Areas and volumes

Quantities of land and earthwork are among the most frequent survey outputs, used for land acquisition, payment, planning of haul and checking of reservoirs. Areas come from coordinates, from offsets to irregular boundaries, or by planimeter or software on a map. Volumes come from cross-sections, spot heights or surface models.

For a polygon defined by coordinates, the coordinate (shoelace) method gives the area exactly. For an irregular boundary measured by offsets at regular intervals from a baseline, the trapezoidal rule treats each strip as a trapezium, while Simpson's one-third rule fits parabolas through sets of three offsets and is more accurate for curved boundaries, but needs an odd number of offsets.

Earthwork volumes for roads, canals and embankments are computed from cross-sectional areas taken at regular stations. The average end area method multiplies the mean of two end areas by the distance between them; it overestimates for shapes that taper. The prismoidal formula, which uses a middle section too, is exact for prismoids and is used for payment where specified. Spot heights on a grid give volumes for site levelling and borrow pits. Software compares two digital terrain models directly. A mass-haul diagram, built from cumulative cut and fill corrected for bulking or shrinkage, shows how far material must travel and where to balance cut against fill.