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

Try Them!
Sector  Segment 

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
You can work out the Area of a Sector by comparing its angle to the angle of a full circle.
Note: we are using radians for the angles.
This is the reasoning:
A circle has an angle of 2π and an Area of:  πr^{2}  
A Sector with an angle of θ (instead of 2π) has an Area of:  (θ/2π) × πr^{2}  
Which can be simplified to:  (θ/2) × r^{2} 
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) 