...I guess the interesting question is, does the UFP lie predominantly in one "hemisphere" or another, or is it split more or less evenly between Alpha and Beta?
That is, does Earth, somewhat reasonably even if not completely canonically established as being at the quadrant border, lie in the middle of those 8,000 ly, or at one end?
There are some onscreen hints that Klingons and Romulans would be in the Beta direction very close to Earth and big enough to block the expansion of the UFP in that direction. Then again, e.g. the star Canopus/Alpha Carinae, some 200 ly into Beta, is mentioned as being UFP territory (wargames in "Ultimate Computer"); that's already something if the UFP is "mere" eight thousand ly across in total.
On a random note, Alpha Carinae is unambiguous for those of us who think the Trek galaxy is identical to our own save for the existence of so many Class M worlds and spatial anomalies and whatnot. Deneb is not, as the sky is full of stars whose name featured this word (meaning "tail"), some of them close to Earth, some far away. Picard may have been far away from Earth in the TNG pilot, then, or relatively close, and the "great unknown" may have remained so because of distance or because of political or astrographical obstacles. Whatever the explanation, in general we can probably rest assured that the UFP is not a sphere - but rather a weird amoeba shape, possibly with detached "islands" of territory as well.