1. Circumference and p
Lesson 7.1.2

2. 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

3. 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.

4. 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.

5. 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…

6. 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…

7. 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.

8. 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.

9. 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

10. 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…

11. 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…

12. 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.

13. 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…

14. 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…

15. 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…

16. 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…

17. 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…

18. 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…

19. 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.