Circle Sector and Segment
There are two main "slices" of a circle:
The Quadrant and Semicircle are two special types of Sector:
Half a circle is
Quarter of a circle is
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:||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)
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)