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2. Geometry Circles 2014-03-31 www.njctl.org

3. Table of Contents Click on a topic to go to that section Parts of a Circle Angles & Arcs Chords, Inscribed Angles & Polygons Tangents & Secants Segments & Circles Equations of a Circle Area of a Sector

4. Parts of a Circle Return to the 

table of 

contents

5. A circle is the set of all points in a 
plane that are a fixed distance from 
a given point in the plane called 
the center. center

6. . . The symbol for a circle is and is named by a capital letter 
placed by the center of the circle. . . (circle A or (circle A or A) A) . . A is a radius of is a radius of A A A radius (plural, radii) is a line 
segment drawn from the center 
of the circle to any point on the 
circle. It follows from the 
definition of a circle that all radii 
of a circle are congruent. B

7. A M is a chord of circle R A chord is a segment that has its 
endpoints on the circle. A A is the diameter of circle C A diameter is a chord that goes 
through the center of the circle. 
All diameters of a circle are 
congruent. Answer & T What are the radii in this diagram? [This object is a pull tab]

8. The relationship between the diameter and the radius T The measure of the diameter, d, is 
twice the measure of the radius, r. A Therefore, or M AC = 5 TC = 10 Answer . A In C then what is the length of If , what is the length of [This object is a pull tab]

9. 1 A diameter of a circle is the longest chord of the circle. True False Answer True [This object is a pull tab]

10. 2 A radius of a circle is a chord of a circle. True False Answer False [This object is a pull tab]

11. 3 Two radii of a circle always equal the length of a diameter of a circle. True False Answer True [This object is a pull tab]

12. 4 If the radius of a circle measures 3.8 meters, what is the measure of the diameter? Answer 7.6 m [This object is a pull tab]

13. 5 How many diameters can be drawn in a circle? A 1 B 2 C 4 D infinitely many Answer D [This object is a pull tab]

14. A secant of a circle is a line that intersects the circle at two points. A line l is a secant of this circle. B A tangent is a line in the plane of 
a circle that intersects the circle 
at exactly one point (the point of 
tangency). l D line k is a tangent E k D is the point of tangency. tangent ray, , and the tangent segment, , 
are also called tangents. They must be part of a 
tangent line. Note: This is not a tangent ray.

15. COPLANAR CIRCLES are two circles in the same plane which 
intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent 
circles. Coplanar circles that have a common center are called 
concentric. . . . . . concentric 
circles tangent 
circles 1 point 2 points no points

16. A Common Tangent is a line, ray, or segment that is tangent to 2 
coplanar circles. Internally tangent (tangent line 
passes 

between them) Externally tangent (tangent line does 

not pass between 

them)

17. 6 How many common tangent lines do the circles have? Answer 4 [This object is a pull tab]

18. 7 How many common tangent lines do the circles have? Answer 1 [This object is a pull tab]

19. 8 How many common tangent lines do the circles have? Answer 2 [This object is a pull tab]

20. 9 How many common tangent lines do the circles have? Answer 0 [This object is a pull tab]

21. Using the diagram below, match the notation with the term that 
best describes it: . B . A . . D . F . C . G E Center Common Tangent Chord Secant Tangent Point of Tangency Radius Diameter Answer secant common tangent center diameter [This object is a pull tab] point of tangency chord radius tangent

22. Angles & Arcs Return to the 

table of 

contents

23. An ARC is an unbroken piece of a circle with endpoints 
on the circle. . A AB Arc of the circle or . B Arcs are measured in two ways: 1) As the measure of the central angle in degrees 2) As the length of the arc itself in linear units (Recall that the measure of the whole circle is 360o.)

24. A central angle is an angle whose vertex is the 
center of the circle. M . . H S Answer , is the central 
angle. In A T Name another central angle. [This object is a pull tab]

25. If is less than 1800, then the points on 
that lie in the interior of form the minor arc with 
endpoints M and H. M MA minor arc . . Answer H S MA Highlight A [This object is a pull tab] T Name another minor arc.

26. major arc M . . H S Answer A T [This object is a pull tab] Points M and A and all points of exterior to 
form a major arc,  Major arcs are the "long way" around 
the circle. Major arcs are greater than 180o. Highlight Major arcs are named by their endpoints and a point on the 
arc. Name another major arc. MSA MSA

27. M . . minor arc H S Answer A T [This object is a pull tab] A semicircle is an arc whose endpoints are the 
endpoints of the diameter. MAT is a semicircle. Highlight the semicircle. Semicircles are named by their endpoints and a point on 

the arc. Name another semicircle.

