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How does ArcView GIS calculate area?
The area of polygons is calculated by summing up all trapezoidal areas
corresponding to individual line-segments of a polygon boundary. Individual trapezoids are formed from start and end points of a segment and their projections on X-axis. The trapezoid area is then calculated as:
0.5 * (X_end - X_start) * (Y_end + Y_start).
(For correctly oriented polygons, the total sum of those is always positive as is the resulting area.)