Once a traverse is adjusted and every corner has a coordinate, the area of the parcel is a closed-form computation with no measurement in it at all. Two methods are in common use. The coordinate method, usually called the shoelace formula, sums cross products around the polygon. The double-meridian-distance method accumulates a meridian distance for each course and multiplies it by the course's latitude.
They are the same computation written two ways, and they always agree on a properly ordered closed figure. That is why running both is worth the extra column: a disagreement is not a choice between two answers, it is a signal that the coordinate list is out of order, not closed, or contains a point entered twice.
Below is one parcel computed both ways, with the sign conventions, the checks, and the correction a curved boundary requires.
The parcel
Five corners, coordinates in feet on a local system, taken from an adjusted boundary traverse. Listed in boundary order, walking the perimeter without crossing it.
| Corner | Northing (ft) | Easting (ft) |
|---|---|---|
| A | 5,000.00 | 5,000.00 |
| B | 5,339.37 | 5,310.87 |
| C | 5,156.68 | 5,644.74 |
| D | 4,704.23 | 5,426.60 |
| E | 4,657.96 | 5,122.27 |
The shoelace method
Twice the signed area is the sum around the polygon of each vertex's easting times the next vertex's northing, less the next easting times this northing. The name comes from the criss-cross pattern the multiplications make when the coordinates are written in two columns.
2A = Σ (Eᵢ · Nᵢ₊₁ − Eᵢ₊₁ · Nᵢ) A = |2A| ⁄ 2| Course | Eᵢ · Nᵢ₊₁ | Eᵢ₊₁ · Nᵢ | Difference |
|---|---|---|---|
| A→B | 26,696,850.00 | 26,554,350.00 | +142,500.00 |
| B→C | 27,386,457.11 | 30,139,355.41 | −2,752,898.30 |
| C→D | 26,554,155.25 | 27,983,239.69 | −1,429,084.44 |
| D→E | 25,276,885.74 | 24,096,336.20 | +1,180,549.53 |
| E→A | 25,611,350.00 | 23,289,800.00 | +2,321,550.00 |
| Σ | - | - | −537,383.21 |
Half of 537,383.21 is 268,691.60 square feet, which is 6.1683 acres. The sign is negative, and on a north-up plan a negative signed double area means the corners were walked clockwise. That sign is diagnostic rather than decorative: reversing the order of the list flips it to +537,383.21 and leaves the magnitude untouched.
The DMD method
Double meridian distance is the older hand method and the one a plat's computation sheet still shows. The meridian distance of a course is the distance from a reference meridian to the midpoint of the course; doubling it removes the fractions. Each course's DMD is built from the previous one.
DMD₁ = dep₁
DMDᵢ = DMDᵢ₋₁ + depᵢ₋₁ + depᵢ
2A = Σ (DMDᵢ · latᵢ)| Course | Latitude (ft) | Departure (ft) | DMD | Double area |
|---|---|---|---|---|
| A-B | +339.37 | +310.87 | 310.87 | +105,499.95 |
| B-C | −182.69 | +333.87 | 955.61 | −174,580.39 |
| C-D | −452.45 | −218.14 | 1,071.34 | −484,727.78 |
| D-E | −46.27 | −304.33 | 548.87 | −25,396.21 |
| E-A | +342.04 | −122.27 | 122.27 | +41,821.23 |
| Σ | 0.00 | 0.00 | - | −537,383.21 |
Identical to the shoelace total, to the penny of a square foot, because the two are the same algebra rearranged. Two checks live in this sheet and neither exists in the shoelace version. The latitudes and the departures must each sum to zero on a closed figure, and the last DMD plus the last departure must come to zero as well - 122.27 − 122.27 here. Both fail loudly if a course is mistyped.
The DMD column itself has a plain meaning worth holding on to. A meridian distance is how far the middle of a course sits from the reference meridian, so DMD times latitude is twice the area of the trapezoid between that course and the meridian. Courses running north contribute positive trapezoids and courses running south contribute negative ones; the parcel is what survives the cancellation. That is why a course running due east or west contributes nothing at all - its latitude is zero - and why an all-cardinal figure collapses to two terms.
Which to use
- Use the shoelace method when you already have coordinates. It is shorter, it needs no running accumulator, and it is what every piece of software computes internally.
- Use DMD when you are working from courses rather than coordinates, or when you need a computation sheet a reviewer can follow line by line. The intermediate columns are the audit trail.
- Use both when the number matters. They agree or they do not, and a disagreement localises the error to the course where the two running totals diverge.
- Use neither on an unadjusted traverse. Area computed from coordinates that do not close is the area of whatever figure the misclosure happens to make, and it is not the parcel.
One caution applies to both methods equally. They compute the area of the figure the coordinates describe, which is a plane figure. On a state plane grid that is a grid area, and converting it to ground area means dividing by the square of the combined factor - a correction of twice the linear parts per million, which at 124 ppm is 0.025 per cent, or about 0.0015 acres on a six-acre parcel.
Curved boundaries
Both methods compute the area of a polygon, so a boundary that follows a circular arc needs a correction. Compute the polygon using the chord, then add or subtract the circular segment between the chord and the arc.
segment area = (R² ⁄ 2) · (Δ − sin Δ) Δ in radiansFor a boundary arc of radius 350 ft through a central angle of 42°, the segment is 3,914 square feet, or 0.0899 acres. On a parcel of six acres that is one and a half percent of the total - far too large to ignore, and exactly the sort of thing that gets ignored when a curve is casually replaced by its chord.
The habit that prevents every problem on this page is the same one: compute the area twice by different routes, state the method on the plat, and treat any disagreement as a data problem rather than a rounding problem until you have proved otherwise.