1. Circles § 11.1 Parts of a Circle § 11.2 Arcs and Central Angles
§ Arcs and Chords
§ Inscribed Polygons
§ Circumference of a Circle
§ Area of a Circle

2. Vocabulary What You'll Learn Parts of a Circle
You will learn to identify and use parts of circles.
Vocabulary
1) circle
2) center
3) radius
4) chord
5) diameter
6) concentric

3. A circle is a special type of geometric figure.
Parts of a Circle
A circle is a special type of geometric figure.
All points on a circle are the same distance from a ___________.
center point O B A
The measure of OA and OB are the same; that is, OA  OB

4. Parts of a Circle
Definition
of a
Circle
A circle is the set of all points in a plane that are a given distance from a given _____ in the plane, called the
______ of the circle.
point
center P
Note: a circle is named by its center. The circle above is named circle P.

5. Parts of a Circle
There are three kinds of segments related to circles.
A ______ is a segment whose endpoints are the center of the circle and a
point on the circle.
radius
A _____ is a segment whose endpoints are on the circle.
chord
A ________ is a chord that contains the center
diameter

6. that passes through the center. chord
Parts of a Circle
Segments of
Circles
radius
chord
diameter R K J K A K T G
From the figures, you can not that the diameter is a special type of _____
that passes through the center.
chord

7. Use Q to determine whether each statement is true or false.
Parts of a Circle
Use Q to determine whether each statement is true or false.
False; C A D B Q
Segment AD does not go through the center Q.
True;
False;

8. All radii of a circle are _________. congruent
Parts of a Circle
Theorem
11-1
11-2
All radii of a circle are _________.
congruent P R S T G
The measure of the diameter d of a circle is twice the
measure of the radius r of the circle.

9. Parts of a Circle Find the value of x in Q. 3x – 5 = 2(17) 3x – 5 = 34B
Find the value of x in Q.
3x – 5 = 2(17) A
3x-5 17
3x – 5 = 34 Q
3x = 39
x = 13 C

10. P has a radius of 5 units, and T has a radius of 3 units.
Parts of a Circle
P has a radius of 5 units, and T has a radius of 3 units. R P Q T B A
If QR = 1, find RT
RT = TQ – QR
RT = 3 – 1
RT = 2
If QR = 1, find PQ
If QR = 1, find AB
PQ = PR – QR
AB = 2(AP) + 2(BT) – 1
PQ = 5 – 1
AB = 2(5) + 2(3) – 1
PQ = 4
AB = – 1
AB = 15

11. Parts of a Circle
P has a radius of 5 units, and T has a radius of 3 units. R P Q T B A
If AR = 2x, find AP in terms of x.
AR = 2(AP)
2x = 2(AP)
x = AP
If TB = 2x, find QB in terms of x.
QB = 2(TB)
QB = 2(2x)
QB = 4x

12. Because all circles have the same shape, any two circles are similar.
Parts of a Circle
Because all circles have the same shape, any two circles are similar.
However, two circles are congruent if and only if (iff) their radii are _________.
congruent
Two circles are concentric if they meet the following three requirements:
They lie in the same plane. S
They have the same center. R T
They have radii of different lengths.
Circle R with radius RT and
circle R with radius RS are
concentric circles.

13. Arcs and Central Angles
What You'll Learn
You will learn to identify major arcs, minor arcs, and semicircles
and find the measures of arcs and central angles.
Vocabulary
1) central angle
2) arcs
3) minor arc
4) major arc
5) semicircle
6) adjacent arc

14. Arcs and Central Angles
A ____________ is formed when the two sides of an angle meet at the center of a circle.
central angle
Each side intersects a point on the circle, dividing it into ____ that are
curved lines.
arcs
There are three types of arcs: T S
A _________ is part of the circle in the interior of
the central angle with measure less than 180°.
minor arc
central
angle R
A _________ is part of the circle in the exterior of
the central angle.
major arc
Semicircles
__________ are congruent arcs whose endpoints
lie on the diameter of the circle.

15. Arcs and Central Angles
Types of
Arcs
minor arc PG
major arc PRG
semicircle PRT K T P R G K P G K P G R
Note that for circle K, two letters are used to name the minor arc, but three letters are
used to name the major arc and semicircle. These letters for naming arcs help us
trace the set of points in the arc. In this way, there is no confusion about which arc
is being considered.

16. Arcs and Central Angles
Depending on the central angle, each type of arc is measured in the
following way.
Definition
of Arc
Measure
1) The degree measure of a minor arc is the degree measure of its central angle.
2) The degree measure of a major arc is 360 minus the degree measure of its central angle.
3) The degree measure of a semicircle is 180.

