10-6: Circles and Arcs
Goal: Be able to find the measures of central angles
and arcs, and to find the circumference and arc length
of circles.
circle – the set of all points ______________
equidistant
center
from a given point, the __________.
This is A
(circle A)
A
center
radius – a segment that has one __________
endpoint
center and the other endpoint
at the __________
circle
on the _________.
diameter – a segment that contains the
center of a circle and has both
__________
endpoints the circle.
__________on
C
A B is a radius.
A C , A D are also radii.
A
B
CD
D
is a diameter.
central angle – an angle whose _________
vertex
center of the circle.
is at the __________
C
CAB
A
B
is a central angle.
 B A D is a central angle.
D
There are ________
360 degrees in a circle.
The following information was determined from a survey determining how people really spend their
time.
How People Spend Their Time
7%
31%
18%
Sleep
Other
Food
Work
Entertainment
20%
Must Do
15%
9%
Directions: Find the measure of each central angle to the nearest whole number.
Find the corresponding percent of 360.
1.) Sleep .31(360  )  1 1 2 
4.) Work .20(360  )  7 2 
2.) Other .15(360  )  5 4 
5.) Entertainment .18(360  )  6 5 
3.) Food .09(360  )  3 2 
6.) Must Do .07 (360  )  2 5 
arc – part of a _________.
circle
C
A
E
B
D
is a minor arc.
AB
0   m AB  180 
H
F
G
FGH
is a semicircle.
m FG H  180 
CDE is a major arc.
180   m C D E  360 
***The measure of a
equals
minor arc __________
the measure of its
corresponding
central angle.
______________
Identify the following in circle C.
M
Y
C
W
D
X
X D , D Y , YM , M W , X W
a.) two minor arcs _____________________________
D YW , D YX , YM X , YM D , M W D
b.) two major arcs _____________________________
YM W , M W X , W X Y
c.) two semicircles _____________________________
Find the measure of each arc.
M
84°
Y
C
40°
W
56° 84°
D
56°
84°
X
1 .) X D
m XD  m  D C X  56 
2 .) X Y
m XY  m XD  m D Y
3 .) M W X
so, m XY  56   40   96 
M W X is a semicircle. So, m M W X  180 
so,
m D XM  56   180   236 
4 .) D X M
m DXM  m DX  m XW M
5 .) YM
mYM  180   m XY  180   96   84 
6.) m  X C W m  XCW  m XW  84 
circumference of a circle – the ___________
distance
around the circle.
r
C d
or C  2  r
d
Circles that lie in the same plane and have
concentric circles
the same center are ___________________.
A car has a turning radius of 16.1 feet. The distance between the
two front tires is 4.7 ft. In completing the (outer) turning circle,
how much farther does a tire travel than a tire on the concentric
inner circle?
To find the radius of the inner circle, subtract 4.7 ft from the
turning radius.
circumference of outer circle: C  2  r  2  (16.1)  32.2 
radius of the inner circle: 1 6 .1  4 .7  1 1 .4
circumference of inner circle: C  2 r  2 (11.4)  22.8
The difference of the two distances: 3 2 .2   2 2 .8   9 .4   2 9 .5
A tire on the turning circle travels about 29.5 ft farther than
a tire on the inner circle.
Arc Length – fraction of the_______________
circumference
of a circle.
C
B
length of B C
r
r
 CD ?
2 r
360
A
Is AB

m BC
No.
C
105
105
105
D
B
m A B  m C D , but AB  C D
They are part of different circles
with different radii.
Congruent arcs – arcs that have the _________________
same measure
in the same circle or ___________________.
in congruent circles.
and are __________________
Find the length of the arcs.
C
1 .) B C
5 cm
B
m BC
60
 2 r
360
60
 2  (5) 
360
5
m AD B  360  150  210
cm
B
3
2.) A D B
m ADB
150
A
 2 r
360
210
360
18 cm
 2  (18)  21 cm
D