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8-29-14T1.2g To Find Arc Length And Area Of A Sector

A medicine wheel in Wyoming, used for religious and ceremonial events. There are 32 spokes.

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OPENER: If a bicycle wheel revolves 75 times every minute, through how many degrees will it move in a 2 hour period?

75 r.

1 min.

360

1 r.

60 min.

1 hr.

2 hr.

1 3,240,000

The wheel moves 3,240,000° in a 2 hour period.

1,620,000

1 hr.

1,620,000

1 hr.

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Thurs 9/4: Quiz on•Reference Angles•Rotation•Arc length•Sector area

OPENER # 2

What are the reference angles for the following?

1. 71° 2. 219° 3. 158° 4. 296°

1. 71° 2. 39° 3. 22° 4. 64°

0°

90°

180°

270°

360°

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Active Learning Assignment?

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LESSON: The Arc Length on a circle is the part of the circumference that is between the initial and terminal sides of an angle. To find the Arc Length, what could be important?

These two circles are the same size. The red and blue curves represent the arcs.

Are the arcs the same size? What makes the difference?

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67° 67°

Ah ha! The angle is important. What if we have the same angle on two different circles?Are the arcs the same size? What makes the difference here?

So, the radius makes a difference, too. It there anything else that could influence the size of the arc?

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What are the relationships between the angle size, radius length, and the arc length?

Since it is a direct variation, that is, the larger the angle, the larger the arc and the larger the radius, the larger the arc, then we have the formula for ARC LENGTH of that sector:

s = rӨ (Ө is always in radians)

What if we’re given degrees? Convert it!

180Use or ?

180

Arc length Radius

Angle

*

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Using the formula s = rӨ, try: (1 decimal place)

1. The sector of a circle has a radius of 4 cm and a central angle is 2.4. What is the length of the arc?

9.6 cm. s 4cm 2.4The length is 9.6 cm.

2.4

4 cm

?

s = rӨ

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6.7 m.

s 8m 48

2. The sector of a circle has a radius of 8 m and an angle is 48°. What is the length of the arc? (1 decimal place)

The length is 6.7m.

Try:

8 * 48 * π / 180

137.5

2.4 12in 5in

12in

5in

1802.4

3. If an arc is 12 inches and the radius of a circle is 5 inches, what is the angle? What is the degree measurement?

The angle is 2.4.

The angle is 137.5°.

s = rӨ

s = rӨ

?

180

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The Sector Area of a circle is the part of the circle that is between the initial and terminal sides of an angle. The same factors that influence the arc length influence the sector area.

21The formula is : K r

2( is always in radians.)

Try:

A sector of a circle has a radius of 3 inches and a central angle of 27°. Find the sector area to one decimal place.

The sector area is 2.1 in.2

21K 3 27

2

= 2.1

Area Radius

Angle

2.5 3 27 / 180

*21

K r2

?

180

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1. First, we need to find the angle measure of each sector.

The arc length is 5.9 ft and the sector area is 88.4 ft2.

s 30ft16

= 5.9 ft 21K 30ft

2 16

= 88.4 ft2

3. Find the sector area. (1 dec. pl.)

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32

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The medicine wheel has a radius of 30 feet, with 32 sectors. We want to know the arc length and the sector area.

2. Find the arc length of each sector. (1 dec. pl.)

s = rӨ

21K r

2

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Active Learning Assignment:Written Exercises P 264: 1 – 8

(ALWAYS WRITTEN, UNLESS DIRECTED OTHERWISE)

Thursday 9/4: Quiz on•Reference Angles

•Rotation•Arc length•Sector area

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