Refer to the diagram shown below.
The shaded area is the segment created by a 120°-arc and a chord of. 8√3.
To find the radius, consider the right triangle shown below the circle.
By definition,
cos 30° = (4√3)/r
Therefore
r cos 30° = 4√3
r (√3/2) = 4√3
r = 8
Because the central angle of the segment is 120,° its area is
(120/360)*π*(8²) = (64π)/3 = 67.02 units²
Answer: [tex] \frac{64 \pi }{3} \,\, or \,\, 67.02[/tex]
