1. Geometry 11.1 Circumference and Arc Length
11.1 Warmup
February 23, 2019
Geometry 11.1 Circumference and Arc Length

2. 11.1 Circumference and Arc Length
Geometry
11.1 Circumference and Arc Length

3. Geometry 11.1 Circumference and Arc Length
11.1 Essential Question
How can you find the length of a circular arc?
February 23, 2019
Geometry 11.1 Circumference and Arc Length

4. 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.
arc
February 23, 2019
Geometry 11.1 Circumference and Arc Length

5. Geometry 11.1 Circumference and Arc Length
The circumference is the distance around a circle.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

6. Geometry 11.1 Circumference and Arc Length
Pi π
The ratio of the circumference to the diameter is called pi π.
Use the Greek letter for pi:  d C
February 23, 2019
Geometry 11.1 Circumference and Arc Length

7. Geometry 11.1 Circumference and Arc Length
More About Pi.
  …
The fraction 22/7 is close.
355/113 is closer.
The Hebrews thought it was 3.
Current computer calculations have pi to over 5 trillion places.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

8. Pi to a few decimal places:…
February 23, 2019
Geometry 11.1 Circumference and Arc Length

9. Geometry 11.1 Circumference and Arc Length
Pi is Irrational
 does not repeat.
 does not terminate.
There is NO algebraic technique than can be used to find its value.
Computing  requires the use of calculus.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

10. 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 23, 2019
Geometry 11.1 Circumference and Arc Length

11. Geometry 11.1 Circumference and Arc Length
Example 1
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 π button on your calculator.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

12. 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 arc⁰ to 360. A B
Recall: the measure of the arc is equal to the central angle that intercepts it.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

13. Example 2 Find the length of AB.5
80 L
Write the proportion:
Length of Arc
Central Angle
Circumference of circle
February 23, 2019
Geometry 11.1 Circumference and Arc Length

14. Geometry 11.1 Circumference and Arc Length
Solve the proportion A B 5
80
6.98 L
February 23, 2019
Geometry 11.1 Circumference and Arc Length

15. Your Turn Find the length of AB8
25 x
February 23, 2019
Geometry 11.1 Circumference and Arc Length

16. Geometry 11.1 Circumference and Arc Length
Example 3 A B 10 x 25
The length of AB is 25 and the radius is 10. Find the mAB.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

17. Geometry 11.1 Circumference and Arc Length
Semicircle
Half of a circle.
Its measure is 180.
Its length is one-half the circumference.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

18. Geometry 11.1 Circumference and Arc Length
Example 4
An indoor track at a gym has the design below. What is the length of the entire track?
40 ft
100 ft
February 23, 2019
Geometry 11.1 Circumference and Arc Length

19. Geometry 11.1 Circumference and Arc Length
Example 4 Solution
The ends make one circle.
The diameter is 40 ft.
C = (40)
40 ft
100 ft
February 23, 2019
Geometry 11.1 Circumference and Arc Length

20. Geometry 11.1 Circumference and Arc Length
Example 4 Solution
Add 200 for the straight segments.
Perimeter = 40 ft
Or, ft.
40 ft
100 ft
February 23, 2019
Geometry 11.1 Circumference and Arc Length

21. Geometry 11.1 Circumference and Arc Length
Practice 1
The circumference of a circle is 200 ft. Find its radius.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

22. 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.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

23. 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?
February 23, 2019
Geometry 11.1 Circumference and Arc Length

24. 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.
February 23, 2019
Geometry 11.1 Circumference and Arc Length

25. 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 23, 2019
Geometry 11.1 Circumference and Arc Length

26. Geometry 11.1 Circumference and Arc Length
Homework
February 23, 2019
Geometry 11.1 Circumference and Arc Length