times its circumference. Distance traveled = Number of revolutions ircumference = 6π. in. Step 4 Use unit analysis. hange. inches to feet.. in. ft = 0. ft in. The tire travels approximately 0 feet. in. Using rc Length to Find Distances The curves at the ends of the track shown are 80 arcs of circles. The radius of the arc for a runner on the red path shown is 6.8 meters. bout how far does this runner travel to go once around the track? Round to the nearest tenth of a meter. 6.8 m 44.0 m The path of the runner on the red path is made of two straight sections and two semicircles. To find the total distance, find the sum of the lengths of each part. Distance = Length of each straight section + = (84.9) + ( π 6.8 ) Length of each semicircle The runner on the red path travels about meters. Monitoring rogress 84.9 m Help in English and Spanish at igideasmath.com 6. car tire has a diameter of 8 inches. How many revolutions does the tire make while traveling 00 feet? 7. In Example 4, the radius of the arc for a runner on the blue path is 44.0 meters, as shown in the diagram. bout how far does this runner travel to go once around the track? Round to the nearest tenth of a meter. 64 hapter ircumference, rea, and Volume

5 Measuring ngles in Radians Recall that in a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 60. To see why, consider the diagram. r circle of radius has circumference π, so the arc length of D is m D 60 π. D Recall that all circles are similar and corresponding lengths of similar figures are proportional. ecause m = m D, and D are corresponding arcs. So, you can write the following proportion. rc length of = rc length of r D rc length of = r rc length of D rc length of = r m D 60 π This form of the equation shows that the arc length associated with a central angle is proportional to the radius of the circle. The constant of proportionality, m D 60 π, is defined to be the radian measure of the central angle associated with the arc. In a circle of radius, the radian measure of a given central angle can be thought of as the length of the arc associated with the angle. The radian measure of a complete circle (60 ) is exactly π radians, because the circumference of a circle of radius is exactly π. You can use this fact to convert from degree measure to radian measure and vice versa. ore oncept onverting between Degrees and Radians Degrees to radians Multiply degree measure by π radians, or π radians Radians to degrees Multiply radian measure by 60 π radians, or 80 π radians. onverting between Degree and Radian Measure a. onvert 4 to radians. b. onvert π radians to degrees. a. 4 π radians = π 80 4 radian b. π radians 80 π radians = 70 So, 4 = π 4 radian. So, π radians = 70. Monitoring rogress Help in English and Spanish at igideasmath.com 8. onvert to radians. 9. onvert 4π radians to degrees. Section. ircumference and rc Length 64

6 . Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. OMLETE THE SENTENE The circumference of a circle with diameter d is =.. WRITING Describe the difference between an arc measure and an arc length. Monitoring rogress and Modeling with Mathematics In Exercises 0, find the indicated measure. (See Examples and.). circumference of a circle with a radius of 6 inches 4. diameter of a circle with a circumference of 6 feet. radius of a circle with a circumference of 8π 6. exact circumference of a circle with a diameter of inches 7. arc length of 8 ft 4 8. m DE D 8.7 in. E Q 0 in.. ROLEM SOLVING measuring wheel is used to calculate the length of a path. The diameter of the wheel is 8 inches. The wheel makes 87 complete revolutions along the length of the path. To the nearest foot, how long is the path? (See Example.) 4. ROLEM SOLVING You ride your bicycle 40 meters. How many complete revolutions does the front wheel make? 9. circumference of 0. radius of R F L 7. m G cm 60. ERROR NLYSIS Describe and correct the error in finding the circumference of. 9 in. = πr = π(9) =8π in. R M In Exercises 8, find the perimeter of the shaded region. (See Example 4.). 6. cm. ERROR NLYSIS Describe and correct the error in finding the length of GH. G 7 cm H rc length of GH = m GH πr = 7 π() = 70π cm hapter ircumference, rea, and Volume

