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EnglishSpanishCircles and Circumference6.1How can you find the circumferenceSSTATESTANDARDSof a circle?MA.6.G.4.1MA.6.G.4.2Archimedes was a Greek mathematician, physicist,engineer, and astronomer.Archimedes discovered that in any circle theratio of circumference to diameter is alwaysthe same. Archimedes called this ratio pi, or π(a letter from the Greek alphabet).CircumferenceDiameterπ ——In Activities 1 and 2,you will use the samestrategy Archimedesused to approximate π.1cumferencecireret radiusmadiACTIVITY: Approximating PiWork with a partner. Copy the table. Record your results in the table. Measure the perimeter of the large squarein millimeters. Measure the diameter of the circle inmillimeters. Measure the perimeter of the small squarein millimeters. Calculate the ratios of the two perimeters tothe diameter. Sides ofPolygonThe average of these two ratios is an approximation of 𝛑.LargePerimeterDiameterof Circle46810238Chapter 6Large SquareSmall SquareCircles and AreaSmallPerimeterLarge PerimeterDiameter——Small PerimeterDiameter——Averageof Ratios
EnglishSpanishA page from “SirCumference and theFirst Round Table” byCindy Neuschwander.2ACTIVITY: Approximating PiContinue your approximation of pi. Complete the table using a hexagon(6 sides), an octagon (8 sides), and a decagon (10 sides).Large Hexagona.Small HexagonLarge DecagonLarge OctagonSmall DecagonSmall Octagonb.c.d. From the table, what can you conclude about the value of π ?Explain your reasoning.e. Archimedes calculated the value of π using polygons having 96 sides.Do you think his calculations were more or less accurate than yours?3. IN YOUR OWN WORDS Now that you know an approximation forpi, explain how you can use it to find the circumference of a circle.Write a formula for the circumference C of a circle whose diameteris d. Draw a circle and use your formula to find the circumference.Use what you learned about circles and circumference to completeExercises 10–12 on page 243.Section 6.1Circles and Circumference239
English6.1SpanishLessonLesson TutorialsA circle is the set of all points in a plane that are the same distance froma point called the center.circleKey Vocabularycircle, p. 240center, p. 240radius, p. 240diameter, p. 240circumference, p. 241pi, p. 241semicircle, p. 242centerThe radius is the distancefrom the center to anypoint on the circle.The diameter is thedistance across the circlethrough the center.Radius and DiameterThe diameter d of a circle is twice the radius r. The radius r ofa circle is one-half the diameter d.WordsAlgebraEXAMPLE1Diameter: d 2rd2Radius: r —Finding a Radius and a Diametera. The diameter of a circle is20 feet. Find the radius.b. The radius of a circle is7 meters. Find the diameter.20 ft7md2Radius of a circle —202Substitute 20 for d. 2(7)Substitute 7 for r. 10Divide. 14Multiply.r —The radius is 10 feet.d 2rDiameter of a circleThe diameter is 14 meters.1. The diameter of a circle is 16 centimeters. Find the radius.Exercises 4–9240Chapter 62. The radius of a circle is 9 yards. Find the diameter.Circles and Area
EnglishSpanishThe distance around a circle is called the circumference. The ratiocircumferencediameter—— is the same for every circle and is represented by the Greek227letter π, called pi. The value of π can be approximated as 3.14 or —.Study TipWhen the radius ordiameter is a multipleof 7, it is easier to use227— as the estimate ofπ.Circumference of a CircleWordsAlgebraEXAMPLE2CThe circumference C of a circle isequal to π times the diameter d orπ times twice the radius r.C πdordrC 2π rFinding Circumferences of Circlesa. Find the circumference of the flying disc. Use 3.14 for 𝛑.C 2π r5 in. Write formula for circumference. 2 3.14 5Substitute 3.14 for π and 5 for r. 31.4Multiply.The circumference is about 31.4 inches.227b. Find the circumference of the watch face. Use — for 𝛑.C πd22728 mmWrite formula for circumference. 227 — 28Substitute — for π and 28 for d. 88Multiply.The circumference is about 88 millimeters.227Find the circumference of the object. Use 3.14 or — for 𝛑.Exercises 10–133.4.5.2 cmm14 ftSection 6.19 in.Circles and Circumference241
EnglishSpanishEXAMPLE3Standardized Test PracticeThe diameter of the new roll of caution tape decreases 3.25 inchesafter a construction worker uses some of the tape. Which is the bestestimate of the circumference of the roll after the decrease?10 in.A 9 inches B 16 inches C 21 inches D 30 inches After the decrease, the diameter of the roll is 10 3.25 6.75 inches.C πdWrite formula for circumference. 3.14 6.75 Substitute 3.14 for π and 6.75 for d. 3 7Round 3.14 down to 3. Round 6.75 up to 7. 21Multiply.The correct answer is C .6. WHAT IF? In Example 3, the diameter of the roll of tape decreases5.75 inches. Estimate the circumference after the decrease.EXAMPLE4Finding the Perimeter of a Semicircular RegionA semicircle is one-half of a circle. Findthe perimeter of the semicircular region.The straight side is 6 meters long. The distancearound the curved part is half the circumferenceof a circle with a diameter of 6 meters.πd2C —6mDivide the circumference by 2. 3.14 62 —Substitute 3.14 for π and 6 for d. 9.42Simplify.So, the perimeter is about 6 9.42 15.42 meters.Find the perimeter of the semicircular region.Exercises 15 and 167.8.2 ft242Chapter 6Circles and Area7 cm9.15 in.
