1 lesson 7.1.2 circumference and. 2 lesson 7.1.2 circumference and california standards: algebra and...

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Lesson 7.1.2Lesson 7.1.2

Circumference and Circumference and

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Lesson

7.1.2 Circumference and Circumference and

California Standards:Algebra and Functions 3.1Use variables in expressions describing geometric quantities (for example, P = 2w + 2l, A = ½bh, 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 3.2Express in symbolic form simple relationships arising from geometry.

Measurement and Geometry 1.1Understand the concept of a constant such as ; know the formulas for the circumference and area of a circle.

Measurement and Geometry 1.2Know 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: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 ()• estimate

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Lesson


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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The Circumference is Another Important Measurement

Lesson


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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Example 1

Solution follows…

Lesson


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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Guided Practice

Solution follows…

Lesson


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. 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?

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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3.14

Circumference Is Related to Diameter

Lesson


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.

Its circumference is about 3.14.

If the diameter is doubled, the circumference also doubles.

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Lesson


In fact, circumference ÷ diameter for any circle is always 3.1415926535897…

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:

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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You Can Use in a Formula to Find Circumference

Lesson


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 = 2r

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Example 2

Solution follows…

Lesson


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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Guided Practice

Solution follows…

Lesson


4. d = 20 in. 5. d = 13 ft.

6. d = 3.5 m 7. r = 5 ft.

8. r = 26 cm 9. 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 m 10 ft

52 cm 12.6 cm

20 in.

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You Can Often Use a Rounded Value of

Lesson


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 3.142 as the value of .

If the question uses fractions, you should use . 227

You should normally use a rounded value of in real-life situations.

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Example 3

Solution follows…

Lesson


Solution

Alicia has a circular swimming pool. The diameter of the pool is 30 ft. What is the circumference of Alicia’s 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 Alicia’s pool is 94.2 ft.

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Example 4

Solution follows…

Lesson


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 = 2r, and estimate as :227

C = 2 × × 3227

= 2 × × 227

134

= = 57228

1437

37= 20 So the circumference of the circle is 20 in.3

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10. 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


In Exercises 10–15, calculate the circumference of a circle with the given diameter or radius. Use = 3.14.

C = d = 3.14 × 15

= 47.1 cm

C = 2r = 2 × 3.14 × 11

= 69.1 yd

C = d = 3.14 × 42

= 131.9 in.

C = 2r = 2 × 3.14 × 6

= 37.7 m

C = d = 3.14 × 9.2

= 28.9 ft

C = 2r = 2 × 3.14 × 13.5

= 84.8 m

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Guided Practice

Lesson


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 = .22

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58

411

37

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517

13 C = 2r

= 2 × ×13

= 2 yd221

227

C = d = ×

58

= 1 m2728

227

C = 2r

= 2 × ×4

11

= 2 in27

227

C = 2r

= 2 × × 1237

= 78 m649

227

C = d = × 9

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= 28 ft4063

227

C = d = × 1

517

= 4 ft1198

227

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Independent Practice

Solution follows…

Lesson


In Exercises 1–9, use a suitable estimate of to find the circumference of a circle with the given diameter or radius.

1. r = ft 2. r = 2.75 cm

3. d = 6.3 m 4. d = 13 in.

5. r = m 6. d = 0.75 yd

7. r = 3.3 ft 8. d = m

9. r = 8.1 in.

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711

43

5 ft521

4 m421

17.3 cm

19.8 m 40.8 in.

4 m 2.4 yd

20.7 ft

50.9 in.

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Independent Practice

Solution follows…

Lesson


ircumference

10. Complete the sentence by filling in the blanks.

is the ratio of a circle’s 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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Lesson


Round UpRound 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.

1 lesson 7.1.2 circumference and. 2 lesson 7.1.2 circumference and california standards: algebra and...

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