This is a REGULAR OCTAGON so all sides and all angles are congruent.

The distance between 2 parallel sides of an octagon is twice the measure of its apothem.

The APOTHEM is the ALTITUDE drawn from the APEX of the octagon to each base of the 8 isosceles triangles in the octagon.

The APOTHEM is also opposite the 60o angle of any of the 2 right triangles, formed when the altitude is drawn, and represents the longer leg too

The length of the APOTHEM/ALTITUDE/LONGER LEG (opposite the 60o angle) is the length of the SHORTER Leg * 

Since the length of one of the sides is y, the shorter leg (SL) of any one of the 2 right triangles, in this case, is ,

so the length of the APOTHEM is: . 

Now, with the distance being twice the length of the APOTHEM, we get:  

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