Question

given the diagonal of a square, how do i determine its perimeter?

Answer 1

Divide the diagonal by sqrt 2. That will be the side. Multiply that by 4. That will be the perimeter.

Answer 2

If you like direct formulas, then you can do as Mertsj said. If you want to derive the result yourself, then you can use the pythagorean theorem to calculate the side length \(x\) given by \(\large x^2+x^2=d^2 \implies x=\frac{d}{\sqrt{2}}\), where \(d\) is the diagonal.
Since the perimeter of a square is the sum of its sides (which are 4 sides). The perimeter equals to \(4x\).