Triangle Area from Two Vectors

The diagonal of a parallelogram cuts it into two congruent triangles, so a triangle built on two vectors has exactly half the cross-product area: ½|a_xb_y − a_yb_x|. Legs of a = (4, 0) m and b = (0, 3) m give ½ × 12 = 6 m², matching the familiar half-base-times-height. But the formula does not care about right angles: a = (6, 2) m and b = (1, 5) m give ½|30 − 2| = 14 m² with no height ever measured.

Written for three arbitrary corner points rather than two vectors, this becomes the shoelace formula — so named because the cross-multiplied terms lace together like a boot — published by A. L. F. Meister in 1769 and later generalized by Gauss to polygons of any number of sides. Surveyors computed parcel areas with it for two centuries before computers, and GIS software still does. Because the absolute value discards the sign, the area alone cannot be run backwards to recover a component; that sign, however, is what lets the polygon version cancel overlapping strips automatically and return the correct area for any simple polygon, convex or not.