a line segment has no area. By definition a line segment is a part of a line, thus it has no area
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The segment is called a secant.The area bounded by a secant and its arc is a sector.
A segment of a circle is an area enclosed by a chord and an arc.
A slice, a segment or a sector.----------------A segment is the area of a circle between the chord and the arc. A "sector" is what you are looking for. See the links below.
The term bisector means that bi is the two of something (a cycle with two wheels is a bicycle, for instance). Therefore a bisector will split a segment, area, angle, into two equal parts.