DIAMETER:
P
Distance
across the
circle through
its center
Also known as the
longest chord.
RADIUS:
P
Distance
from the
center to
point on
circle
Formula
Radius = ½ diameter
or
Diameter = 2r
D = ?
r = ?
r = ?
D = ?
Secant Line:
intersects the
circle at
exactly TWO
points
Tangent Line:
a LINE that intersects the circle
exactly ONE time
Forms a 90°angle
with one radius
Point of Tangency:
The point where the
tangent intersects
the circle
Name the term that best describes the notation.
Central Angles
An angle whose vertex is
at the center of the circle
Semicircle: An Arc that equals 180°
D
E
To name:
use 3 letters
P
F
EDF
THINGS TO KNOW AND REMEMBER
ALWAYS
A circle has 360 degrees
A semicircle has 180 degrees
Vertical Angles are CONGRUENT
Linear Pairs are SUPPLEMENTARY
Formula
measure Arc = measure Central
Angle
Find the measures.
EB is a diameter.
m AB = 96°
A
E
m ACB= 264°
Q
m AE = 84°
96
B
C
Tell me the measure of the following arcs.
AC is a diameter.
m DAB =240
m BCA = 260
D
C
140
R
40
100
80
B
A
Using Properties of Tangents
HK and HG are tangent to F. Find HG.
HK = HG
2 segments tangent to
 from same ext. point
 segments .
5a – 32 = 4 + 2a
Substitute 5a – 32 for
HK and 4 + 2a for HG.
3a – 32 = 4
Subtract 2a from both sides.
3a = 36
a = 12
HG = 4 + 2(12)
= 28
Add 32 to both sides.
Divide both sides by 3.
Substitute 12 for a.
Simplify.
Applying Congruent Angles, Arcs, and Chords
TV  WS. Find mWS.
TV  WS
mTV = mWS
9n – 11 = 7n + 11
2n = 22
 chords have  arcs.
Def. of  arcs
Substitute the given measures.
Subtract 7n and add 11 to both sides.
Divide both sides by 2.
n = 11
mWS = 7(11) + 11 Substitute 11 for n.
Simplify.
= 88°
Example 3B: Applying Congruent Angles, Arcs, and
Chords
C  J, and mGCD  mNJM. Find NM.
GD  NM
GCD  NJM
GD  NM
 arcs have  chords.
GD = NM
Def. of  chords
Find QR to the nearest tenth.
Step 1 Draw radius PQ.
PQ = 20
Radii of a  are .
Step 2 Use the Pythagorean Theorem.
TQ2 + PT2 = PQ2
Substitute 10 for PT and 20 for PQ.
TQ2 + 102 = 202
Subtract 102 from both sides.
TQ2 = 300
TQ  17.3
Take the square root of both sides.
Step 3 Find QR.
QR = 2(17.3) = 34.6
PS  QR , so PS bisects QR.
The circle graph shows the types of cuisine
available in a city. Find mTRQ.
158.4
Inscribed Angle
Inscribed Angle = intercepted Arc/2
160
80
The inscribed angle is
half of the
intercepted angle
Find the value of x and y.
120

x
y

= 120 
= 60
In J, m3 = 5x and m 4 = 2x + 9.
Find the value of x.
Q
5x = 2x + 9
3x = + 9
x=3
D
T
3
J
4
U
Example 4
In K, GH is a diameter and mGNH = 4x – 14.
Find the value of x.
4x – 14 = 90
4x = 104
G
x = 26
H
K
N
Example 5 Solve for x
and z.
2x +18 + 22x – 6 = 180
24x +12 = 180
24x = 168
x=7
z + 85 = 180
z = 95
z
2x + 18
22x – 6
85
1. Solve for arc ABC
244
2. Solve for x and y.
x = 105
y = 100
Vertex is INSIDE the Circle NOT
at the Center
Arc+Arc
ANGLE =
2
Ex. 1 Solve for x
180 – 88
84  x
92 
2
84
184  84  x
x = 100
88
92
X
Ex. 2 Solve for x.
360 – 89 – 93 – 45
133  45
x
2
133
x = 89
93
xº
89
45
Vertex is OUTside the Circle
ANGLE =
Large Arc 

2
 Small Arc 
Ex. 3 Solve for x.
65  15
x
2
x
15°
65°
x = 25
Ex. 4 Solve for x.
27°
x
70  x
27 
2
70°
54  70  x
x = 16
Ex. 5 Solve for x.
360 – 260
260  100
x
2
100
x = 80
x
Warm up: Solve for x
124◦
1.)
2.)
53
70◦
145
x
18◦
x
3.)
260◦
80
x
4.)
70
110◦
x
20◦
Circumference, Arc
Length, Area, and
Area of Sectors
Find the EXACT circumference.
1. r = 14 feet
2. d = 15 miles
C  2 14 
C  28 ft
C   15 
C  15 miles
Ex 3 and 4: Find the circumference. Round to
the nearest tenths.
C  2 14.3 
C    33 
C  89.8 mm
C  103.7 yd
Arc Length
The distance along the curved line making
the arc (NOT a degree amount)

Arc Length
 measure of arc 
Arc Length  
2 r

360


Ex 5. Find the Arc Length
Round to the nearest hundredths
 measure of arc 
Arc Length  
2 r

360


 70 
Arc Length  
2  8 

 360 

8m
Arc Length = 9.77 m
70
Ex 6. Find the exact Arc
Length.
 measure of arc 
Arc Length  
2 r

360


 120 
Arc Length  
2  5 

 360 

10
Arc Length =
 in
3
Ex 7. What happens to the arc length if
the radius were to be doubled? Halved?
 measure of arc 
Arc Length  
2 r

360


20
Doubled

3
5
Halved 
3

Area of Circles
The amount of space occupied.

r
A=
2
r
Find the EXACT area.
2
A    29
8. r = 29 feet
A  841 ft
9. d = 44 miles
 44 
A 
 2 
2
2
A  484 mi
2
10 and 11
Find the area. Round to the nearest tenths.
A   7.6 
 53 
A 
 2 
2
2
A  181.5 yd
2
2
A  2206.2 cm
Area of a Sector
the region bounded by two radii of the circle
and their intercepted arc.
Area of a Sector
 measure of arc  2
A

r

360


Example 12
Find the area of the sector to the nearest hundredths.
R
60
Q
2
 60 
A

6



 360 
A  18.85 cm2
Example 13
Find the exact area of the sector.
6 cm
7 cm
Q
R
2
 120 
A
 7

 360 
120
49
2
A
cm
3
Example 14
Area of minor segment =
(Area of sector) – (Area of triangle)
R


mRQ 2   1


Area of minor segment =
r   b  h
 360
  2



12 yd
Q
 90
1

2
=
 (12)    (12)(12) 
 360
 2

=113.10  72
Area of minor segment =41.10 yd
2