Each side of a regular octagon is y cm long find the distance in cm between any two parallel sides of this octagon

We  can fing this using some trig

We have a right triangle  with one leg  = y/2

And the angle   opposite this side = 22.5°

And   1/2 of the distance between the parallel sides is the other leg....and the angle opposite this side   is  67.5°

So....using the Law of Sines....we can find half of the distance as

(y/2)sin (67.5)  / sin (22.5)  =  (y/2) ( 1 + √2)  =  [ ( 1 + √2) / 2 ]   y

So...twice the distance is   (1 + √2) y  = distance between the parallel sides