So I recently re-watched First Contact. I noticed Picard telling Lily Sloane that the Federation is 8,000 light years across, however in Voyager it's estimated a trip of 70,000 ly would take about 75 years at maximum warp to complete. So doesn't that mean if one wants to travel across the Federation it would take them ~8,5 to cover the distance. Is the Federation really that big or am I missing something? If the Federation really is that big how can it be successfully managed and defended?
Welll it all depends on which warp speed figure you use.
If using VOY about 8 years
Doing the Maths from TMP 4 days to travel 16 ly, it would be in the region of 4 years
If using TOS figures about 8 days
In TNG to traverse the intergaltic void some 2.7million ly, about 300 years.
Plain and simple the writers of VOY ignored any previous established figures, in order to make the jounrey seem longer than it should have been.