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Area by coordinates

Shoelace and DMD, side by side — because when they disagree, your vertex order is wrong.

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Vertices — one per line, in boundary orderlabel, northing, easting — e.g. A, 0.00, 0.00. The label is optional. Commas, tabs or spaces all work.

The list must run around the boundary — every vertex adjacent to the two it touches on the ground. The figure is closed for you, so do not repeat the first point at the end. Order matters far more than direction: walking the parcel backwards flips the sign of the double area and changes nothing else, but visiting the corners out of sequence produces a bow-tie and an area that means nothing.

Area (acres): 40.000
Area (ft²): 1,742,400
Perimeter: 5,280.00 ft
Vertices: 4 vertices
Walked direction: Clockwise
Shoelace vs DMD: Agree

Double-meridian-distance computation sheet
Course	Latitude	Departure	DMD	DMD × latitude
A–B	1,320.000	0.000	0.000	0.00
B–C	0.000	1,320.000	1,320.000	0.00
C–D	-1,320.000	0.000	2,640.000	-3,484,800.00
D–A	0.000	-1,320.000	1,320.000	0.00
Σ = 2A	—	—	—	-3,484,800.00
\|2A\| ÷ 2	—	—	—	1,742,400.00

Agreement check: shoelace 1,742,400.00 ft² against DMD 1,742,400.00 ft². The two methods match, so the vertex list is in boundary order.

The latitude and departure columns are the components of each course between consecutive vertices, and the DMD column is the running meridian distance: DMD₁ = dep₁, then DMDᵢ = DMDᵢ₋₁ + depᵢ₋₁ + depᵢ. The last DMD, plus the last departure, comes back to the first departure with the opposite sign — the arithmetic check a hand-worked sheet ends on.

Curved boundary

A shoelace area is the area of the straight-line figure through the coordinates. When one boundary is an arc, the coordinate list gives you the long chord, and the area between that chord and the arc — the circular segment — has to be put back by hand.

Curve radius (ft)R of the boundary arc

Central angle Δ40-00-00 or 40 — degrees, minutes, seconds

Stand inside the parcel and look at the boundary. If the arc swings AWAY from you — the parcel is fatter than the chord suggests — the segment is land you own and the segment area is ADDED to the shoelace area. If the arc swings TOWARD you, the segment is land outside the boundary that the chord wrongly included, and it is SUBTRACTED. On a cul-de-sac frontage the arc almost always bulges into the lot, so the correction is usually a subtraction.

Segment area (ft²): 1,729.50
Segment area (acres): 0.040
Δ used: 40°00′00″
Radius (ft): 250.00
Corrected area (ft²): 1,744,130
Corrected area (acres): 40.040

Segment area is R²/2 · (Δ − sin Δ), with Δ in radians. The corrected figure above is the shoelace area plus this segment. A parcel with more than one curved boundary needs one segment computed per curve, each added or subtracted on its own.

Once a parcel is reduced to coordinates, its area is arithmetic. This page does that arithmetic twice — once by the coordinate (shoelace) sum and once by the double-meridian-distance sheet a plat's computation page shows — and reports whether the two agree. They always do when the vertices are in boundary order, and only then, which is what makes the comparison worth printing rather than hiding.

The shoelace formula

Take the vertices in order around the boundary and pair each one with the next, closing back to the first. Each pair contributes a cross product; the sum of the cross products is twice the signed area of the figure. The name comes from the way the terms lace across each other when the coordinate pairs are written in two columns.

2A = Σ (Eᵢ · Nᵢ₊₁ − Eᵢ₊₁ · Nᵢ)
A  = |2A| / 2

Indices run around the polygon, with the last vertex paired back to the first.

The sign is not noise. A negative double area means the vertices were walked clockwise on a north-up plan, a positive one that they were walked counter-clockwise. Direction is a free choice — a deed can be written either way and the area is the same — so the calculator takes the absolute value and reports the direction separately. What the sign cannot survive is a list that is out of order. Visiting the corners in the wrong sequence produces a self-intersecting bow-tie whose two lobes have opposite signs and partly cancel, and the number that falls out is not the area of anything.

A double area of exactly zero is degenerate: every point lies on one line, or two corners are the same point. The calculator reports that as an error rather than printing a confident zero.

The double-meridian-distance method

The DMD method predates coordinates and is still the form a computation sheet is laid out in. The meridian distance of a course is the distance from a reference meridian to the midpoint of that course; doubling it clears the fractions, and the doubled value carries forward course to course by simple addition.

DMD₁ = dep₁
DMDᵢ = DMDᵢ₋₁ + depᵢ₋₁ + depᵢ
2A  = Σ (DMDᵢ · latᵢ)

The reference meridian is taken through the first vertex, which is what makes DMD₁ equal the first departure.

