Definition: The number of square units it takes to exactly fill a sector of a circle.

Try this Drag one of the orange dots that define the endpoints of the sector.
The sector area is recalculated as you drag.

What the formulae are doing is taking the area of the whole circle, and then taking a fraction of that depending on what fraction of the circle the sector fills.
So for example, if the central angle was 90°, then the sector would have an area equal to one quarter of the whole circle.

where: L is the arc length.
R is the radius of the circle of which the sector is part.

Sector area is proportional to arc length

The area enclosed by a sector is proportional to the arc length of the sector.
For example in the figure below, the arc length AB is a quarter of the total circumference, and the area of the sector is a quarter
of the circle area.
Similarly below, the arc length is half the circumference, and the area id half the total circle.
You can experiment with other proportions in the applet at the top of the page.