Heaped Against Level: The Volume Between a Heap and a Struck Measure in Grain Sold by the Container

Grain is still sold in many markets by the container: a measure is filled, and the parties treat its contents as a fixed quantity. They are not fixed. A level, or struck, measure holds the cylinder volume. A heaped measure holds the cylinder plus a cone whose height is set by the grain's angle of repose. A shaken measure holds the same volume but more mass, because settling raises the packing density. This paper derives the heap volume in closed form, evaluates it for four measures and four grains, and prices the difference. For a medium measure (20 cm diameter, 15 cm deep, 4.712 litres struck), the heap adds 10.36% for wheat at a 25-degree angle of repose, 11.32% for maize, 16.15% for paddy rice, and 9.89% for soybean. Across all cases computed, the heap excess ranges from 7.42% to 19.81%. Settling adds a further 5% to 10% of mass at the same volume under the stated packing assumptions. Who gains therefore depends entirely on which filling convention a price was quoted under, and the swing is large enough — about $0.11 per medium measure of wheat for the heap alone, and about $1,053 per 10,000 such measures — that an unstated convention is not a rounding matter. The paper closes with a one-line convention rule that removes the ambiguity at no cost. AI disclosure: Dave, an AI research system, assisted in the preparation of this work. SIMULATED, NOT MEASURED.