1. 11.1 Circumference and Arc Length
Geometry
EQ: How can you find the length of a circular arc?

2. Geometry 11.1 Circumference and Arc Length
Goals
Find the circumference of a circle.
Find the length of an arc.
Solve problems using circumference and arc length.
February 18, 2019
Geometry 11.1 Circumference and Arc Length

3. Geometry 11.1 Circumference and Arc Length
The circumference is the distance around a circle.
February 18, 2019
Geometry 11.1 Circumference and Arc Length

4. Geometry 11.1 Circumference and Arc LengthPi
The ratio of the circumference to the diameter is called pi.
Use the Greek letter for pi:  d C
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Geometry 11.1 Circumference and Arc Length

5. Geometry 11.1 Circumference and Arc Length
More About Pi.
  …
The fraction 22/7 is close.
355/113 is closer.
Current computer calculations have pi to over 5 trillion places.
February 18, 2019
Geometry 11.1 Circumference and Arc Length

6. Pi to a few decimal places:…
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Geometry 11.1 Circumference and Arc Length

7. Geometry 11.1 Circumference and Arc Length
Pi is Irrational
 does not repeat.
 does not terminate.
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Geometry 11.1 Circumference and Arc Length

8. Geometry 11.1 Circumference and Arc Length
You can find the circumference if you know the diameter (d) or the radius (r) of the circle.
February 18, 2019
Geometry 11.1 Circumference and Arc Length

9. Geometry 11.1 Circumference and Arc Length
Example
Find the circumference of a circle with a radius of 8.
This is the circumference “in terms of pi”.
When computing C in decimal, use 3.14 or the pi button.
February 18, 2019
Geometry 11.1 Circumference and Arc Length

10. Geometry 11.1 Circumference and Arc Length
The ratio of the length of an arc to the circumference of a circle is equal to the ratio of the measure of the arc to 360. A B
Recall: the measure of the arc is equal to the central angle that intercepts it.
February 18, 2019
Geometry 11.1 Circumference and Arc Length

11. Example Find the length of AB.5
80 L
Write the proportion:
Length of Arc
Central Angle
Circumference of circle
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Geometry 11.1 Circumference and Arc Length

12. Geometry 11.1 Circumference and Arc Length
Solve the proportion A B 5
80
6.98 L
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Geometry 11.1 Circumference and Arc Length

13. Your Turn Find the length of AB8
25 x
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Geometry 11.1 Circumference and Arc Length

14. Geometry 11.1 Circumference and Arc Length
Example A B 10 x 25
The length of AB is 25 and the radius is 10. Find the mAB.
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Geometry 11.1 Circumference and Arc Length

15. Geometry 11.1 Circumference and Arc Length
Semicircle
Half of a circle.
Its measure is 180.
Its length is one-half the circumference.
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Geometry 11.1 Circumference and Arc Length

16. Geometry 11.1 Circumference and Arc Length
Problem 1
An indoor track at a gym has the design below. What is the length of the entire track?
40 ft
100 ft
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Geometry 11.1 Circumference and Arc Length

17. Geometry 11.1 Circumference and Arc Length
Problem 1 Solution
The ends make one circle.
The diameter is 40 ft.
C = (40)
40 ft
100 ft
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Geometry 11.1 Circumference and Arc Length

18. Geometry 11.1 Circumference and Arc Length
Problem 1 Solution
Add 200 for the straight segments.
Perimeter = 40 ft
Or, ft.
40 ft
100 ft
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Geometry 11.1 Circumference and Arc Length

19. Geometry 11.1 Circumference and Arc Length
Problem 2
Using a GPS, it is found that Tombstone is at 31.7 N latitude, and Show Low is at 34.3 N. Both are at 110 W longitude.
If the radius of the earth is 3963 miles, how far apart are these two towns?
34.3 ?
31.7
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Geometry 11.1 Circumference and Arc Length

20. Geometry 11.1 Circumference and Arc Length
Simplified
34.3
2.6
31.7
3963 mi
Equator
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Geometry 11.1 Circumference and Arc Length

