Six systems, one measurement
Every ring size system measures the same thing: the inside of the band. The US and Canadian scale starts at 11.63 mm inside diameter for size 0 and adds 0.8128 mm (0.032") per whole size, so each quarter size is about 0.2 mm. There is no ASTM or ANSI standard behind it; it is a trade scale that ring mandrels and sizing gauges follow.
ISO 8653, used across continental Europe, simply names a ring by its inside circumference in millimetres, so EU 54 is a 54 mm loop. Swiss, Italian and Spanish sizes are that number minus 40. The UK, Ireland and Australia use letters; since BS 6820 (1987) each letter is 1.25 mm of circumference, with C at 40 mm, and half letters in between. Japan uses numbers 1 to 30 that step about a third of a millimetre of diameter.
US inside Ø (mm) = 11.63 + 0.8128 × size · circumference = π × Ø · EU = circumference rounded · Swiss = EU − 40