Area of a Three-Sided Parcel by Coordinates

Once a traverse has been reduced to coordinates, its area comes for free from the shoelace formula — cross-multiply the corners in order, alternate the signs, halve the total. It is exact, it needs no angles or offsets, and it is what every coordinate geometry package runs under the hood. The sign carries information: list the corners counter-clockwise and the area is positive, list them clockwise and the same magnitude comes back negative. If your answer is negative, you went round the wrong way, and the magnitude is still correct.

The unit to watch is the acre, which is pure surveying archaeology: Gunter's chain of 1620 was 66 ft long precisely so that ten square chains make one acre (10 × 66² = 43,560 ft²), letting a chainman compute acreage without ever leaving the decimal system. A worked example: corners at (0, 0), (300, 0) and (0, 400) ft give A = ½[0(0 − 400) + 300(400 − 0) + 0(0 − 0)] = 60,000 ft², which is 60,000/43,560 = 1.377 acres. For parcels with more than three corners, split them into triangles and add — or run the same shoelace pattern with more terms.