Your best bet is to get an atlas (google search for "cia world fact book"
That will tell you the area in miles of all these countries. Then add up the areas from as many countries are in the empire. If there's half a country, only add half. Probably no need to add luxembourg, malta, et al.
After all, the difference between 2 000 000 and 2 500 000 mi^2 is pretty substantial
Or, if you're feeling really snazzy
I think the world's circumference at the equator is 25000 miles. I don't know the surface area equation any more, but just solve for area (or google search for world surface area [4/3 pi r ^3 is volume?) and get your handy dandy map, and figure out what pct of the world it covers. Then again, that's needlessly complex.