Since most ML barrels are Octagon in shape, that leaves 4 pairs of parallel " flats" around the barrel. " Across the flats" is the measurement of the width of the barrel, from one parallel flat to its opposite number. When it comes to inletting a barrel into the stock, this is the sizing measurement that is most important to a builder, as it tells you how much wood you have to remove from the stock to allow the barrel to seatdown in the stock's barrel mortise.
Barrel measurement also becomes a safety issue when determining how much steel you want around a bore of a stated dimension. Too little, and you risk blowing the barrel up. There can be too much, if the barrel makes the gun weigh so much, or shift the balance so far forward that the gun becomes difficult to hold to shoot.
Then, you will see " swamped " barrels, where the width of the barrel across the flats narrows towards the middle of the length of the barrel, and then increases as you go toward the muzzle. Swamping is, and was, done to relieve some weight, and to improve balance of longer barreled rifles. A well designed and made swamped-barreled rifle is a joy to hold and shoot.
You want enough metal around the breech of the barrel so that you can contain the higher breech pressures when the gun is fired. As the ball moves down the barrel, the need for the same amount of metal lessens, as the pressure drops as the distance from the breechplug, to the ball increases.
You will find tapered barrels, in addition to straight, and Swamped barrels being used. And of course, there are half octagon, half round barrels out there as well. Finally, you will find barrels that are octagon at the breech, but are tapered to round a foot or so down the barrel, and the tapering continues to the muzzle.