We want the area of the SEGMENT, which is the region
inside the red boundary.

The SECTOR consists of both the triangle and the SECTOR.
The triangle is equilateral because all its sides are 8.
Therefore the angle is 60° or  radians.

The area of the whole SECTOR is given by the formula:

 and ,



There is a formula for the area of the equilateral triangle 
in terms of its sides which all equal s.  It is



Since the sides are 8, s=8, the area of the equilateral
triangle is 







Edwin

The area of your segment is  of the area of the circle with the radius 8 MINUS the area of the equilateral triangle with the side length 8, i.e.

the area of the segment =  -  =  -  =  -  = 5.78 square inches (approximately).