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.

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.

circular sector area

This is the reasoning:

A circle has an angle of 2π and an Area of:   πr2
A Sector with an angle of θ (instead of 2π) has 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)

 

circular sector arc length

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:

circular segment area

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)