# 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

## 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.

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)

## Arc Length

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)

## 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 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)