21. Geometry 11.1 Circumference and Arc Length
Problem 2 Solution
Using a GPS, it is found that Tombstone is at 31.7 N latitude, and Show Low is at 34.3 N. Both are at 110 W longitude.
If the radius of the earth is 3963 miles, how far apart are these two towns?
Change in latitude
34.3 – 31.7 = 2.6
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Geometry 11.1 Circumference and Arc Length

22. Geometry 11.1 Circumference and Arc Length
Problem 2
180 miles. How good is this?
This is a good estimate.
180
160
120 80 40
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Geometry 11.1 Circumference and Arc Length

23. Converting between Degrees to Radians
The Radian Measure of a central angle can be thought of as the length of the associated arc. This concept will be explored more thoroughly in Trig.
EX: convert 30 degrees to radians.
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Geometry 11.1 Circumference and Arc Length

24. Geometry 11.1 Circumference and Arc Length
Practice 1
The circumference of a circle is 200 ft. Find its radius.
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Geometry 11.1 Circumference and Arc Length

25. Geometry 11.1 Circumference and Arc Length
Practice 2
The radius of a circle is 16 in and central angle ACB measures 42. Find the length of arc AB.
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Geometry 11.1 Circumference and Arc Length

26. Geometry 11.1 Circumference and Arc Length
Practice 3
An arc of a circle has a length of 12 cm. Its central angle is 15. What is the radius of the circle?
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Geometry 11.1 Circumference and Arc Length

27. Geometry 11.1 Circumference and Arc Length
Practice 4
A circle has a diameter of 20 cm. An arc on the circle is 10 cm in length. Find the measure of the central angle.
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Geometry 11.1 Circumference and Arc Length

28. Geometry 11.1 Circumference and Arc Length
Practice 5
At an altitude of 237 miles, a space ship travels in orbit an angle of 2. How many miles is this? (Recall that the radius of the earth is 3963 mi.)
February 18, 2019
Geometry 11.1 Circumference and Arc Length

29. Geometry 11.1 Circumference and Arc Length
Practice 5
At an altitude of 237 miles, a space ship travels in orbit an angle of 2. How many miles is this? (Recall that the radius of the earth is 3963 mi.)
237
3963 2
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Geometry 11.1 Circumference and Arc Length

30. Geometry 11.1 Circumference and Arc Length
Practice 5
237
3963 2
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Geometry 11.1 Circumference and Arc Length

31. Geometry 11.1 Circumference and Arc Length
Problem
A string is tied snuggly around the Earth, from pole to pole. The radius of the Earth is approximately 3963 miles.
3963 mi
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Geometry 11.1 Circumference and Arc Length

32. Geometry 11.1 Circumference and Arc Length
Problem continued
A second string is tied around the Earth so that is stands off the surface exactly 1 inch for the entire circumference.
1 inch
3963 mi
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Geometry 11.1 Circumference and Arc Length

33. Geometry 11.1 Circumference and Arc Length
Question
How much longer is the second string than the first one?
1 inch
3963 mi
Useful info:
1 mi = 5280 ft
1 ft = 12 in
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Geometry 11.1 Circumference and Arc Length

34. String Around the Earth Solution
How much longer is the second string than the first one?
1 inch
3963 mi
Useful info:
1 mi = 5280 ft
1 ft = 12 in
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Geometry 11.1 Circumference and Arc Length

35. Geometry 11.1 Circumference and Arc Length
Solution
First, change 3963 mi to inches.
3963 mi  5280 ft/mi  12 in/ft = in
That’s the radius of the inner string.
The radius of the outer string is in
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Geometry 11.1 Circumference and Arc Length

36. Find both circumferences
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Geometry 11.1 Circumference and Arc Length

37. Now find the difference
6.3 inches
That’s right! A string standing one inch off the surface of the Earth, all 24,900 miles of it, is only 6.3 inches longer than the snug-fitting one.
Believe it or Not!
February 18, 2019
Geometry 11.1 Circumference and Arc Length

38. Geometry 11.1 Circumference and Arc Length
Summary
The length of an arc on a circle is proportional to the measure of the arc and to the central angle.
A semicircle has 180 and it’s length is half the circumference.
February 18, 2019
Geometry 11.1 Circumference and Arc Length