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Well, if you are going to have 6 chickens you prolly ought to have a MINIMUM run size of at least 6x10 or 4x15. Six feet is on the borderline of too wide to have convenient hinged-top access; four feet is fairly well within the realm of having it be reasonable that way. But, if you are going to go with the minimum-size run, then I'd say you kind of have your choice.
If OTOH you want to make your run any bigger than that -- to give them more-ample space, to reduce future sanitation problems, or to allow room for more possible chickens in the future -- then it will almost right away get too big to be really feasible for hinged-top access, and you'd almost HAVE to go to walk-in height.
Note that hinged-top designs (where the run is only 3' tall or so) really don't save you THAT much money. They force you to put a good strong wire top on and it really is safer if it's also fairly small mesh (whereas walk-in size runs can be topless, or net-topped, or 2x4-wire-topped), and all they save you is a *little* money on shorter posts plus the cost of 3' x total-length-of-run-fence fencing material. THey also make it quite a lot harder to add sand or gravel to the run if you ever want to someday.
I'm not against them, as such, and I think that in some circumstances they make very good sense. I'm just trying to outline what I see as the sometimes-not-obvious pros and cons.
Note that there are really THREE (not just two) types of run options -- rectangular wood-framed short (3-4' high) run with hinged top or side panels to allow access, or rectangular wood-framed walk-in height run, or hoop-style walk-in height run (cattle panels are more durable but pvc-with-wire-over-it is also an option)
Good luck, have fun,
Pat