Circle Sector and Segment
SlicesThere are two main "slices" of a circle:

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Sector  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 SectorYou 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. 

This is the reasoning:
Area of Sector = ½ × θ × r^{2} (when θ is in radians) Area of Sector = ½ × (θ × π/180) × r^{2} (when θ is in degrees) 
Arc LengthBy 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 SegmentThe 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) × r^{2} (when θ is in radians) Area of Segment = ½ × ( (θ × π/180)  sin θ) × r^{2} (when θ is in degrees) 