sectors, segments, & annuli
DESCRIPTION
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 PresentationTRANSCRIPT
![Page 1: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/1.jpg)
Sectors, Segments, & Annuli
Parts of Circles(and yes, you need to know this)
![Page 2: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/2.jpg)
We start with a circle
Then…
![Page 3: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/3.jpg)
Sector
![Page 4: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/4.jpg)
Segment
![Page 5: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/5.jpg)
Annulus
![Page 6: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/6.jpg)
Sector
![Page 7: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/7.jpg)
Segment
![Page 8: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/8.jpg)
Annulus
![Page 9: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/9.jpg)
How do we find the areas of these?
![Page 10: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/10.jpg)
We know the area of a circle
So…
r = radius
A = πr2
![Page 11: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/11.jpg)
Sector
r = radiusα
Degrees: 360˚ - αRadians: 2π - α
![Page 12: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/12.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/13.jpg)
Sector
rα
Radians:A = πr2 (α/2π)= πr2 (α/2π)
=(α/2) r2
![Page 14: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/14.jpg)
Sector
rα
Radians:A = ½ r2α
Degrees:A = (α/360) πr2
![Page 15: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/15.jpg)
What’s the area of this sector?
5
90°
Hint: 90° is the same as π/2 radians
![Page 16: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/16.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/17.jpg)
Segment
α r = radius
Degrees: 360˚ - αRadians: 2π - α
![Page 18: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/18.jpg)
Segment
α r
Radians:A = ½ r2α
Degrees:A = (α/360) πr2
Then…
![Page 19: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/19.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/20.jpg)
What’s the area of this segment?
120° 5
8
Hint: 120° is the same as 2π/3 radians
![Page 21: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/21.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/22.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/23.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/24.jpg)
Annulus
r1
r2
Area of outside circle:A = πr1
2
Area of inside circle:A = πr2
2
![Page 25: Sectors, Segments, & Annuli](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/25.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/26.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/27.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022062310/56815c3f550346895dca3d22/html5/thumbnails/28.jpg)
Questions?
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And now for the homework…