What is the relationship between a shotgun's barrel diameter and the gauge of the shotgun?

In order of biggest to smallest - 8, 10, 12, 20 bore, then .410 as the smallest. The 410 is .410 inches. Do not know the mathematical relationship between gauge and barrel diameter.

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The relationship is INVERSE- the smaller the number, the bigger the bore. The gauge of a gun barrel is the number of lead balls that fit the barrel that weigh 1 pound. So 12 lead balls that fit a 12 g weigh one pound, 20 lead balls that fit a 20 g weigh 1 pound, etc. .410 is actually a caliber, not a gauge- if converted to gauge would be about 69 gauge.