CIRCLES AND ARCSChapter 10.6

Circle A set of all points equidistant from the

center

Center

Circle A circle is named by the center

Circle P (P)

P

Diameter A segment that contains the center of a

circle and has both endpoints on the circle.

Diameter

Radius A segment that has one endpoint at the

center of the circle and the other on the circle.

Radius

Congruent Circles Congruent circles have the congruent

radii

P Q

Central Angle An angle whose vertex is the center of

the circle.

Central Angle

Arc Part of a circle. From point to point on

the outside of the circle.

Arc

Semicircle An arc that’s half of the circle.

SemicircleHas a measure of 1800

1800

Minor Arc A minor arc is smaller than half the

circle.

Minor Arc

400

Same measure as the corresponding interior angle

Major Arc A major arc is larger than half the circle.

Major Arc360 minus the minor arc

400

3200

Practice 1Name 3 of the following in A.1. the minor arcs2. the major arcs3. the semicircles

Adjacent Arcs Adjacent arcs are arcs of the same circle that

have exactly one point in common.

Arc Addition Postulate The measure of the arc formed by two adjacent

arcs is the sum of the measure of the two arcs.

400 7001100

Practice 2 Find the measure of each arc in R.1. UT2. UV3. VUT4. ST5. VS

Practice 3 Find each indicated measure for D.

1. mEDI2.3. mIDH4.

Circumference The distance around the circle A measure of length

Circumference The circumference of a circle is π times the

diameter (a = πd) or 2 times π and the radius (a = 2πr).

Diameter

Circumference Example:

D = 4

C = d= 4

or = 12.52

Circumference Example:

C = 2r= 2(5)

or = 31.4r = 5= 10

Practice 4 Find the circumference of each circle.

Leave your answer in terms of .

1. 2.

Arc Length The length of an arc is calculated using

the equation:

600

measure of the arc________________360 * circumference

Arc Length The length of an arc is calculated using

the equation:

600

measure of the arc________________360 * d

Arc Length The length of an arc is calculated using

the equation:

600

measure of the arc________________360 * 2r

Arc Length________________measure of the arc

360 * d

600

7

Arc Length________________ 60

360 * 7

600

7

Arc Length________________ 1

6 * 22

600

7

= 3.67

Practice 5 Find the length of each darkened arc.

Leave your answer in terms of .

1. 2.

Area of a Circle The product of π and the square of the

radius.

A = r2

Radius

Area of a Circle Example:

A = r2

= 52

or = 78.54r = 5= 25

Practice 6 Find the area of a circle:

1. 6 in. radius

2. 10 cm. radius

3. 12 ft. diameter

Sector of a Circle A sector of a circle is a region bounded

by an arc of the circle and the two radii to the arc’s endpoints.

You name a sector using the two endpoints with the center of the circle in the middle.

Sector of a Circle Sector is the area of part of the circle

Area of blue section

Area of Sector of a Circle The area of a sector is:

measure of the arc________________360 * r2

Sector of a Circle Find the area of the sector

600

12

Arc Length________________measure of the arc

360 * r2

600

12

Arc Length________________ 60

360 * 122

600

12

Arc Length________________ 1

6 * 144

600

12

= 24

Segment of a Circle Part of a circle bounded by an arc and

the segment joining its endpoints

Area of a Segment of a Circle Equal to the area of the sector minus the

area of a triangle who both use the center and the two endpoints of the segment.

Sector – Triangle = Segment

Area of a Segment of a Circle

- =

Area of a Segment of a Circle Find the area of the segment.

600

12

Area of a Segment of a Circle Separate the triangle and the sector

600

12600

12

Area of a Segment of a Circle Find the area of both figures

600

12600

12

Area of Sector

600

12

________________ 60360 * 122

= 24

Area of Triangle

600

6ð3

Find the altitude 12

or 10.4Find the base

6

Area of Triangle

600

12

10.4

6

a = ½bh

= ½(12)(10.4)= 62.4

Area of a Segment of a Circle Subtract the triangle from the Sector

24 62.4

-24 62.4 = 13