C
Parts of a Circle
Circle – set of all
points _________
equidistant
from a given point
called the _____
center
of the circle.
Symbol:
C
CHORD:
A segment
whose
endpoints
are on the
circle
Radius
RADIUS:
P
Distance from
the center to
point on
circle
Diameter
P
DIAMETER:
Distance
across the
circle
through its
center
Also known as the
longest chord.
D = ?
r = ?
r = ?
D = ?
Use P to determine whether each
statement is true or false.
Q
1. RT is a diameter. False R
2. PS is a radius. True
P
3. QT is a chord. True
T
S
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 a radius
Point of Tangency:
The point where the
tangent intersects
the circle
Name the term that best describes the notation.
Central Angle :
An Angle whose vertex is at the center of the circle
A
Major Arc
Minor Arc
More than 180°
Less than 180°
P
ACB
To name: use
3 letters
AB
C
B
APB is a Central Angle
To name: use
2 letters
Semicircle: An Arc that equals 180°
E
D
To name: use
3 letters
EDF
P
F
THINGS TO KNOW AND
REMEMBER ALWAYS
A circle has 360 degrees
A semicircle has 180 degrees
Vertical Angles are Equal
measure of an arc = measure of central angle
A
E
Q
m AB = 96°
m ACB = 264°
m AE = 84°
96
B
C
Arc Addition Postulate
A
C
B
m ABC = m AB + m BC
Tell me the measure of the following arcs.
m DAB = 240
m BCA = 260
D
C
140
R
40
100
80
B
A
Congruent Arcs have the same measure and
MUST come from the same circle or of
congruent circles.
C
B
45
A
45
D
110
Arc length is proportional to “r”
Warm up
Central Angle
Angle = Arc
Inscribed Angle
• Angle where the vertex in
ON the circle
Inscribed Angle
ARC
ANGLE =
2
Intercepted Arc
Inscribed Angle 
2
160

80
The arc is
twice as big as
the angle!!
Find the value of x and y.
120

x
y


Examples
1. If mJK = 80 and JMK = 2x – 4, find x.
x = 22
2. If mMKS = 56, find m MS.
112 
J
K
Q
M
S
Find the measure of DOG and DIG
72˚
D
If two inscribed
angles intercept
the same arc,
then they are
congruent.
G
O
I
If all the vertices of a polygon
touch the edge of the circle, the
polygon is INSCRIBED and the
circle is CIRCUMSCRIBED.
Quadrilateral inscribed in a circle:
opposite angles are
SUPPLEMENTARY
B
A
D
C
mA  mC  180
mB  mD  180
If a right triangle is inscribed in a
circle then the hypotenuse is the
diameter of the circle.
Example 3
In J, m3 = 5x and m 4 = 2x + 9.
Find the value of x.
Q
D
x=3
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
x = 26
H
K
G
N
Bonus: What type of triangle is this? Why?
Example 5 Find y and z.
z
110
110 + y =180
y
y = 70
z + 85 = 180
z = 95
85
Warm Up
1. Solve for arc ABC
244
2. Solve for x and y.
x = 105
y = 100
Wheel of Formulas!!
Vertex is INSIDE the Circle
NOT at the Center
Arc+Arc
ANGLE =
2
Ex. 1 Solve for
x
84
88
X
x = 100
Ex. 2 Solve for x.
93
xº
x = 89
89
45
Vertex is OUTside the Circle
Large Arc  Small Arc
ANGLE =
2
Ex. 3 Solve for x.
x
15°
65°
x = 25
Ex. 4 Solve for x.
27°
x
70°
x = 16
Ex. 5 Solve for x.
x
x = 80
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 INside 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>
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
of Circles
2 Types of Answers
Rounded
• Type the Pi
button on your
calculator
• Toggle your
answer
• Round
Exact
• Type the Pi
button on your
calculator
• Pi will be in your
answer
• TI 36X Pro gives
exact answers
Circumference
The distance around a circle
Circumference
C  2r
or
C  d
Find the EXACT circumference.
1. r = 14 feet
C  214
2. d = 15 miles
C  15
28 ft
15 miles
Ex 3 and 4: Find the circumference.
Round to the nearest tenth.
C  214.3
C  33
89.8 mm
103.7 yd
5. A circular flower garden has a radius
of 3 feet. Find the circumference of the
garden to the nearest hundredths.
C  2r
C  23
C = 18.85 ft
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 6. 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 7. 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 8 Find the radius. Round to the nearest
hundredth.
»
Arc Length of AB = 3.82m

60
3.82 
2 r
360
1375.2  60  2 r
1375.2  120 r
11.46   r
r  3.65 m
Ex 9 Find the circumference. Round to
the nearest hundredth.
»
Arc Length of AB = 32.11in

80
32.11 
2 r
360
11,559.6   80  2 r
11,559.6   80  C
C  144.50in
Ex 10 Find the radius of the unshaded region.
Round to the nearest tenth.
»
Arc Length of AB = 10cm

75
10 
2 r
360
3600  75  2 r
3600  150 r
24   r
r  7.6cm