Areas of Circles, Sectors, and Segments 11'6

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Areas of Circles, Sectors, and Segments11.6
GEC Chpt 11 Notes\Geometry Formulas for Ch. 11.doc
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11.6 Areas of Circles and Regions of Circles
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Circumference

• The circumference of a circle equals
• C2?r C ?d

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Circle Measures
A whole circle has 360 a circumference of 2?r
an area of ?r2
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Circle Measures
A whole circle has 360 a circumference of 2?r
and an area of ?r2.
An arc that measures 45 intercepts an arc that
is how long, and an area that is how big?
How do we divide up the circle, its degrees,
circumference, or area?
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Arc Length Corollary

• In a circle, the ratio of the length of a given
  arc to the circumference is equal to the
  ratio of the measure of the arc to 360

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Arc Length Corollary



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GSP Arcs

• ..\..\..\..\..\Geometry\Heath geometry\HChpt
  10-Circles\GCP_Notes\arcs 2003.gsp

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Postulate

• The ratio of the degree measure m of the
  central angle of a sector to 360 degrees is the
  same ratio of the area of the sector to the area
  of the circle, that is,

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Theorem

• In a circle of radius r, the Area A of a sector
  whose arc has degree measure m is given by
• I think of this theorem as Aproportion of slice
  times area of circle.

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Example

• Find the area of the sector given that the arc
  length is 30 degrees and the radius is 2 units.

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Definition

• A sector of a circle is a region bounded by two
  radii of the circle and the arc intercepted by
  those radii.

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Corollary

• Corollary The area of a semicircular region of
  the radius r is

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Area of a Segment

• Definition A segment of a circle is the region
  bounded by a chord and its minor (or major) arc.

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Area of a Segment of a Circle
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Area of a Triangle with an Inscribed Circle

• Theorem Where P represents the perimeter of a
  triangle and r represents the length of the
  radius of its inscribed circle, the area of the
  triangle is given by

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Example
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Homework

• HW 11_6.gsp

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