Of course the quad will only be directional on one
band - or it might point different ways on 20 and 17m,
depending on the tuning of the parasitic elements.
For multi-band use (at least over a 2 : 1 range or so)
a single wire loop should work, and it can be fed with
either 300 or 450 ohm line. For multi-band gain I'd
try something like a lazy-H or some sort of curtain.
But, to answer your question, there are "standard"
formulas for quad dimensions, and they may work to
some extent, but design is really a lot more complicated.
I recommend the quad design series at www.cebik.com
(for example, "new quad studies".) This includes a
set of basic programs to calculate optimized dimensions
based on the desired frequency and wire size.