Cut and Fill Calculator: End Area vs Prismoidal

Cross-section areas from levels

For horizontal natural ground and a horizontal formation, let signed depth be $d=z_g-z_d$. The separate cut and fill depths are $h_c=\max(d,0)$ and $h_f=\max(-d,0)$. Each trapezoidal area is $A=b h+m h^2$, where $b$ is the full formation width and $m$ is the horizontal/vertical side slope. This is the Iowa DOT 5A level-ground geometry, using full width $b$ rather than its half-width notation.

Average end area and prismoidal integration

For each surveyed interval of length $\Delta x$, $V_{EA}=\Delta x(A_i+A_{i+1})/2$ (FHWA Earthwork Design). Sum every interval, separately for cut and fill. For a panel spanning two intervals, $L=2\Delta x$ and $V_P=L(A_0+4A_m+A_2)/6$ (Deakin & Hunter, 2009, equations 1–2 and Table 1). The middle area is an independently supplied or level-derived section, not the mean of the end areas.

The signed correction reported here is $C=\sum V_{EA}-\sum V_P$. For one panel using all three surveyed sections in the AEA total, $C=\Delta x(A_0-2A_m+A_2)/6$. If AEA instead omits the surveyed midpoint and uses only the panel ends, its correction is $L(A_0-2A_m+A_2)/3$. These conventions differ; this tool consistently compares methods using all available survey data. Subtract the displayed correction from AEA to obtain the displayed prismoidal volume. It can have either sign.

Four-point pad grid

Follow OTELP, Manual on Land and Water Management, printed p.86 (PDF p.86). In each cell, $H_c$ is the sum of positive corner depths and $H_f$ is the sum of absolute negative corner depths. For rectangular cell area $A_g$, $V_c=A_g H_c^2/[4(H_c+H_f)]$ and $V_f=A_g H_f^2/[4(H_c+H_f)]$. Return zero when all depths are zero. Replacing square spacing $L^2$ by rectangular cell area is the geometric extension used here. With one sign only this becomes area times mean corner depth; with mixed signs it is an approximate partition. Cells are numbered southwest, southeast, northwest, northeast.

Material states and mass haul

Let $k=V_{compacted}/V_{bank}$ and $s=V_{loose}/V_{bank}$. Bank demand is $V_f/k$, and net bank balance is $V_c-V_f/k$. Borrow and surplus are the negative and positive parts of that balance. Loose cut is $sV_c$; loose off-site movement is $s|V_c-V_f/k|$, assuming borrow has the same factors. The route sketch accumulates cut minus bank-equivalent fill at panel boundaries for both integration methods. Positive slope means net material production; negative slope means net consumption. It is a coarse diagnostic, not a contract mass-haul plan (FHWA CFLHD Earthwork Representation, printed p.28).

The numeric balance chart uses section index S on the route: physical station is S times the entered survey spacing. Grid mode uses cell boundary order, not distance or a physical haul path. The chart uses the selected volume method and the viewer converts its bank volume axis between metric and US units.

In measured-area mode, transition detection uses the entered cut and fill vectors. It conservatively flags a panel when both cut and fill occur and either area's presence changes across its sections. Persistent cut and fill at every section are not by themselves a longitudinal transition. Area data alone cannot locate the zero-depth point; survey and split the reach there.

The pad plan classifies each whole cell from its four corner depths: amber C is cut only, blue F is fill only, purple M is mixed cut and fill, and gray Z means all depths are zero. A zero signed mean does not make a mixed cell zero. These categories do not locate the cut/fill boundary within a mixed cell.

All sections are perpendicular to a straight alignment at equal station spacing. One prismoidal panel requires three sections; two require five. Natural ground is horizontal across each level-based section, the formation is horizontal across its width, and design grade is linear longitudinally. Side slopes are constant separately in cut and fill. Use measured areas for side-hill or irregular sections.

The pad is a horizontal rectangular footprint covered by exactly two by two cells. Spot levels use a common datum. The nine levels and two spacings define its survey resolution; do not extrapolate beyond the footprint.

All excavation is suitable for reuse. Factors are user-entered project assumptions, not material recommendations; defaults 0.90 compacted/bank and 1.25 loose/bank illustrate shrink and swell. No topsoil, waste fraction, unsuitable material, stripping, overbreak, foundation settlement, or stockpile loss is deducted. Borrow has the same material factors as on-site cut. Volume changes represent density changes; these quantities are not masses.

Use for preliminary quantities and method comparisons on short reaches and small rectangular pads. Prismoidal integration is exact for a prismoid and for area varying as a polynomial up to degree three; arbitrary surveyed terrain is only approximated. A cut/fill change inside a panel creates a break in the separate area functions: place the zero-depth section at a panel boundary or evaluate shorter reaches. Midpoint survey data is required; the formulas cannot recover missing terrain.

Mixed grid cells use the OTELP four-point approximation, which conserves signed volume but does not reconstruct the actual zero contour. Refine the grid for broken terrain or quantity takeoff. The route mass-haul sketch has only panel-boundary ordinates and cannot show intermediate peaks, haul distances, free haul, overhaul, or sequencing. It does not meet the detail required for a contract mass-haul plan. The pad ledger cell order is bookkeeping, not a haul route.

The calculator follows geometric methods rather than a jurisdiction-specific payment rule. Check the applicable project specifications and survey/measurement requirements. It provides no slope stability, shoring, drainage, or excavation safety design. Unit fields are converted by the viewer; all input elevations must share a datum.