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Circumference and p Lesson 7.1.2

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**Circumference and p 7.1.2 California Standards: What it means for you:**

Lesson 7.1.2 Circumference and p California Standards: Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (for example, P = 2w + 2l, A = ½bh, C = pd — 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 p; know the formulas for the circumference and area of a circle. Measurement and Geometry 1.2 Know common estimates of p (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: You’ll 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 (p) estimate

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Lesson 7.1.2 Circumference and p In this Lesson, you’ll 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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Lesson 7.1.2 Circumference and p The Circumference is Another Important Measurement 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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Lesson 7.1.2 Circumference and p Example 1 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? Solution 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. Solution follows…

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**Circumference and p 7.1.2 Guided Practice**

Lesson 7.1.2 Circumference and p Guided Practice 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? 150 yd × 2 = 300 yd 2. Judy’s scarf is 60 cm long. The scarf will go around a telephone pole exactly 3 times. What is the circumference of the pole? 60 cm ÷ 3 = 20 cm 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? 5 m ÷ 5 = 1 m Solution follows…

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**Circumference and p 7.1.2 Circumference Is Related to Diameter**

Lesson 7.1.2 Circumference and p Circumference Is Related to Diameter You’ve 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. If the diameter is doubled, the circumference also doubles. 3.14 1 Its circumference is about 3.14.

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Lesson 7.1.2 Circumference and p 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. There’s 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: p p is kind of tricky — it’s a symbol, so it looks like a variable. This is one time where a symbol doesn’t stand for an unknown. It’s just the name of a special number.

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**Circumference and p 7.1.2 C = pd C = 2pr**

Lesson 7.1.2 Circumference and p You Can Use p in a Formula to Find Circumference Using C for circumference and d for diameter, you can use p to make a formula for the circumference of a circle: C = pd This formula is often written using r for radius, like this: C = 2pr

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Lesson 7.1.2 Circumference and p Example 2 A circle has a diameter of 12 inches. What is its exact circumference? Solution You can use the formula C = pd to solve this question. C = p × 12 = 12p in. You can leave p in an exact answer, because it is a number. Solution follows…

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**Circumference and p 7.1.2 Guided Practice**

Lesson 7.1.2 Circumference and p Guided Practice In Exercises 4–9, find the exact circumference of a circle with the given diameter or radius. Remember: C = pd 4. d = 20 in d = 13 ft. 6. d = 3.5 m 7. r = 5 ft. 8. r = 26 cm 9. r = 6.3 cm 20p in. 13p ft 3.5p m 10p ft 52p cm 12.6p cm Solution follows…

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**Circumference and p 7.1.2 You Can Often Use a Rounded Value of p**

Lesson 7.1.2 Circumference and p You Can Often Use a Rounded Value of p If an exercise asks you to give the exact circumference, you should always leave p in the answer. Numbers with p in them are not very easy to work with, so it is more usual to estimate the circumference using a rounded value of p. If the radius or diameter is given as a decimal, you should use 3.14 or as the value of p. 22 7 If the question uses fractions, you should use You should normally use a rounded value of p in real-life situations.

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Lesson 7.1.2 Circumference and p Example 3 Alicia has a circular swimming pool. The diameter of the pool is 30 ft. What is the circumference of Alicia’s pool? Solution Using the formula C = pd, and estimating p as 3.14, you can solve the question: C = 3.14 × 30 = 94.2 The circumference of Alicia’s pool is 94.2 ft. Solution follows…

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Lesson 7.1.2 Circumference and p Example 4 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? 1 4 Solution 22 7 You can use the formula C = 2pr, and estimate p as : C = 2 × × 3 22 7 1 4 = 2 × × 22 7 13 4 = = 572 28 143 7 3 7 = 20 So the circumference of the circle is in. 3 7 Solution follows…

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**Circumference and p 7.1.2 Guided Practice**

Lesson 7.1.2 Circumference and p Guided Practice In Exercises 10–15, calculate the circumference of a circle with the given diameter or radius. Use p = 3.14. 10. d = 15 cm r = 11 yd 12. d = 42 in r = 6 m 14. d = 9.2 ft r = 13.5 m C = pd = 3.14 × 15 = 47.1 cm C = 2pr = 2 × 3.14 × 11 = 69.1 yd C = pd = 3.14 × 42 = in. C = 2pr = 2 × 3.14 × 6 = 37.7 m C = pd = 3.14 × 9.2 = 28.9 ft C = 2pr = 2 × 3.14 × 13.5 = 84.8 m Solution follows…

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**Circumference and p 7.1.2 Guided Practice**

Lesson 7.1.2 Circumference and p Guided Practice In Exercises 16–21, calculate the circumference of a circle with the given diameter or radius. Use p = . 22 7 16. r = yd d = m 18. r = in r = m 20. d = ft d = ft 1 3 5 8 C = 2pr = 2 × × 1 3 = yd 2 21 22 7 C = pd = × 5 8 = m 27 28 22 7 4 11 3 7 C = 2pr = 2 × × 4 11 = in 2 7 22 C = 2pr = 2 × × 12 3 7 = m 6 49 22 1 9 5 17 C = pd = × 9 1 9 = ft 40 63 22 7 C = pd = × 1 5 17 = ft 119 8 22 7 Solution follows…

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**Circumference and p 7.1.2 Independent Practice**

Lesson 7.1.2 Circumference and p Independent Practice In Exercises 1–9, use a suitable estimate of p to find the circumference of a circle with the given diameter or radius. 1. r = ft r = 2.75 cm 3. d = 6.3 m d = 13 in. 5. r = m d = 0.75 yd 7. r = 3.3 ft d = m 9. r = 8.1 in. 5 6 5 ft 5 21 17.3 cm 19.8 m 40.8 in. 7 11 4 m 2.4 yd 4 3 4 m 4 21 20.7 ft 50.9 in. Solution follows…

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**Circumference and p 7.1.2 Independent Practice**

Lesson 7.1.2 Circumference and p Independent Practice 10. Complete the sentence by filling in the blanks. p is the ratio of a circle’s c _ _ _ _ _ _ _ _ _ _ _ _ to its d _ _ _ _ _ _ _. ircumference 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. Solution follows…

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**Circumference and p 7.1.2 Round Up**

Lesson 7.1.2 Circumference and p Round Up p is a really useful and important number. It might look a little strange, but remember, it is just a fixed number. You will see p again many times in math, including in the next Lesson of this book.

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