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.

The Quadrant and Semicircle are two special types of 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.

This is the reasoning:

Area of Sector = ½ × θ × r^{2}

= ½ × (θ × π/180) × r^{2} (if θ is in degrees)

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)

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:

Area of Segment = ½ × (θ - sin θ) × r^{2}

= ½ × ( (θ × π/180) - sin θ) × r^{2} (if θ is in degrees)