7 x + y = In Exercises and 6, find the circumference of the circle with the given equation. Write the circumference in terms of π (x + ) + (y ) = 9 7. USING STRUTURE semicircle has endpoints In Exercises 9, convert the angle measure. (See Example.) (, ) and (, 8). Find the arc length of the semicircle. is an arc on a circle with radius r. 8. RESONING EF 9. onvert 70 to radians. Let x be the measure of EF. Describe the effect on the length of EF if you (a) double the radius of the circle, and (b) double the measure of EF. 0. onvert 00 to radians. π. onvert radians to degrees. 9. MKING N RGUMENT Your friend claims that it is π 8 possible for two arcs with the same measure to have different arc lengths. Is your friend correct? Explain your reasoning.. onvert radian to degrees.. ROLEM SOLVING The London Eye is a Ferris wheel in London, England, that travels at a speed of 0.6 meter per second. How many minutes does it take the London Eye to complete one full revolution? 67. m 0. ROLEM SOLVING Over 000 years ago, the Greek scholar Eratosthenes estimated Earth s circumference by assuming that the Sun s rays were parallel. He chose a day when the Sun shone straight down into a well in the city of Syene. t noon, he measured the angle the Sun s rays made with a vertical stick in the city of lexandria. Eratosthenes assumed that the distance from Syene to lexandria was equal to about 7 miles. Explain how Eratosthenes was able to use this information to estimate Earth s circumference. Then estimate Earth s circumference. t ligh m = 7. lexandria sun stick t ligh sun well 4. ROLEM SOLVING You are planning to plant a circular garden adjacent to one of the corners of a building, as shown. You can use up to 8 feet of fence to make a border around the garden. What radius (in feet) can the garden have? hoose all that apply. Explain your reasoning. Syene center of Earth Not drawn to scale. NLYZING RELTIONSHIS In, the ratio of the length of Q to the length of RS is to. What is the ratio of m Q to m RS? 4 to to to 4 D to. NLYZING RELTIONSHIS 4 arc in and a D 0 0 arc in have the same length. What is the ratio of the radius r of to the radius r of? Explain your reasoning. Section. int_math_pe_0.indd 64 ircumference and rc Length 64 /0/ : M

8 . ROLEM SOLVING How many revolutions does the smaller gear complete during a single revolution of the larger gear? 7 4. USING STRUTURE Find the circumference of each circle. a. a circle circumscribed about a right triangle whose legs are inches and 6 inches long b. a circle circumscribed about a square with a side length of 6 centimeters c. a circle inscribed in an equilateral triangle with a side length of 9 inches. REWRITING FORMUL Write a formula in terms of the measure θ of the central angle (in radians) that can be used to find the length of an arc of a circle. Then use this formula to find the length of an arc of a circle with a radius of 4 inches and a central angle of π 4 radians. 6. HOW DO YOU SEE IT? ompare the circumference of to the length of DE. Explain your reasoning. 7. MKING N RGUMENT In the diagram, the measure of the red shaded angle is 0. The arc length a is. Your classmate claims that it is possible to find the circumference of the blue circle without finding the radius of either circle. Is your classmate correct? Explain your reasoning. r r a D E 8. MODELING WITH MTHEMTIS What is the measure (in radians) of the angle formed by the hands of a clock at each time? Explain your reasoning. a. :0 p.m. b. : p.m. 9. MTHEMTIL ONNETIONS The sum of the circumferences of circles,, and is 6π. Find. x x 40. THOUGHT ROVOKING Is π a rational number? ompare the rational number to π. Find a different rational number that is even closer to π. 4. ROOF The circles in the diagram are concentric and FG GH. rove that JK and NG have the same length. L K M N F J x G H 4. REETED RESONING is divided into four congruent segments, and semicircles with radius r are drawn. r a. What is the sum of the four arc lengths? b. What would the sum of the arc lengths be if was divided into 8 congruent segments? 6 congruent segments? n congruent segments? Explain your reasoning. Maintaining Mathematical roficiency Find the area of the polygon with the given vertices. (Skills Review Handbook) Reviewing what you learned in previous grades and lessons 4. X(, 4), Y(8, ), Z(, ) 44. L(, ), M(4, ), N(4, ), (, ) 646 hapter ircumference, rea, and Volume