chapter 10.6. circle a set of all points equidistant from the center

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 1

Name 3 of the following in A.1. the minor arcs

2. the major arcs

3. 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. UT

2. UV

3. VUT

4. ST

5. VS

Practice 3

Find each indicated measure for D.

1. mEDI

2.

3. mIDH

4.

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 arc360 * d

600

7

Arc Length

________________ 60360 * 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 arc360 * r2

600

12

Arc Length

________________ 60360 * 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