EnglishSpanishExercises6.1Help with Homework1. VOCABULARY What is the relationship between the radius and the diameterof a circle?2. WHICH ONE DOESN’T BELONG? Which phrase does not belong with the otherthree? Explain your reasoning.π times twice the radiusthe distance around a circleπ times the diameterthe distance from the center to any point on the circle3. OPEN-ENDED Choose a real-life circular object. Explain why you might need toknow its circumference. Then find the circumference.6) 39 (- 3) 3 (- 9) 4 (- 1)9 (-Find the radius of the button.15.4.6.28 mm5 cm31in.2Find the diameter of the object.7.8.6 cm9.0.8 ft2 in.227Find the circumference of the pizza. Use 3.14 or — for 𝛑.2 10.11.10 in.12.7 in.18 in.Section 6.1Circles and Circumference243
EnglishSpanish13. SINKHOLE A circular sinkhole has a radius of 12 meters. A week later, ithas a diameter of 48 meters. How much greater is the circumference ofthe sinkhole compared to the previous week?14. REASONING Consider the circles A, B, C, and D.BACD2 ft8 ft50 in.10 in.a. Copy and complete the table.b. Which circle has the greatestcircumference?c. Which circle has the leastcircumference?CircleARadiusDiameterBC10 inches8 feet50 inches2 feetFind the perimeter of the window.4 15.16.20 cm3 ftFind the circumferences of both circles.17.18.5 cm19.9 ft5 cm22 m2.5 ft20. WIRE A wire is bent to form four semicircles. How long is the wire?32 cm32 cm32 cm32 cmcircumferencediameter21. CRITICAL THINKING Because the ratio —— is the same for everycircumferenceradiuscircle, is the ratio —— the same for every circle? Explain.244Chapter 6Circles and AreaD
EnglishSpanish22. AROUND THE WORLD “Lines” of latitude on Earthare actually circles. The Tropic of Cancer isthe northernmost line of latitude at which theSun appears directly overhead at noon. TheTropic of Cancer has a radius of 5854 kilometers.Tropic ofCancerTo qualify for an around-the-world speed record,a pilot must cover a distance no less than thecircumference of the Tropic of Cancer, cross allmeridians, and end on the same airfield wherehe started.a. What is the minimum distance that a pilotmust fly to qualify for an around-the-worldspeed record?EquatorMeridianb. RESEARCH Estimate the time it would take for a pilotto qualify for the speed record.23. BICYCLE Bicycles in the late 1800s looked very different thanthey do today.60 in.9 in.a. How many rotations does each tire make after traveling600 feet? Round your answers to the nearest whole number.b. Would you rather ride a bicycle made with two large wheelsor two small wheels? Explain.24.The length of the minute hand is150% of the length of the hour hand.36 mma. What distance will the tip of the minutehand move in 45 minutes? Explain howyou found your answer.b. In 1 hour, how much farther does the tip ofthe minute hand move than the tip of thehour hand? Explain how you found your answer.Find the perimeter of the figure. SKILLS REVIEW HANDBOOK26.25.6m5m4 ft27.16 in.12 in.9m7 ft12 in.25 in.28. MULTIPLE CHOICE What is the median of the data set?SECTION 5.512, 25, 16, 9, 5, 22, 27, 20A 7 B 16 C 17 Section 6.1D 18 Circles and Circumference245
The radius of a circle is 9 yards. Find the diameter. A circle is the set of all points in a plane that are the same distance from a point called the center. circle center The radius is the distance from the center to any point on the circle. The diameter is the distance across the circle through the center. Key Vocabulary circle, p. 240 center .
