Question

if area of sector of a circle is equal to corresponding segment of that sector show that the corresponding arc is a semicircle

Answer 1

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.

If you know the central angle
Area = π r 2

C
360


where:
C is the central angle in degrees
r is the radius of the circle of which the sector is part.
π is Pi, approximately 3.142

This is the method used in the animation above.
Calculator
If you know the arc length
Area =
R L
2

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.