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Sectors, Segments, & Annuli Parts of Circles (and yes, you need to know this)

Sectors, Segments, & Annuli

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Sectors, Segments, & Annuli. Parts of Circles (and yes, you need to know this). We start with a circle. Then…. Sector. Segment. Annulus. Sector. Segment. Annulus. How do we find the areas of these?. We know the area of a circle. r. = radius. A = π r 2. So…. Sector. r. - PowerPoint PPT Presentation

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Page 1: Sectors, Segments, & Annuli

Sectors, Segments, & Annuli

Parts of Circles(and yes, you need to know this)

Page 2: Sectors, Segments, & Annuli

We start with a circle

Then…

Page 3: Sectors, Segments, & Annuli

Sector

Page 4: Sectors, Segments, & Annuli

Segment

Page 5: Sectors, Segments, & Annuli

Annulus

Page 6: Sectors, Segments, & Annuli

Sector

Page 7: Sectors, Segments, & Annuli

Segment

Page 8: Sectors, Segments, & Annuli

Annulus

Page 9: Sectors, Segments, & Annuli

How do we find the areas of these?

Page 10: Sectors, Segments, & Annuli

We know the area of a circle

So…

r = radius

A = πr2

Page 11: Sectors, Segments, & Annuli

Sector

r = radiusα

Degrees: 360˚ - αRadians: 2π - α

Page 12: Sectors, Segments, & Annuli

Sector

r = radiusα

Degrees: 360˚ - αRadians: 2π - α

Area of the Circle:A = πr2

Ratio of Sector to Circle:(degrees) α/360(radians) α/2π

Page 13: Sectors, Segments, & Annuli

Sector

Radians:A = πr2 (α/2π)= πr2 (α/2π)

=(α/2) r2

Page 14: Sectors, Segments, & Annuli

Sector

Radians:A = ½ r2α

Degrees:A = (α/360) πr2

Page 15: Sectors, Segments, & Annuli

What’s the area of this sector?

5

90°

Hint: 90° is the same as π/2 radians

Page 16: Sectors, Segments, & Annuli

What’s the area of this sector?

5

90°

Area of the Circle:A = 52π

Ratio of the Sector:R = 90°/360° orR = (π/2)(1/2π)

Area of the Sector:A = 52π(π/2)(1/2π)A = 52π(90°/360°)

Answer:A = 25π/4

Page 17: Sectors, Segments, & Annuli

Segment

α r = radius

Degrees: 360˚ - αRadians: 2π - α

Page 18: Sectors, Segments, & Annuli

Segment

α r

Radians:A = ½ r2α

Degrees:A = (α/360) πr2

Then…

Page 19: Sectors, Segments, & Annuli

Segment

α r

Area of the Segment:A = ½ r2α – ½ bh (radians)A = (α/360) πr2 – ½ bh (degrees)

bh

Area of the Segment =Area of the Sector – Area of the Triangle

Page 20: Sectors, Segments, & Annuli

What’s the area of this segment?

120° 5

8

Hint: 120° is the same as 2π/3 radians

Page 21: Sectors, Segments, & Annuli

What’s the area of this segment?

120° 5

8

Area of the circle:A = 52π

Ratio of the sector:R = 120°/360° orR = (2π/3)(1/2π)

R = 1/3

Area of the sector:A = 25π(1/3)A = 25π/3

Then…

Page 22: Sectors, Segments, & Annuli

5

Area of the Triangle:A = ½ bhA = ½ (8)(3) = ½ 24 = 12

4h

Area of the Segment =Area of the Sector – Area of the Triangle

4

h2 = 52 – 42

h2 = 25 – 16h2 = 9h = 3

So…

What’s the area of this segment?

Page 23: Sectors, Segments, & Annuli

Area of the Segment:A = As - At

A = 25π/3 – 12 = (25π/3) – (36/3)

Area of the Sector = 25π/3Area of the Triangle = 12

What’s the area of this segment?

Answer:A = (25π – 36)/3

Page 24: Sectors, Segments, & Annuli

Annulus

r1

r2

Area of outside circle:A = πr1

2

Area of inside circle:A = πr2

2

Page 25: Sectors, Segments, & Annuli

Annulus

r1

r2

Area of Annulus =Area of Outside Circle – Area of Inside Circle

Area of Annulus:A = πr1

2 – πr22

Page 26: Sectors, Segments, & Annuli

3

2

What’s the Area of this annulus?Area of outside circle:A = 32π = 9π

Area of inside circle:A = 22π = 4π

Area of Annulus =Area of Outside Circle – Area of Inside Circle

So…

Page 27: Sectors, Segments, & Annuli

3

2

What’s the Area of this annulus?

Area of Annulus :A = Ao - Ai

A = 9π – 4π

Answer:A = 5π

Page 28: Sectors, Segments, & Annuli

Questions?