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Sunday, May 31, 2015

Circle Sector and Segment

Circle Sector and Segment
Slices
There are two main "slices" of a circle:
·         The "pizza" slice is called a Sector.
·         And the slice made by a chord is called a Segment.

SECTOR OF A CIRCLE
Common Sectors
The Quadrant and Semicircle are two special types of Sector:

Half a circle is
 
a
 Semicircle.

Quarter of a circle is
a
 Quadrant.
Area of a Sector


A circle has an angle of 2π and an Area of:

πr2
A Sector with an angle of θ (instead of 2π) has an Area of:

(θ/2π) × πr2
Which can be simplified to:

(θ/2) × r2

Area of Sector = ½ × θ × r2   (when θ is in radians)
Area of Sector = ½ × (θ × π/180) × r2   (when θ is in degrees)

Arc Length
By the same reasoning, the arc length (of a Sector or Segment) is:
L = θ × r   (when θ is in radians)
L = (θ × π/180) × r   (when θ is in degrees)

Area of Segment
The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here).
There is a lengthy reason, but the result is a slight modification of the Sector formula:
Area of Segment = ½ × (θ - sin θ) × r2   (when θ is in radians)
Area of Segment = ½ × ( (θ × π/180) - sin θ) × r2   (when θ is in degrees)

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