Arc Length, Sectors, SectionsArc Length, Sectors, Sections

GeometryGeometry

Arc Lengths and Areas of SectorsArc Lengths and Areas of Sectors

Important to know!!Important to know!!

In a circle, the measure of the In a circle, the measure of the central angle equals the measure central angle equals the measure of its corresponding arcof its corresponding arc

110⁰

That means if the angle is 110⁰

Then the measure of the arcright across from it is also 110⁰

Let’s Try anotherLet’s Try another

AB

70⁰

What is the measure of arc AB?70⁰

Another Another

A

B

C

D

E

F

120⁰ 100⁰

Is <AB central? YES

What is the measure of ACB?

240⁰

If DF is the diameter, what is the measure of <EF? 80⁰

What is the measure of arc EF?

Also 80⁰

Arc LengthArc Length

The length of part of the The length of part of the circumference. circumference. The length of the arc depends on what two things?

1) The measure of the arc.2) The size of the circle.

An arc length measures distance while the measure of an arc is in degrees.

Sector of a circleSector of a circle

A region bounded by 2 radii and an A region bounded by 2 radii and an arc.arc.

.

Minor ArcMinor Arc•Use 2 lettersUse 2 letters•Angle is less than or equal to 180Angle is less than or equal to 180

XX YY

ZZ

120°120°99

XYZ Major ArcMajor Arc•Use 3 lettersUse 3 letters•Angle is greater than Angle is greater than 180180

XZ

m XZ = m<XCZ = 120o

The measure of arc XZ equals the measure of angle XCZ

C

Central Angle: Any angle whose vertex is the center of the circle

Termin

ology

Portions of a Circle: Determine the Arc measure based on the portion given.

A. B. C. D.

¼ of a circle: ½ of a circle: 1/3 of circumference : 6π out of a total 36π on the circle: ¼ ● 360

90o

90o

½ ● 360

180o

180o

1/3 ● 360

120o

120o

1/6 ● 360

60o

60o

Area of a Sector Area of a Sector FormulaFormula

ѳ 360

Area of a sector =

measure of the central angle or arc

The fraction of the circle!

The area of the entire circle!

.

Arc Length FormulaArc Length Formula

2Πrѳ360

Arc Length =

measure of the central angle or arc

The fraction of the circle!

The circumference of the entire circle!

.

90

6

m AOB

radius

240

12

m AOB

radius

300

12

m AOB

radius

120

2.4

m AOB

radius

108

10 2

m AOB

radius

Find the length of AB and the area of sector AOB.

1. 2. 3. 4. 5.

Length of AB Length of AB Length of AB Length of AB Length of AB

Area of sector AOB Area of sector AOB Area of sector AOB Area of sector AOB Area of sector AOB

Fraction of circle:

¼

90o

90o

6

Fraction ● circumference

¼ ● 12π

3π units

Fraction ● area

¼ ● 36π

9π units2

28.26

240o

240o

12

Fraction of circle:

2/3

Fraction ● circumference

2/3 ● 24π

16π units

Fraction ● area

2/3 ● 144π

96π units2

301.44

300o

300o 12

Fraction of circle:

5/6

5/6 ● 24π

20π units

Fraction ● area

5/6 ● 144π

120π units2

376.8

A

AB

B

O OAO

B

120o

120o

2.4AO

B

Fraction of circle:

1/3

Fraction ● circumference

1/3 ● 4.8π

1.6π units

Fraction ● area

1/3 ● 5.76π

1.92π units2

6.03

108o

108o

10√2A

B

O

Fraction of circle:

3/10

3/10 ● 20√2π

6√2π units

Fraction ● area

3/10 ● 200π

60π units2

188.4

6. The area of sector AOB is 48π and 270m AOB . Find the radius of ○O.

m

360πr2Area of a sector =

270

360πr248π =

3

4r248 =

4

3

4

3

16

r264 =

r = 8

9

4 40m AOB 7. The area of sector AOB is and . Find the radius of ○O.

m

360πr2Area of a sector =

40

360πr2 π =

9

41

9r2 =

9

4

9

1

9

1

r2 =81

4

r = 9

2

SectionSectionssLet’s talk Let’s talk

pizzapizza

AREA OF SECTIONAREA OF SECTION = = AREA OF SECTOR – AREA OF SECTOR – AREA OF AREA OF TRIANGLETRIANGLE

¼ ¼ ππ r² - r² - ½ bh½ bh

Area of sectionArea of section = = area of sector – area of sector – area of area of triangletriangle ¼ ¼ ππ r² - r² - ½ bh½ bh

1010A OF = ½∙10∙10=A OF = ½∙10∙10= 5050

A OF SECTION = A OF SECTION =

2525ππ - 50 - 50A of circle = A of circle = 100100ππ

A OF = ¼ 100A OF = ¼ 100ππ == 2525ππ

60˚

8 612

60 430

OO

O

8. 9. 11.

Find the area of the shaded region. Point O marks the center of the circle.

10.

160

3π units2 9π - 18 units2 24π - 36√3 units2 8π - 8√3 units2

Some common fractions and Some common fractions and measures!measures!

Arc or Central Arc or Central Angle MeasureAngle Measure

Fraction of the Fraction of the CircleCircle

Arc or Central Arc or Central Angle MeasureAngle Measure

Fraction of the Fraction of the CircleCircle

3636oo 108108oo

1/61/6 5/65/6

120120oo 2/32/3

3030oo 11/1211/12

1/81/8 5/85/8

1/10

1/3

1/12

3/10

60o

45o

300o

240o

225o

330o