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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.

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

Quarter of a circle is called a Quadrant.

Half a circle is called a Semicircle.

 

Area of a Sector

You can work out the Area of a Sector by comparing its angle to the angle of a full circle.

Note: I am using radians for the angles.

circular sector area

This is the reasoning:

  • A circle has an angle of 2π and an Area of: πr2
  • So a Sector with an angle of θ (instead of 2π) must have an area of: (θ/2π) × πr2
  • Which can be simplified to: (θ/2) × r2

Area of Sector = ½ × θ × r2

= ½ × (θ × π/180) × r2   (if θ is in degrees)

 

circular sector arc length

Arc Length of Sector or Segment

By the same reasoning, the arc length (of a Sector or Segment) is:

Arc Length "L" = θ × r

= (θ × π/180) × r   (if 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 derivation, but the result is a slight modification of the Sector formula:

circular segment area

Area of Segment = ½ × (θ - sin θ) × r2

= ½ × ( (θ × π/180) - sin θ) × r2   (if θ is in degrees)

 

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