Grain Cone Volume

A cone holds exactly one third of the cylinder that encloses it, and the reason this page exists is that almost nobody guesses that correctly. Asked to estimate the peak above a level fill, most people reach for something near a half, which overstates the grain by fifty percent. The one-third result is ancient — Archimedes proved the cone-to-cylinder ratio and thought it worth recording — and it applies to any cone regardless of how steep it is.

The peak's height is not free to choose: grain will not hold a slope steeper than its angle of repose. That angle is a property of the seed, running about 16° for whole soybeans, 21° for wheat, and 23–27° for corn and canola, with dockage and damp grain raising it a little and smooth round seed lowering it. So the cone above a bin is roughly $h = (d/2)\tan\theta$, which for a 9.1 m (30 ft) bin of wheat is a peak of about 1.75 m and something over 300 bushels of grain — a quantity large enough that leaving it out of an inventory is a real error, and small enough that people do.

The same equation covers a ground pile, where the diameter is measured at the base and the peak height at the centre. Ground piles depart from the ideal cone more than bin peaks do, because they are usually built by a conveyor that moves, so measuring a mean diameter across two directions is worth the extra minute. A pile with a flattened top is not a cone at all but a frustum, and treating it as one overstates the volume.

What the cone means for storage matters more than its volume. Filling a bin from a central spout sorts the grain: heavy sound kernels roll to the outside of the peak, while fines, chaff and broken kernels concentrate in a core directly under the spout. That core is denser than the surrounding grain, so aeration air flows around it rather than through it, and it is where spoilage begins in almost every bin that spoils. Levelling the peak after filling, or coring the bin by drawing a few tonnes out of the centre, redistributes that column and is among the cheapest storage insurance available.