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	Segment

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.

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:

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 (when θ is in radians)

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

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)

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)