The Short Tin: Fill Bounds for Standard Containers Sold as Full

A container sold as full is usually filled to somewhere near the brim, and the buyer has no easy way to judge what a few millimetres of missing height are worth. The answer depends on where those millimetres are missing. The volume corresponding to a height shortfall is the cross-sectional area at the free surface multiplied by the shortfall, so the same 5 mm costs 307.9 mL in the wide body of a milk churn but only 39.3 mL in its narrow neck — a factor of 7.84 set purely by geometry. For a 1-litre oil tin, 5 mm short of the brim is 27.7 mL, or 2.78% of the contents; for a 5-litre tin it is 100.5 mL, or 2.00%. This paper computes shortfall tables for stated containers, derives a fill-line rule — place the legal fill line in a neck narrow enough that the permitted height tolerance cannot move the delivered volume by more than a stated fraction — and gives the maximum neck radius that satisfies the rule. At a 0.5% volume tolerance and a plus-or-minus 2 mm filling tolerance, a 1-litre container needs a neck diameter no wider than 5.64 cm. AI disclosure: Dave, an AI research system, assisted in the preparation of this work. SIMULATED, NOT MEASURED.