17. Arcs and Central Angles
In P, find the following measures:
= APM P H A M T
46°
80°
= 46°
APT
APT
= 80°
= 360° – (MPA + APT)
= 360° – (46° + 80°)
= 360° – (126°)
= 234°

18. Arcs and Central Angles
In P, AM and AT are examples of ________ arcs.
adjacent P H A M T
46°
80°
Adjacent arcs have exactly one point in common.
For AM and AT, the common point is __. A
Adjacent arcs can also be added.
Postulate
11-1
Arc
Addition
The sum of the measures of two adjacent arcs is the measure
of the arc formed by the adjacent arcs. C Q P R
If Q is a point of PR, then
mPQ
+ mQR =
mPQR

19. Arcs and Central Angles
In P, RT is a diameter.
Find mQT.
mQT + mQR = mTQR R P S Q T
75°
65°
mQT ° = 180°
mQT = 105°
Find mSTQ.
mSTQ + mQR + mRS = 360°
mSTQ ° + 65° = 360°
mSTQ ° = 360°
mSTQ = 220°

20. Arcs and Central Angles
Suppose there are two concentric circles
with ASD forming two minor arcs,
BC and
AD.
60° A D B
Are the two arcs congruent? S C
Although BC and AD each measure 60°, they are not congruent.
The arcs are in circles with different radii, so they have different lengths.
However, in a circle, or in congruent circles, two arcs are congruent if they
have the same measure.

21. Arcs and Central Angles
Theorem
11-3
In a circle or in congruent circles, two minor arcs are congruent
if and only if (iff) their corresponding central angles are congruent. Z Y X W Q
WX  YZ
iff
60°
60°
mWQX = mYQZ

22. Arcs and Central Angles
In M, WS and RT are diameters, mWMT = 125, mRK = 14.
Find mRS.
WMT  RMS
Vertical angles are congruent T W S K M R
WMT = RMS
Definition of congruent angles
mWT = mRS
Theorem 11-3
125 = mRS
Substitution
Find mKS.
Find mTS.
KS + RK = RS
TS + RS = 180
KS = 125
TS = 180
KS = 111
TS = 55

23. Arcs and Central Angles
Twenty-two percent of all teens ages 12 through 17 work either full or part-time.
Source: ICRs TeenEXCEL survey for Merrill Lynch
Teens at Work
The circle graph shows the number of hours they work per week.
Find the measure of each central angle.
1 – 5:
= 72
20% of 360
6 – 10:
= 83
23% of 360
11 – 20:
= 119
33% of 360
21 – 30:
= 50
14% of 360
30 or More:
= 36
10% of 360

24. Vocabulary What You'll Learn Arcs and Chords
You will learn to identify and use the relationships among
arcs, chords, and diameters.
Vocabulary
Nothing New!

25. In circle P, each chord joins two points on a circle.
Arcs and Chords
In circle P, each chord joins two points on a circle.
Between the two points, an arc forms along the circle.
By Theorem 11-3, AD and BC are congruent
because their corresponding central angles are
_____________, and therefore congruent.
vertical angles P A C
By the SAS Theorem, it could be shown that
ΔAPD  ΔCPB.
Therefore, AD and BC are _________.
congruent B D
The following theorem describes the relationship between two congruent
minor arcs and their corresponding chords. S

26. In a circle or in congruent circles, two minor arcs are congruent
Arcs and Chords
Theorem
11-4
In a circle or in congruent circles, two minor arcs are congruent
if and only if (iff) their corresponding ______ are congruent.
chords A D
AD  BC
iff B
AD  BC C

27. The vertices of isosceles triangle ABC are located on R.
Arcs and Chords
The vertices of isosceles triangle ABC are located on R.
If BA  AC,
identify all congruent arcs. A
BA  AC R C B

28. Hands-On Arcs and Chords Step 1) Use a compass to draw circle on aG H E F
Step 1) Use a compass to draw circle on a
piece of patty paper. Label the center P. Draw a chord that is not
a diameter. Label it EF. P
Step 2) Fold the paper through P so that
E and F coincide. Label this fold
as diameter GH.
Q1: When the paper is folded, how do the lengths of EG and FG compare?
EG  FG
Q2: When the paper is folded, how do the lengths of EH and FH compare?
EG  FG
Q3: What is the relationship between diameter GH and chord EF?
They appear to be perpendicular.

29. Arcs and Chords
Theorem
11-5
In a circle, a diameter bisects a chord and its arc if and only if
(iff) it is perpendicular to the chord. P R D C B A
AR  BR and AD  BD
iff
CD AB
Like an angle, an arc can be bisected.

