What should be the definition of a sector of a circle? For discussion in the community:
We know the difference between circle and disk.

*

*A circle is the set of points equidistant from a given point in the plane e.g. the unit circle is $C= \lbrace (x , y) \in  \mathbb{R}^2 : x^2 + y^2 =1 \rbrace $. A circle has length, called perimeter, but has no area. When we say area of a circle we mean, by convention, the area of the region enclosed by the circle.


*The unit disk $D =  \lbrace (x , y) ∈  \mathbb{R}^2  : x^2 + y^2 ≤ 1 \rbrace$ is the union of the unit circle and its interior. It has area but no length.
In many text books a sector of a circle is defined as follows:

Definition 1. A sector of a circle is the area enclosed by an arc and the two radii joining the centre to the end points of the arc.

My question is:

Should we define, as given above or the definition should follow the criteria as in case of a circle?

In my view the definition of a sector of a circle should be as follows:

Definition 2.  A sector of a circle is the union of an arc and the two radii joining the end points of the arc to the centre.

So a sector has length but no area. By area of a sector we would mean, by convention, the area of the region bounded by the sector.
The advantages of Definition 2 are many; e.g. the definition follows the same pattern as in definitions of circle, rectangle or any other two dimensional closed figures which possess only length. Areas of these figures, by convention, refer to the areas of the regions enclosed by them.
(This discussion is also applicable to definition of a segment of a circle.)
 A: Definitions of a shape by a set of its area verses a definition of a shape by a set of it's perimeter for shapes such as these serve no real advantage over one another.
In conversation, referring to a "circle", without specification, one is simultaneously referring to both a disk and a ring.
For circles, segments, rectangles, etc, if we know the set of points for the perimeter, then we can deduce the set of points of it's interior area or vice versa. In general, from the set of points along a closed one-dimensional curve that does not cross over itself, one can deduce the set of it's interior area and vice versa.
However, if the curve does cross over itself, then there are scenarios where one set can't be deduced without the other.
Take, for instance, the graph of $r=\frac{1}{2}-\cos(\theta)$

By it's area, are we referring to the area within both the bigger and outer loops, or just the outer without the inner? Obviously, it depends on how you specify area. Luckily, once you have specified which, then whatever the case may be, that set can be deduced.
However, if we first start with a set only of area, e.g;

Then in this case, it's perimeter can be deduced, but the original equation we started with is unknown as the smaller loop has now "vanished". I.e; there are an infinite number of curves that have this interior area.
So in general, starting with a set of points that has length but no area is more powerful than starting with just the set of area, because the latter can be deduced from the former.
However, there are many more complicated shapes that need to be first defined by area. Take the Mandelbrot Set.
Semantically battling over whether a circular-sector/segment should be defined by an area or perimeter is pointless, because both definitions can be deduced from each other.
