Ok, I trusted today's version of Merlin and didn't check its math.

A straight calculation of 200 unit sphere gets a radius of 3.6 but extrapolating to 200 hundred billion results in a radius of 3,628 light years. Apologies to my geometry teacher for forgetting what I was taught.

Note that my original statement misquoted Otto Harkamen. He did not say the Federation occupied a sphere of 200 hundred billion cubic light years, but a space of 200 billion cubic light years.  

Now calculations for the Milky Way's volume do not use a sphere calculation but can be "approximated by a disk with a thickness of 1000 light years and a radius of 50,000 light years." This results in a volume of 7.9 trillion light years.

Assuming Piper was using number like this, the Federation could have occupied about a 1/4 of the Milky Way, so need for extra-galactic travel.

Tim