Exact area = 12pi - 9*sqrt(3) square inches
Approximate area = 22.1106545749577 square inches
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Work Shown:
Assuming the chord is horizontal along with the diameter, then this means the two smallest minor arcs are both 30 degrees each. This leaves the northern most arc to be 180-30-30 = 120 degrees in measure
A = area of northern sector (you can think of it as a pizza slize)
A = (angle/360)*pi*r^2
A = (120/360)*pi*6^2
A = 12pi
B = area of triangle
B = (1/2)*side1*side2*sin(included angle)
B = (1/2)*6*6*sin(120)
B = 18*sqrt(3)/2
B = 9*sqrt(3)
C = area of shaded region
C = A - B
C = 12pi - 9*sqrt(3) .... exact area
C = 22.1106545749577 ... use a calculator to get the approximate area