Answer :

The figure is shown below:



Let the side of this hexagon be “x”.


Then, in this hexagon, a total of 6 equilateral triangles can be made, which are:


ΔOED, ΔODC, ΔOCB, ΔOBA, ΔOAF, ΔOFE.


Now, in the figure given below:



In ΔOPF and ΔOPB,


POF = POB (angle in equilateral triangle = 600)


OF = OB (sides of equilateral triangle)


OP is common in both the triangles.


Hence, we can say that ΔOPF ΔOPB



Similarly, we can prove that ΔAPF Δ APB


AP = PO =


Now in, right ΔOPF,


OP2 + PF2 = OF2






b = √3x


Now,


Area of hexagon with side “x” =


Area of big triangle (green shaded) =





Probability of dot is putted in the green part




= 0.083


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