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1 Lesson Circumference and

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2 Lesson Circumference and California Standards: Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (for example, P = 2 w + 2 l, A = ½ bh, C = d the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). Algebra and Functions 3.2 Express in symbolic form simple relationships arising from geometry. Measurement and Geometry 1.1 Understand the concept of a constant such as ; know the formulas for the circumference and area of a circle. Measurement and Geometry 1.2 Know common estimates of (3.14; 22/7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements. What it means for you: Youll learn about a special number which you can use in a formula to find the circumference of a circle. Key words: circumference diameter radius pi ( ) estimate

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3 Lesson Circumference and In this Lesson, youll learn about the distance around the outside of a circle, which is called the circumference. There is a clever way to work out the circumference if you know the radius or diameter of the circle, and it involves a special number.

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4 The Circumference is Another Important Measurement Lesson Circumference and The distance around the edge of a circle is called the circumference. Imagine that a circle is like a loop of string. If you cut the string and laid it out flat, you could measure the length of the string. If you could cut a circle and lay it out in the same way, the length you measured would be the circumference.

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5 Example 1 Solution follows… Lesson Circumference and Solution Norman is walking around the edge of a large, circular rug. By the time he is one-third of the way around it, he has walked 8 feet. What is the circumference of the rug? The circumference is the distance all the way around the rug. Norman has walked one-third of the way around, so the circumference is 3 times the distance he has walked. 3 × 8 = 24 The circumference of the rug is 24 feet.

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6 Guided Practice Solution follows… Lesson Circumference and 1.Jack is running around a circular track. When he is halfway around, he has run 150 yards. What is the circumference of the track? 2.Judys scarf is 60 cm long. The scarf will go around a telephone pole exactly 3 times. What is the circumference of the pole? 3.Jorge is riding his bicycle. When he travels 5 m, his front wheel goes around 5 times. What is the circumference of the wheel? 150 yd × 2 = 300 yd 60 cm ÷ 3 = 20 cm 5 m ÷ 5 = 1 m

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Circumference Is Related to Diameter Lesson Circumference and Youve seen that the diameter is always twice the radius. Diameter and circumference are related too the circumference of a circle is always just over three times its diameter. The diameter of this circle is 1. Its circumference is about If the diameter is doubled, the circumference also doubles. 1

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8 Lesson Circumference and In fact, circumference ÷ diameter for any circle is always … This number is very important in math. The full version of it is a never-ending decimal. Theres a special symbol mathematicians use instead of writing out the full number. The symbol is a Greek letter called pi (pronounced pie). It looks like this: is kind of tricky its a symbol, so it looks like a variable. This is one time where a symbol doesnt stand for an unknown. Its just the name of a special number.

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9 You Can Use in a Formula to Find Circumference Lesson Circumference and Using C for circumference and d for diameter, you can use to make a formula for the circumference of a circle: C = d This formula is often written using r for radius, like this: C = 2 r

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10 Example 2 Solution follows… Lesson Circumference and Solution A circle has a diameter of 12 inches. What is its exact circumference? You can use the formula C = d to solve this question. C = × 12 = 12 in. You can leave in an exact answer, because it is a number.

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11 Guided Practice Solution follows… Lesson Circumference and 4. d = 20 in.5. d = 13 ft. 6. d = 3.5 m7. r = 5 ft. 8. r = 26 cm9. r = 6.3 cm In Exercises 4–9, find the exact circumference of a circle with the given diameter or radius. Remember: C = d 13 ft 3.5 m10 ft 52 cm 12.6 cm 20 in.

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12 You Can Often Use a Rounded Value of Lesson Circumference and If an exercise asks you to give the exact circumference, you should always leave in the answer. Numbers with in them are not very easy to work with, so it is more usual to estimate the circumference using a rounded value of. If the radius or diameter is given as a decimal, you should use 3.14 or as the value of. If the question uses fractions, you should use You should normally use a rounded value of in real-life situations.

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13 Example 3 Solution follows… Lesson Circumference and Solution Alicia has a circular swimming pool. The diameter of the pool is 30 ft. What is the circumference of Alicias pool? Using the formula C = d, and estimating as 3.14, you can solve the question: C = 3.14 × 30 = 94.2 The circumference of Alicias pool is 94.2 ft.

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14 Example 4 Solution follows… Lesson Circumference and Solution T.J. is cutting a circle of paper to use on a poster for a math project. The circle has a radius of 3 in. What is its circumference? You can use the formula C = 2 r, and estimate as : 22 7 C = 2 × × = 2 × × = = 20 So the circumference of the circle is 20 in

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d = 15 cm 11. r = 11 yd 12. d = 42 in. 13. r = 6 m 14. d = 9.2 ft 15. r = 13.5 m Guided Practice Solution follows… Lesson Circumference and In Exercises 10–15, calculate the circumference of a circle with the given diameter or radius. Use = C = d = 3.14 × 15 = 47.1 cm C = 2 r = 2 × 3.14 × 11 = 69.1 yd C = d = 3.14 × 42 = in. C = 2 r = 2 × 3.14 × 6 = 37.7 m C = d = 3.14 × 9.2 = 28.9 ft C = 2 r = 2 × 3.14 × 13.5 = 84.8 m

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16 Solution follows… Guided Practice Lesson Circumference and 16. r = yd 17. d = m 18. r = in. 19. r = 12 m 20. d = 9 ft 21. d = 1 ft In Exercises 16–21, calculate the circumference of a circle with the given diameter or radius. Use = C = 2 r = 2 × × 1 3 = 2 yd C = d = × 5 8 = 1 m C = 2 r = 2 × × 4 11 = 2 in C = 2 r = 2 × × = 78 m C = d = × = 28 ft C = d = × = 4 ft

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17 Independent Practice Solution follows… Lesson Circumference and In Exercises 1–9, use a suitable estimate of to find the circumference of a circle with the given diameter or radius. 1. r = ft2. r = 2.75 cm 3. d = 6.3 m4. d = 13 in. 5. r = m6. d = 0.75 yd 7. r = 3.3 ft8. d = m 9. r = 8.1 in ft m cm 19.8 m 40.8 in. 4 m 2.4 yd 20.7 ft 50.9 in.

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18 Independent Practice Solution follows… Lesson Circumference and ircumference 10.Complete the sentence by filling in the blanks. is the ratio of a circles c _ _ _ _ _ _ _ _ _ _ _ _ to its d _ _ _ _ _ _ _. iameter 11.A good estimate of the height of an elephant from its foot to its shoulder is twice the circumference of its (roughly circular) foot. A zoologist in Africa finds an elephant footprint with a 12-inch diameter. Estimate how tall the elephant that left the footprint is. About 75 in.

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19 Lesson Circumference and Round Up is a really useful and important number. You will see again many times in math, including in the next Lesson of this book. It might look a little strange, but remember, it is just a fixed number.

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