Immaculata Week 2013 July 29—August 2, 2013

1. Immaculata Week 2013July 29—August 2, 2013 Area of Circle, Sector and Segment SECONDARY LEVEL • Session #2 • Presented by: Dr. Del Ferster

2. What’s in store for today? • Today, we’ll look at finding area of a circle, and extending that idea to: • Find the area of a sector of a circle. • Find the area of a segment of a circle. • We’ll take a look at using trigonometry to help to find the area of a triangle. • Of course, I’ll have some problems for us to solve. • And again, as usual, I’ve collected some handouts that you’re welcome to have.

3. Let’s see what you remember about area and circles.

4. Area of a Circle • The area of a circle is  times the square of the radius or A = r2 • Of course, some problems might try to be tricky and give you the diameter, or maybe even the circumference of the circle.

5. What is a SECTOR of a circle? • A SECTOR of a circle is a part of a circle comprised by two radii and the arc that those radii intercept. • We can name a sector by using the endpoint of one radius, the center of the circle, and the endpoint of the other radius. • For those of you who remember trivial pursuit, it’s the shape of the pieces you tried to earn by answering questions. A arc radius B O radius • Sector AOB of Circle O

6. More on Sectors • The area of a sector is a fractional part of the area of a circle. • The ratio of a sector’s area to a circle’s area is:

7. Area of a Sector Formula measure of the central angle or arc The area of the entire circle m˚ πr2 Area of a sector = 360 . The fraction of the circle

8. Let’s look at some problems We’ll focus on circles and sectors for now.

9. Recall the following facts • Area of a circle • Area of the sector, with a central angle of m degrees. (think of this as a fractional portion of the entire circle)

10. Example 1 • The radius of the circle is 10 cm. • Angle AOB is 450 • What is the area of the sector AOB? Leave your answer in terms of

11. Example 2 • Use the diagram shown to find the following. Leave your answers in terms of • 1. The area of the circle. • 2. The area of the red sector. • 3. The area of the blue sector

12. Example 3 Bob is hungry for pizza, however, due to the sequester, the radius of the pizzas have been reduced to 6 inches. Darn sequester!! Bob’s slice of the pizza has an area of HOW MANY SLICES WAS THE PIZZA CUT INTO? What was the size of the central angle for Bob’s slice of pizza?

13. Example 4 • The area of sector AOB is • The measure of • Find the length of the radius of circle O. A O B

14. Example 5 • The area of the BIG SECTOR (the yellow one) is • The measure of the central angle for the purple sector is Find the length of the radius of circle O. A O B

15. Let’s take a look at SEGMENTS We’ll start with a definition and look at some problems.

16. A SEGMENT of a Circle • A segment of a circle is a region bounded by an arc and its chord

17. radius radius For those of you who enjoy pie, an alternate look at Area of a Segment A segment is the portion of the circle with the radii connected at their endpoints and the central arc. (The crust of the pie) slice.) The red section is the segment.

18. Finding the area of a segment • To find the area of a segment for a minor arc, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed.

19. Doesn’t sound too bad, does it? • Unfortunately, finding the area of the triangle might not be as easy as you might think.  • Remember that when we use the formula to find the area of a triangle. • Base and height MUST be perpendicular to each other!!

20. radius radius Example 1: An easy one Find the area of a segment of a circle whose radius is 10 cm, and central angle is 90°. Step 1: Find the area of the sector. Step 2: Find the area of the triangle. Area = ½ bh Step 3: Find the area of the segment.

21. Why was that one easy? • Yep, the measure of the central angle was 90 degrees, that made the length of the radius applicable for both base and height for our triangle. • Sadly, when the measure of the central angle isn’t 90 degrees, it’s more of a pain to calculate the area of the triangle.

22. Example #2: A bit tougher Find the area of segment LNM to the nearest hundredth. Step 1 Find the area of sector LNM. 6 cm

23. Example #2: Continued Let’s take a closer look at that triangle. Warning, we’re about to use 30-60-90 triangle ratios. Step 2:Find the area of ∆LNM. Draw altitude--h 6 6

24. Example #2: Concluded Step 3: area of segment = area of sector LNM – area of ∆LNM cm 6 cm

25. YIKES…that was about as much fun as a root canal.  • That was tough! • Is there another way to find the area of the triangle?

26. Yes, but it might not be as pretty as you wished. • We can make use of trigonometry to find the area of the triangle formed by 2 radii with a given central angle measure, let’s call it • Finding the area of a triangle knowing 2 sides (both of length r) and the included angle, • Here’s a link, if you’re so inclined, that explains the derivation of the formula SHOW ME THAT WORK

27. Let’s do one that way. Shall we? • Find the area of a segment of a circle whose diameter is 24 inches and whose central angle is 20 degrees. • Grab your pencil, we’ll do this one on the chalkboard. 

28. PROBLEMS Grab your pens and pencils, and let’s do some math! Feel free to work by yourself or with a partner. 

29. 3. Find the area of the shaded region. Problem #1 • Find the area of the shaded region. Leave your answer in terms of

30. Problem #2 • The radius of the circle is 12 in. • What is the area of sector ACB. Leave your answer in terms of

31. Problem #3 • Find the area of the indicated segment. Leave your answer in terms of

32. Problem #4 • Find the area of the red portion of the sign. Leave your answer in terms of

33. Problem #5 • The circle graph shows the results of a survey about the favorite types of muffins. Find the central angle measure of the sector for each type of muffin. • Explain your solution.

34. Problem #6 • Find the area of the record. THAT’s the black part, for you ITUNES people.  3cm 1cm 5cm

35. Problem #7 • Find the shaded area if r=10. (Note: r is the length of the radius of the small circles.) Leave your answer in terms of

36. Problem #8 • Find the area of sector ACB. Leave your answer in terms of

37. Problem #9 • Find the area of the shaded segment. Round your final answer to the nearest hundredth.

38. 30 4 Problem #10 • Find the area of the shaded region. Point O is the center of the circle, and the radius is 4 units. O

39. Problem #11 • The rectangular wall shown above has an entrance cut into it. You want to paint the wall. To the nearest square foot, what is the area of the region that you need to paint? If one gallon of paint will cover 200 square feet of surface area, how many gallons of paint will you need to do the project?

40. Problem #12 • The diagram shown above, models a rectangular rear window of a car. This fancy car has a rear window wiper that can rotate through 160 degrees. What percentage of the rear window is cleaned by the wiper.

41. ANSWERS TO PROBLEMS

42. ANSWERS TO PROBLEMS (Continued)

43. Wrapping it up for today? • Questions or comments • Remember, the process is the most important thing. Don’t memorize, UNDERSTAND! • Challenge your students, support them, and share their successes!!

44. The last slide!!! • Thanks for your attention, participation, and energy. (I know it’s a long time to sit! ) • Grab your shades and your beach gear and head out! • If I can be of help during the school year, please don’t hesitate to let me know • EMAIL: • Here at Immaculata: • dferster@immaculata.edu • My home email: • delferst@gmail.com • PHONE: (610) 369-7344 (HOME) • (610) 698-7615 (CELL)