I've had interesting discussions over the years about the old question, "Are there more acres in a hilly field than in a flat field?" At one point I did a lot of research in hope of doing a Farm Journal story that would answer the question. I found experts at Iowa State University, along with some geo-mapping guys with the federal goverment, and they pretty much confirmed that, in theory, a hilly field has more acres than a flat field.
Their proof involved a LOT of mathematics, but their explanation was simple enough for me to understand: Imagine a 4-foot by 4-foot blanket spread on a flat floor. Draw a chalkline around the perimeter of the blanket. Place a basketball or large object under the center of the blanket, and the edge of the blanket no longer touches the chalked outline.
So, their conclusion was that, back when the government calculated acres by running a gizmo over printed aerial maps that measured only the borders of the fields in two dimensions, there was no way to allow for additional acres created by the third dimension of altitude.
They all agreed that it would take some big hills with steep sides and long slopes to add significant acres to a field. But the differences were mathematically calculable and they were confident that acres calculated from 2-dimensional aerial maps were at least slightly "off" to the actual surface acres in 3-D.
Things got complicated when they took into consideration that the original survey of land in the United States was done over hill and dale using mechanical measuring devices, which SHOULD have accounted for the third dimension. I'm not a surveyor or civil engineer--perhaps there's a way to adjust for those variations.
But that brings us to why I brought this topic to an "In The Shop" blog: perhaps this helps explain why some yield monitors, acre counters and GPS-guided steering systems come up with "odd" acres compared to the field acres fathers and grandfathers have used for years. I've been involved with discussions by experts in precision farming/GPS mapping who believed the differences were due to machine-based mapping systems following the exact terrain, compared to aerial maps in two dimensions that couldn't account for subtle increases in acreage in hilly fields.
I'll leave the arguing about precise acreage in hilly fields to people smarter than me. What I can confirm is that there is more corn in a crooked row than in a straight row. Maybe only a dozen or more kernels in a quarter mile row, but enough so I always used "increased yields" as justification for never being able to plant a perfectly straight row.