HINT: Pi =3.14

GeometryGeometry

Lessons 8.5 and 8.6Lessons 8.5 and 8.6

Area of Circles, Sectors and Area of Circles, Sectors and SegmentsSegments

HW: 8.5/1-8, 12-14HW: 8.5/1-8, 12-14

8.6/1-128.6/1-12

3in

C ≈ 18.8in

Circumference

Finding a formulae for the area of a circle

πr

r

Area of Rectangle= Base x Height

Finding the area given the diameter

The radius of a circle is half of its diameter, or

We can substitute this into the formula

A = πr2

r = d2

3 m

Find the area of the circle below:

2A r

How to find the area of a circle given the radius

Substitute the values for and r.

23.14 3 3.14 9A 228.26 mA

2A r6 cmr

How to find the area of a circle given the diameter

Given the diameter: Radius = diameter divided by 2

Substitute the values for and r.

12 cm

23.14 6 3.14 36A 2113.04 cmA

Find the area of the square and subtract the area of the circle.

How would you find the shaded area?

The area of a circle

Use π = 3.14 to find the area of the following circles:

A = πr22 cm

= 3.14 × 22

= 12.56 cm2

A = πr210 m

= 3.14 × 52

= 78.5 m2

A = πr2

23 mm = 3.14 × 232

= 1661.06 mm2

A = πr278 cm

= 3.14 × 392

= 4775.94 cm2

Find the area of this shape

Use π = 3.14 to find area of this shape.

The area of this shape is made up of the area of a circle of diameter 13 cm and the area of a rectangle of width 6 cm and length 13 cm.

6 cm13 cm Area of circle = 3.14 × 6.52

= 132.665 cm2

Area of rectangle = 6 × 13

= 78 cm2

Total area = 132.665 + 78

= 210.665 cm2

The circumference of a circle is 18. Find the area of the circle, leave answer in terms of .C = 2

r18 = 2r

9 = r

A = r2

A = 92

A = 81

Arc Length ReminderArc Length Reminder

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.

Arc Length FormulaArc Length Formula

m3602πrArc Length =

measure of the central angle or arc

The fraction of the circle!

The circumference of the entire circle!

.

Find the arc length. Give answers in terms of and rounded to the nearest hundredth.

REVIEW: Finding Arc Length

FG

Use formula for area of sector.

5.96 cm 18.71 cm

Substitute 8 for r and 134 for m.

Simplify.

SectorsSectorsLet’s talk Let’s talk pizzapizza

Sector of a circleSector of a circle

A region bounded by 2 radii and A region bounded by 2 radii and their intercepted arc.their intercepted arc.

.

Area of a Sector Area of a Sector FormulaFormula

m˚

360πr2Area of a sector =

measure of the central angle or arc

The fraction of the circle!

The area of the entire circle!

.

Find the area of the sector. Give answers in terms of and rounded to the nearest hundredth.

Finding the Area of a Sector

sector HGJ

Use formula for area of sector.

Substitute 12 for r and 131 for m.

= 52.4 m2 164.54 m2 Simplify.

Find the area of the sector. Give answers in terms of and rounded to the nearest hundredth.

Finding the Area of a Sector

sector ABC

Use formula for area of sector.

Substitute 5 for r and 25 for m.

1.74 ft2 5.45 ft2 Simplify.

9

4 40m AOB 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

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

A O

B

Find the area of each sector. Give your answer in terms of and rounded to the nearest hundredth.

sector JKL

Use formula for area of sector.

Substitute 16 for r and 36 for m.

= 25.6 in2 80.42 in2 Simplify.

Finding the Area of a Sector

A windshield wiper blade is 18 inches long. To the nearest square inch, what is the area covered by the blade as it rotates through an angle of 122°?

Automobile Application

Use formula for area of sector.

r = 18 in. m˚ = 122˚

345 in2 Simplify.

A segment of a circle is a region bounded by an arc and its chord.

AREA OF SEGMENT

10 =½∙10∙10=5=½∙10∙10=500

Area of Area of Segment = 25Segment = 25ππ

- 50- 50

= ¼100= ¼100ππ= = 2525ππ

Check It Out!

Find the area of segment RST to the nearest hundredth.

Use formula for area of sector.

Substitute 4 for r and 90 for m.

= 4 m2 Simplify.

Step 1 Find the area of sector RST.

Simplify.

Step 2 Find the area of ∆RST.

Check It Out! Continued

Find the area of segment RST to the nearest hundredth.

ST = 4 m, and RS = 4m.

= 8 m2

Step 3

area of segment = area of sector RST – area of ∆RST

4.57 m2

Find the area of segment RST to the nearest hundredth.

Check It Out! Continued

= 4 – 8

Lesson Quiz: Part I

Find each measure. Give answers in terms of and rounded to the nearest hundredth.

1. area of sector LQM

2.5 in. 7.85 in.

7.5 in2 23.56 in2

2. length of NP

Lesson Quiz: Part II

3. The gear of a grandfather clock has a radius of 3 in. To the nearest tenth of an inch, what distance does the gear cover when it rotates through an angle of 88°?

4.6 in.

4. Find the area of segment GHJ to the nearest hundredth.

55.94 m2

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