What is the formulae for calculating he length of a side of an octogon 50" in diameter?

I won't use formulae, but only common logic.

The diameter is 50", so the radius is 25" right?

That is the length from the centre to the vertex of one side. The angle between two closest radius of these is 45 degrees.

So, using the cosine rule, we can find the length of the side, taking a triangle from the octagon, as having two sides of 25" and an angle of 45 degrees between them.

I get 19".

That is of course assuming it is a regular octagon.

Of course, if not, there is not enough information given.