Computation of Areas & Volumes Notes

Last reviewed 16 Sept 2026 · 8 min read

Areas from field measurements

Regular figures

Triangle: $A = \frac{1}{2} bh$ , or Heron's formula $A = s (s - a) (s - b) (s - c)$ , $s = \frac{a + b + c}{2}$ , or $A = \frac{1}{2} ab sin C$ .
Trapezium: $A = \frac{( a + b )}{2} h$ .
Areas of chain surveys are computed by dividing the main framework into triangles, and the strips between survey lines and irregular boundaries by offsets.

Offsets to irregular boundaries

Offsets $O_{1}, O_{2}, \dots, O_{n}$ taken at a regular interval $d$ along a survey line:

FormulaRules for areas from offsets

Mid-ordinate rule: $A = d (h_{1} + h_{2} + \dots + h_{n - 1})$ , where $h$ = ordinates at mid-points of divisions

Average ordinate rule: $A = \frac{O _{1} + O _{2} + \dots + O _{n}}{n} \times L$ ( $L$ = length of base)

Trapezoidal rule:

A = d [\frac{O _{1} + O _{n}}{2} + O_{2} + O_{3} + \dots + O_{n - 1}]

Simpson's one-third rule (boundary assumed parabolic; odd number of ordinates, even number of divisions):

A = \frac{d}{3} [O_{1} + O_{n} + 4 (O_{2} + O_{4} + \dots) + 2 (O_{3} + O_{5} + \dots)]

Simpson's rule is more accurate for curved boundaries; if the number of ordinates is even, the last division is computed separately by the trapezoidal rule.
The trapezoidal rule assumes straight-line boundaries between offsets.

Areas from coordinates

FormulaCoordinate method (shoelace formula)

For a closed figure with vertices $(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})$ in order:

A = \frac{1}{2} i = 1 \sum n (x_{i} y_{i + 1} - x_{i + 1} y_{i})

(with $x_{n + 1} = x_{1}$ , $y_{n + 1} = y_{1}$ ), or equivalently $A = \frac{1}{2} ∣ \sum x_{i} (y_{i + 1} - y_{i - 1}) ∣$ .

Areas from latitudes and departures

Double meridian distance (DMD) method:

Meridian distance of a line — perpendicular distance of its mid-point from the reference meridian.
DMD of the first line = its departure.
DMD of any line = DMD of the preceding line + departure of the preceding line + departure of the line itself.
DMD of the last line = its departure with opposite sign (check).

2 A = \sum (DMD \times latitude) \Rightarrow A = \frac{1}{2} \sum D M D \times L

The double parallel distance (DPD) method uses departures and latitudes interchanged.

Areas from plans

Counting squares — a transparent sheet of squares placed over the plan; full squares plus estimated fractions.
Division into triangles, trapezoids or strips measured on the plan.
Computational/digital — digitising the boundary in CAD/GIS software.
Planimeter — mechanical or digital instrument measuring area by tracing the boundary.

Amsler polar planimeter

FormulaPlanimeter

A = M (F - I \pm 10 N + C)

$F$ = final reading, $I$ = initial reading (each reading has 4 digits from the wheel, drum and vernier)
$N$ = number of complete revolutions of the dial (+ if the zero mark passes the index clockwise, − if anticlockwise)
$M$ = multiplying constant (area per revolution of the wheel) — set by the tracing arm length
$C$ = constant added only when the anchor point is inside the figure; $10 M C$ = area of the zero circle (circle traced without rotation of the wheel)

The tracing point is moved clockwise around the figure. A digital planimeter displays area directly.

Volumes

Cross-sections for earthwork

For roads, railways and canals, cross-sections are taken at intervals $d$ along the centre line; formation width $b$ , side slopes $n$ horizontal : 1 vertical.

FormulaAreas of cross-sections

Level section (ground level across): $A = (b + nh) h$

Two-level section (ground with transverse slope 1 in $m$ , depth $h$ at centre):

Side widths: $w_{1} = (\frac{b}{2} + nh) \frac{m}{m - n}$ , $w_{2} = (\frac{b}{2} + nh) \frac{m}{m + n}$
Area: $A = \frac{1}{n} \frac{( \frac{b}{2} + nh ) ^{2} m ^{2}}{m ^{2} - n ^{2}} - \frac{b ^{2}}{4 n}$

Three-level section (depths $h_{1}$ , $h$ , $h_{2}$ at left edge, centre and right edge of formation; side widths $w_{1}$ , $w_{2}$ ):

A = \frac{h}{2} (w_{1} + w_{2}) + \frac{b}{4} (h_{1} + h_{2})

Multi-level (irregular) sections: by the coordinate method.

Volume from cross-sections

FormulaEarthwork volume formulas

Trapezoidal (average end area) formula:

V = d [\frac{A _{1} + A _{n}}{2} + A_{2} + A_{3} + \dots + A_{n - 1}]

Prismoidal formula (odd number of sections):

V = \frac{d}{3} [A_{1} + A_{n} + 4 (A_{2} + A_{4} + \dots) + 2 (A_{3} + A_{5} + \dots)]

Prismoidal correction (difference between trapezoidal and prismoidal volumes) for level sections between two cross-sections with depths $h_{1}$ and $h_{2}$ :

C_{p} = \frac{d n}{6} (h_{1} - h_{2})^{2}

(subtracted from the trapezoidal volume.)

The trapezoidal formula generally overestimates volumes; the prismoidal formula is more accurate (exact for prismoids).
Curvature correction (Pappus' theorem) applies when the centre line is curved and sections are unsymmetrical.

Volume from spot levels (borrow pits)

The area is divided into squares (or rectangles) or triangles of equal area; depths of cut/fill are found at corners.

FormulaSpot level (unit area) method

Rectangles (each of area $A$ ):

V = \frac{A}{4} (\sum h_{1} + 2 \sum h_{2} + 3 \sum h_{3} + 4 \sum h_{4})

$h_{1}$ = depths at corners common to one rectangle; $h_{2}$ — common to two; $h_{3}$ — three; $h_{4}$ — four.

Triangles (each of area $A$ ):

V = \frac{A}{3} \sum n h

( $n$ = number of triangles sharing that corner.)

Volume from contours

Volumes of reservoirs, hills and large excavations are computed from areas enclosed by contours using the trapezoidal or prismoidal formula with the contour interval as $d$ (see Contouring).

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Computation of Areas & Volumes