Chapter 10: Area

10.6

Circles & Arcs

Definitions• circle: set of all points equidistant from a given

point

• center: point that is equidistant from the circle

• radius: segment that has one endpoint at the center and the other endpoint on the circle

• congruent circles: have congruent radii

Definitions

• diameter: segment that contains the center of a circle and has both endpoints on the circle

• central angle: angle whose vertex is the center of the circle

Example 1• In the circle graph from a hour-by-hour survey

of 3600 people, what is the measure in degrees of the entertainment part?

other 15%

entertainment 18%

must do 7%

work 20%food 9%

sleep 31%

Definitions

• arc: a piece of a circle

• semicircle: half of a circle

• minor arc: smaller than a semicircle

• major arc: greater than a semicircle

Example 2• Identify the

following:

• the minor arcs

• the semicircles

• the major arcs that contain point A

O

A

E

C

D

Adjacent Arcs

• arcs of the same circle that have exactly one point in common

• can add the measures of adjacent arcs like adjacent angles

Arc Addition Postulate

• The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

O

B

C

A

mBCmABmABC

Example 3

• Find the measure of each arc:

• BC

• BD

• ABC

• AB

58

32

O

B

C

A

D

Circumference of a Circle

• The circumference of a circle is π times the diameter.

rdC 2

Concentric Circles

• circles that lie in the same plane and have the same center

Example 4• A car has a turning radius of 16.1 ft. 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?

Example 4a• The diameter of a bicycle wheel is 22 in. To the

nearest whole number, how many revolutions does the wheel make when the bicycle travels 100 ft?

Arc Length• The length of an arc of a circle is the product of

the ratio (measure of the arc / 360) and the circumference of the circle.

OB

A

Example 5

• Find the length of the arc XY.

16 inO Y

X

Example 5

• Find the length of arc XY.

240

15 cm

O

P

Y

X

Homework

• P. 569

• 2-8 even, 18, 20, 22, 28, 30, 36, 53