28. Measurement By A Central Angle The measure of a minor arc is the measure of its central angle. The measure of the major arc is 3600 minus the measure of the 
central angle. B 400 400 . A G 3600 - 400 = 3200 D

29. The Length of the Arc Itself (AKA - Arc Length) Arc length is a portion of the circumference of a circle. Arc Length Corollary - In a circle, the ratio of the length of 
a given arc to the circumference is equal to the ratio of the 
measure of the 

arc to 3600. C arc length of CT = CT 3600 or A r . = T CT CT arc length of 3600

30. EXAMPLE A In , the central angle is 600 and the radius is 8 cm. Find the length of CT . C = CT arc length of CT 3600 Answer . 8 cm = 600 3600 A 600 8.38 cm T [This object is a pull tab]

31. EXAMPLE A A In , the central angle is 400 and the length of 
is 4.19 in. Find the circumference of In , the central angle is 400 and the length of 
is 4.19 in. Find the circumference of SY SY A. A. S SY arc length of = SY 4.19 in 3600 = 4.19 400 A 400 Answer 3600 = 4.19 1 9 Y = 37.71 in [This object is a pull tab]

32. 10 In circle C where is a diameter, find B 1350 D C 15 in A Answer [This object is a pull tab]

33. 11 In circle C, where is a diameter, find B 1350 D C Answer 15 in A [This object is a pull tab]

34. 12 In circle C, where is a diameter, find B 1350 D C 15 in A Answer [This object is a pull tab]

35. 13 In circle C can it be assumed that AB is a diameter? B Yes No 1350 Answer D Yes C [This object is a pull tab] A

36. 14 Find the length of A B 450 3 cm C Answer [This object is a pull tab]

37. 15 Find the circumference of circle T. T 750 6.82 cm Answer [This object is a pull tab]

38. 16 In circle T, WY & XZ are diameters. WY = XZ = 6. If XY = , what is the length of YZ? 1400 A X W B Answer T C A D [This object is a pull tab] Y Z

39. ADJACENT ARCS Adjacent arcs: two arcs of the same circle are adjacent if they 
have a common endpoint. Just as with adjacent angles, measures of adjacent arcs can be 
added to find the measure of the arc formed by the adjacent arcs. . C . + = A . T

40. EXAMPLE A result of a survey about the ages of people in a city are shown. 
Find the indicated measures. T S 1. 2. 3. 4. >65 = 600 + 800 = 1400 300 900 U 1000 17-44 1000 + 300 = 1300 Answer 800 600 = 600 + 800 + 900 = 2300 45-64 = 3600 - 900 = 2700 R V 15-17 [This object is a pull tab]

41. Match the type of arc and it's measure to the given arcs below: T Q 1200 600 800 Arc labels and measurements in 
the box are infinitely cloned so 
they can be pulled up and 
matched with the arc. Teacher Notes R S [This object is a pull tab] minor arc major arc semicircle 1600 800 1200 1800 2400

42. CONGRUENT CIRCLES & ARCS Two circles are congruent if they have the same radius. Two arcs are congruent if they have the same measure and they 
are arcs of the same circle or congruent circles. T R D E U S 550 550 F C & have the same 
measure, but are not congruent 
because they are arcs of circles 
that are not congruent. because they are in the 
same circle and

43. A 17 B True 700 1800 400 False C D True Answer [This object is a pull tab]

44. 18 M L True False 850 P N False Answer [This object is a pull tab]

45. 19 Circle P has a radius of 3 and has a measure of . What is the length of ? 900 A A B P C D B A Answer [This object is a pull tab]

46. 20 Two concentric circles always have congruent radii. True False Answer False [This object is a pull tab]

47. 21 If two circles have the same center, they are congruent. True False False Answer [This object is a pull tab]

48. 22 Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece? Answer [This object is a pull tab]

49. Chords, Inscribed 

Angles & Polygons Return to the 

table of 

contents

50. When a minor arc and a chord have the same endpoints, we call 
the arc The Arc of the Chord. P . is the arc of C Q **Recall the definition of a chord - 
a segment with endpoints on the 
circle.