The arithmetic behind acreage from coordinates is short — one loop over the boundary vertices, called the shoelace formula — but the arithmetic is not where the difficulty lives. The difficulty is that the formula assumes the coordinates sit on a flat plane, and latitude and longitude do not. This guide covers both: the method, worked through with real numbers, and the assumptions that decide whether the result is accurate to a tenth of a percent or wrong by several.
List the boundary vertices in order — clockwise or anticlockwise, but never mixed — and repeat the first vertex at the end to close the ring. Then sum the cross products:
The absolute value is there because the sign of the sum tells you the winding direction, clockwise or anticlockwise, not the size of the area. If your vertices are in the wrong order the formula still returns a number, and that number is meaningless — the boundary crosses itself. Checking that the polygon is traced as a simple closed ring is part of the job, not a formality.
Take a four-sided parcel with projected coordinates in metres, listed anticlockwise:
(0, 0), (100, 0), (120, 80), (10, 90).
| Vertex pair | xᵢ · yᵢ₊₁ | xᵢ₊₁ · yᵢ | Difference |
|---|---|---|---|
| (0,0) → (100,0) | 0 × 0 = 0 | 100 × 0 = 0 | 0 |
| (100,0) → (120,80) | 100 × 80 = 8,000 | 120 × 0 = 0 | 8,000 |
| (120,80) → (10,90) | 120 × 90 = 10,800 | 10 × 80 = 800 | 10,000 |
| (10,90) → (0,0) | 10 × 0 = 0 | 0 × 90 = 0 | 0 |
| Sum of differences | 18,000 | ||
Area = ½ × 18,000 = 9,000 m². That is 0.9 hectares, or 2.2239 acres, since one acre is 4,046.8564224 m². In imperial units, 9,000 m² is 96,875.2 ft², and divided by 43,560 that gives the same 2.2239 acres.
Grid area and ground area are not the same thing, and the gap between them comes from the projection. A projected coordinate is a position on a developable surface that has been flattened, which means the plane is either slightly stretched or slightly compressed relative to the ground. The ratio is the scale factor, and area scales with its square.
For a UTM coordinate the scale factor is 0.9996 on the central meridian and rises towards the zone edge, so the plane is compressed in the middle and grid area runs about 0.08% smaller than ground area at the centre, crossing over to slightly larger further out. On a one-acre lot that is a few square feet. On a 640-acre section it is most of an acre — small in percentage terms and large in the currency of a property transaction.
The larger problem is what happens if you feed latitude and longitude into the formula directly. Degrees are not a linear distance: a degree of longitude is about 111.32 km at the equator and about 78.85 km at 45° of latitude. Applying the shoelace formula to raw degrees produces a number with units of square degrees, and converting that to area with a single constant factor is only approximately right over a small area at a known latitude. For anything bigger than a lot, or anything that will be compared with a deed, project first.
A useful sanity check. A one-acre square is 208.71 ft on a side, or about 63.6 m. A section is one square mile — 640 acres, 2.59 km². If a computed acreage implies a shape that does not match the parcel you can see, the fault is almost always vertex order or a projection mismatch, not the shoelace formula.
Coordinate lists arrive in whatever order the source produced. Some start at the point of beginning and run clockwise as deeds usually do; some are sorted by nothing at all. Two diagnostics will catch most problems before the area is reported. First, plot the shape — a self-intersecting polygon is obvious on a plot and invisible in the numbers. Second, compare the perimeter computed from the coordinate list with the sum of the stated boundary lengths; if they differ, a vertex is missing or duplicated.
It is also worth recording the units and the projection with the result. An area quoted as "2.22 acres" without saying whether the coordinates were in State Plane metres, UTM metres or survey feet is only three digits of context away from being unusable.
| From | To | Multiply by |
|---|---|---|
| Square metres | Acres | 0.000247105381 |
| Square metres | Hectares | 0.0001 |
| Square feet | Acres | 0.0000229568411 |
| Acres | Hectares | 0.40468564224 |
| Hectares | Acres | 2.4710538147 |
| Square miles | Acres | 640 |
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