Earthwork volumes
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Estimating dirt: average end area and the prismoidal correction, stockpile volume, and the swell and shrinkage between bank, loose and compacted.
Earthwork Volume by Average End Area
Volume between two cross sections, averaging their end areas over the distance between them; the answer is reported in cubic yards.
Earthwork Volume by the Prismoidal Formula
Simpson's rule applied to earthwork, weighting the middle cross section four times the ends; the answer is reported in cubic yards.
Stockpile Volume (Truncated Pyramid)
Volume of a flat-topped stockpile or borrow pit from its base area, top area and height; the answer is reported in cubic yards.
Swell: Loose Volume from Bank Volume
Converts undisturbed bank volume into the loose volume the same soil occupies once excavated, using its percent swell.
Shrinkage: Compacted Volume from Bank Volume
Converts bank volume into the compacted volume it fills in an engineered embankment, using the soil's percent shrinkage.
How they fit together
Volume between two cross-sections is normally the average of their areas times the distance between them. That is exact only where the section changes linearly; where it does not, the prismoidal formula weights the midpoint area four times and gets closer. Average end area consistently overestimates on a curved transition, which is part of why it survives — the contractor is rarely the one who complains.
The expensive mistake is quoting one volume for three different states of the same soil. A cubic metre in the ground becomes roughly 1.25 loose in the truck and about 0.9 once compacted into the embankment. Bid bank volume as truck loads and you undercount the haul by a quarter; place loose volume as compacted fill and you come up short.