Cut and fill is earthwork’s ledger: excavation (cut) on one side, embankment (fill) on the other, tallied along an alignment from surveyed cross-sections. Each station contributes a chainage and its cut and fill areas; integrating the areas between stations gives segment volumes, and the running sum nets out to the answer the estimate hinges on — surplus material to haul off, or a deficit that must come from borrow.
Two integration rules serve different section behavior. The average-end-area method takes each segment as length × the mean of its two end areas; the prismoidal method applies Simpson’s rule across equally spaced station triples and is more accurate when the cross-section varies non-linearly. Three stations at 0/20/40 m with cut areas 10, 12, and 14 m² show them agreeing — 220 + 260 = 480 m³ by end areas, and 40 × (10 + 4×12 + 14)/6 = 480 m³ prismoidal — precisely because this section tapers linearly.
The balance is honest only after a unit correction: fill is measured compacted while cut is measured in place, so the required fill is converted back to bank volume through a compaction (shrinkage) factor, V_bank = V_fill / factor, with typical soil values of 0.80–0.95. The net is then V_cut − V_fill/factor — comparing both sides of the ledger in the same in-situ cubic metres.