SOLUTION: find the area of the largest square that can be cut from a sector of a circle radius 8 cm and a central 120 degree

Let the square be ABCD.

Side of the square = AB = PQ = 2OQ.

Since triangle OBQ is a 30°,60°,90° triangle, its longer
leg is  times its shorter leg.

We let its shorter leg BQ be x, then its longer leg OQ
is .

Since  = OQ = PQ/2 = AB/2,

BC = 2∙OQ = .

OC = radius = 8

Using Pythagorean theorem on right triangle OQC

OQ² + QC² = OC²



Solve that and get 



So one side of the square is twice that, or 2x, which is

 = side of square.

Therefore the area of the square is



or 



Edwin