1. GSE Geometry: Arc, Angles, and Area.
Objectives
SWBAT understand that circles are similar IOT find circumference and areas of circles
SWBAT apply relationships among inscribed angles, radii, chords, tangents and secants IOT solve for unknown quantities in a circle.
SWBAT apply the relationship between central, inscribed, and circumscribed angles IOT solve for unknown quantities in a circle.
SWBAT define radian measure and derive the formula for area of a sector IOT solve circle problems.

2. Group Check
30
80
100
180
100
260

3. Circumference, Arc Length, Area, and Area of Sectors

4. An Application Problem for Area

5. A couple of questions concerning Arcs, Angles, and Measurements
Arcs and Angles
A circle is divided into three arcs of measures 'x', '2x' and '3x.' What is the value of 'x'?

6. A couple of questions concerning Arcs, Angles, and Measurements
Arcs and Lengths

7. The distance around a circle
Circumference
The distance around a circle

8. Circumference or Twinkle, Twinkle Little Star
Circumference Equals 2 pi r or

9. 2 Types of Answers Exact Rounded Pi will be in your answer
Type the Pi button on your calculator
Toggle your answer
Do NOT write Pi in your answer
Exact
Pi will be in your answer

10. Guided Practice: Find the exact measure.
r = 14 feet
d = 15 miles

11. Guided Practice: Find the approximate measure
33 yd
14.3 mm

12. Arc Length
The distance along the curved line making the arc (NOT a degree amount) 

13. Arc Measure vs Arc Length
Measure will be in degrees.
Length will be in a “distance” unit of measure EX:

14. Arc Length

15. Guided Practice: Find the Arc Length Round to the nearest hundredths8m 
70

16. Guided Practice: Find the exact Arc Length.

17. Guided Practice: What happens to the arc length if the radius were to be doubled? Halved?

18. Guided Practice: Find the perimeter of the region.

19. The amount of space occupied.
Area of Circles
The amount of space occupied. r
A = pr2 

20. Find the EXACT area.
8. r = 29 feet
9. d = 44 miles

21. 10 and 11 Find the area. Round to the nearest tenths.
53 cm
7.6 yd

22. Area of a Sector
the region bounded by two radii of the circle and their intercepted arc.

23. Area of a Sector

24. Find the area of the sector to the nearest hundredths.
Guided Practice
Find the area of the sector to the nearest hundredths. R
6 cm
60 Q
A  cm2

25. Find the exact area of the sector.
Guided Practice
Find the exact area of the sector.
6 cm
120
7 cm Q R

26. (Area of sector) – (Area of triangle)
Guided Practice
Area of minor segment =
(Area of sector) – (Area of triangle) R Q
12 yd

27. Inscribed Angles and Inscribed Quadrilaterals

28. Central Angle
Angle = Arc

29. Inscribed Angle
Angle where the vertex is ON the circle

30. Inscribed Angle

31. 160
The arc is twice as big as the angle!!
80

32. 120  = 120  = 60  Guided Practice: Find the value of x and y. x

33. Guided Practice
1. If mJK = 80 and <JMK = 2x – 4, find x.
x = 22
2. If m<MKS = 56, find m MS.
112  M Q K S J

34. Guided Practice: Find the measure of <DOG and <DIG
72˚
If two inscribed angles intercept the same arc, then they are congruent. G O I

35. If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.

36. Quadrilateral inscribed in a circle: opposite angles are SUPPLEMENTARY

37. If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle. What would be the converse of this statement?
diameter

38. x = 3 5x = 2x + 9 3x = + 9 In J, m<3 = 5x and m<4 = 2x + 9. Q D
Guided Practice:
In J, m<3 = 5x and m<4 = 2x + 9.
Find the value of x. Why can we set these angles equal to each other? 3 Q D J T U 4
5x = 2x + 9
3x = + 9
x = 3

39. x = 26 4x – 14 = 90 4x = 104 Guided Practice: H K N G
In K, GH is a diameter and m<GNH = 4x – 14. Find the value of x. What do we automatically know about this triangle?
4x – 14 = 90 H K G N
4x = 104
x = 26

40. Guided Practice: Solve for x and z.85
2x x – 6 = 180
24x +12 = 180
22x – 6
24x = 168
x = 7
z + 85 = 180
z = 95

41. GSE Geometry: Arc, Angles, and Area.
Objectives
SWBAT understand that circles are similar IOT find circumference and areas of circles
SWBAT apply relationships among inscribed angles, radii, chords, tangents and secants IOT solve for unknown quantities in a circle.
SWBAT apply the relationship between central, inscribed, and circumscribed angles IOT solve for unknown quantities in a circle.
SWBAT define radian measure and derive the formula for area of a sector IOT solve circle problems.

42. 244 x = 105 y = 100 Group Check 1. Solve for arc ABC
2. Solve for x and y.
x = 105
y = 100

43. Secant and Tangent Angles
Vertex is INSIDE OR OUTSIDE the circle

44. Vertex is INSIDE the Circle NOT at the Center

45. Guided Practice Solve for x
180 – 88 X
88
84 92
x = 100

46. x = 89 360 – 89 – 93 – 45 133 Guided Practice Solve for x. 93 xº 45
89
133
x = 89

47. Vertex is OUTside the Circle

48. Guided Practice Solve for x.
15°
x = 25o
65°

49. Guided Practice Solve for x.
27° x
70°
x = 16

50. Guided Practice Solve for x.
360 – 260
260°
100 x
x = 80

51. Tune: If You’re Happy and You Know It
If the vertex is ON the circle half the arc. <clap, clap>
If the vertex is IN the circle half the sum. <clap, clap>
But if the vertex is OUTside, then you’re in for a ride, cause it’s half of the difference anyway. <clap, clap>

52. Exit Ticket: Solve for x
1.)
124◦
2.)
70◦ x
18◦ x
4.)
3.)
260◦ x
110◦
20◦ x