Euphonium mouthpieces come in 3 sizes. There is the small, tenor-trombone-size shank, which fits a Yamaha 321 and most of the old American baritone/euphonium instruments. The middle size, which is called the Euro shank, fits the old Bessons (pre 1974) and most Willsons. And the large, bass-trombone-size shank fits most modern euphoniums.
Speaking a little roughly, if you measure the small end of the mouthpiece, you'll find this:
Small, tenor trombone-size shank: 7/16" or 11mm
Large, bass trombone-size shank: 1/2" or 12mm
The medium (European) shank measures about halfway between the other 2, or 15/32" or 11.5mm
I checked with a AAA battery, and the diameter is very close to the end of the small shank mouthpiece. The AAA battery ALMOST fits inside the end of the large shank.