30. Find the measure of AB in K.
Arcs and Chords
Find the measure of AB in K.
Theorem 11-5
Substitution B C A K D 7

31. Find the measure of KM in K if ML = 16.
Arcs and Chords
Find the measure of KM in K if ML = 16.
Pythagorean Theorem
Given; Theorem 11-5 M K L N 6

32. Vocabulary What You'll Learn Inscribed Polygons
You will learn to inscribe regular polygons in circles and explore
the relationship between the length of a chord and its distance
from the center of the circle.
Vocabulary
1) circumscribed
2) inscribed

33. Definition of Inscribed Polygon
Inscribed Polygons
When the table’s top is open, its circular top is said to be ____________
about the square.
circumscribed
We also say that the square is ________ in the circle.
inscribed
Definition of Inscribed Polygon
A polygon is inscribed in a circle if and only if every vertex of the polygon lies on the circle.

34. Some regular polygons can be
Inscribed Polygons
Some regular polygons can be
constructed by inscribing them in circles. B A
Inscribe a regular hexagon, labeling
the vertices, A, B, C, D, E, and F. C P F
Construct a perpendicular segment
from the center to each chord.
From our study of “regular polygons,” we know that the chords AB, BC, CD, DE, and EF are
_________ D E
congruent
From the same study, we also know that all of the perpendicular segments,
called ________, are _________.
apothems
congruent
Make a conjecture about the relationship between the measure of the chords
and the distance from the chords to the center.
The chords are congruent because the distances from the center of the
circle are congruent.

35. In a circle or in congruent circles, two chords are congruent
Inscribed Polygons
Theorem
11-6
In a circle or in congruent circles, two chords are congruent
if and only if they are __________ from the center.
equidistant B L M
AD  BC
iff P A
LP  PM C D

36. In circle O, O is the midpoint of AB.
Inscribed Polygons O S T B A C R
In circle O, O is the midpoint of AB.
If CR = -3x and ST = 4x,
find x
definition of midpoint
Theorem 11-6
substitution

37. Circumference of a Circle
What You'll Learn
You will learn to solve problems involving circumferences of
cirlces.
Vocabulary
1) circumference
2) pi (π)

38. Circumference of a Circle
An in-line skate advertises “80-mm clear wheels.”
The description “80-mm” refers to the diameter of the skates’ wheels.
As the wheels of an in-line skate complete one revolution, the distance traveled is the same as the circumference of the wheel.
Just as the perimeter of a polygon is the distance around the polygon,
the circumference of a circle is the ______________________.
distance around the circle

39. Circumference of a Circle
On a sheet of paper, create a table similar to the one below:
Circumference
Diameter
Ratio:
C/D
Result
Example Data
271 86
271 ÷ 86
Go to this website to Collect Data from various Circles.
Go to this website to Analyze your Data.

40. Circumference of a Circle
In the previous activity, the ratio of the circumference C of a circle to its
diameter d appears to be a fixed number slightly greater than 3, regardless
of the size of the circle.
The ratio of the circumference of a circle to its diameter is always fixed and
equals an irrational number called __ or __. pi π
Thus, ____ = __, π

41. Circumference of a Circle
Theorem
11-7
Circumference
of a Circle
If a circle has a circumference of C units and a radius of r
units, then C = ____
or C = ___ d r C

42. Circumference of a Circle

43. Circumference of a Circle

44. Vocabulary What You'll Learn Area of a Circle
You will learn to solve problems involving areas and sectors
of circles.
Vocabulary
1) sector

45. Area of a Circle
The space enclosed inside a circle is its area.
By slicing a circle into equal pie-shaped pieces as shown below, you can
rearrange the pieces into an approximate rectangle.
Note that the length along the top and bottom of this rectangle equals the
_____________ of the circle, ____.
circumference
So, each “length” of this approximate rectangle is half the circumference, or __

46. Area of a Circle
The “width” of the approximate rectangle is the radius r of the circle.
Recall that the area of a rectangle is the product of its length and width.
Therefore, the area of this approximate rectangle is (π r)r or ___.

47. If a circle has an area of A square units and a radius of
Area of a Circle
Theorem
11-8
Area
of a Circle
If a circle has an area of A square units and a radius of
r units, then A = ___ r

48. You could calculate the area if you only knew the radius.
Area of a Circle
Find the area of the circle whose circumference is meters.
Round to the nearest hundredth.
Use your knowledge
of circumference.
Solve for radius, r.
You could calculate the area if you only knew the radius.
Any ideas?

49. Area of a Sector of a Circle
Area of a Circle
Theorem 11-9
Area of a Sector of a Circle
If a sector of a circle has an area of A square units, a central angle measurement of N degrees, and a radius of
r units, then

50. Area of a Circle