Each course contributes the area of a trapezoid between it and the meridian, positive when the course runs north and negative when it runs south. The northbound and southbound trapezoids overlap everywhere outside the parcel and cancel there exactly, leaving the enclosed area. The recurrence is the whole method: no course needs its own distance from the meridian computed, because the previous DMD plus the two departures gets you there.

Why both are computed, and what a disagreement means

The two methods are the same sum rearranged, so on a valid vertex list they agree to the last digit the floating-point arithmetic can carry. That makes their agreement a weak check of the arithmetic and a strong check of the input. If this page ever reports a disagreement larger than a rounding epsilon, the arithmetic is not what changed — the vertex list is not a simple closed boundary, and the figure being measured is not the parcel you meant. Re-order the points so that each one is adjacent to the two corners it actually touches on the ground, then compute again.

The check earns its place because a mis-ordered list has no other symptom. The figure still closes, the perimeter still adds up, and the area is plausible.

Acres and hectares, exactly

An acre is 43,560 square feet, exactly and by definition. It comes from the chain: Gunter's chain is 66 ft, a square chain is 4,356 ft², and ten square chains make an acre. That is why an old description in chains converts to acreage without a decimal in sight — 10 chains by 10 chains is exactly 10 acres.

1 acre    = 10 square chains = 43,560 ft²
1 hectare = 10,000 m²
1 square mile = 640 acres

All three are exact definitions, not rounded conversions.

The 43,560 relationship holds in whichever foot the coordinates are in, because both sides are squared in the same foot. That does not make the acre unit-free: an acre in square meters depends on which foot you started from, and the two US feet give answers 4 parts per million apart. Acres are reported here only for foot coordinates and hectares only for metric ones, because a 0.4 ha parcel relabelled as 0.4 ac is a plat that has to be redrawn.

PLSS aliquot checks

In the rectangular system a nominal section is a mile square: 5,280 ft on a side, 640 acres. The aliquot parts divide it by halves, so the arithmetic is worth memorising as a check on any computed area in that country.

A quarter-quarter section is 1,320 ft square — 1,742,400 ft², which is 40.000 acres exactly. That is the default parcel on this page, and its perimeter is exactly one mile.
A quarter section is 2,640 ft square: 160 acres.
A half-quarter (a 1,320 by 2,640 ft strip) is 80 acres.
A sixteenth of a quarter-quarter — 330 ft square — is 2.5 acres.

Real sections are not nominal. Sections closing on the north and west boundaries of a township absorb the accumulated excess or deficiency in their lots, and government lots are patented at their measured acreage rather than at 40. An aliquot check confirms your arithmetic, not the parcel; the Manual of Surveying Instructions is the authority on which is which.

Curved boundaries

A coordinate list has no curves in it. Two points on an arc define the long chord, and a shoelace area computed through them is the area of the straight-line figure, short or long by the circular segment between the chord and the arc.

segment area = R²/2 · (Δ − sin Δ) Δ in radians

The area between an arc and its long chord, for a central angle Δ on radius R.

The sign is decided by geometry, not by formula. If the arc swings away from the interior of the parcel, the segment is land inside the boundary that the chord left out, and it is added. If the arc swings into the parcel — the usual case on a cul-de-sac frontage — the chord included land that is not there, and the segment is subtracted. Compute one segment per curved boundary and apply each correction on its own; there is no shortcut that handles several at once.

Questions

Do I have to repeat the first vertex at the end of the list?

No. The figure is closed for you — the last vertex is joined back to the first automatically. Repeating the first point adds a zero-length course, which does not change the area but does add a meaningless row to the DMD sheet and inflates the vertex count.

Does it matter whether I list the corners clockwise or counter-clockwise?

No. The direction only flips the sign of the double area, and the area reported is the absolute value. The direction is shown separately because it is a useful sanity check against the drawing. What does matter is the sequence: every vertex has to be adjacent to the two corners it touches on the ground.

Why does the calculator show both a shoelace area and a DMD area?

Because they are the same sum computed two different ways, so their agreement is a check on the input rather than on the arithmetic. On a valid boundary they match to the last digit. A disagreement means the vertex list is not in boundary order — a fault that produces no other visible symptom, since a mis-ordered figure still closes and still has a plausible perimeter.

Can I use State Plane coordinates directly?

You can, and the area you get is a grid area. Grid areas are smaller or larger than ground areas by roughly the square of the combined factor, which at a combined factor of 0.9999 is about 200 ppm — 0.008 acres on a 40 acre parcel. If the deed area has to be a ground area, scale the coordinates to ground first, or compute the grid area and correct it by dividing by the combined factor squared.

How do I handle a parcel with a curved boundary?

Compute the area through the curve's endpoints, which gives you the figure bounded by the long chord, then add or subtract the circular segment for each arc. The helper on this page computes the segment from the radius and the central angle. Add it when the arc bulges out of the parcel and subtract it when the arc bulges in.

Is anything I type sent anywhere?

No. The whole computation runs in your browser. There is no server call, no logging and no storage, which is also why the page keeps working with no network connection once it has loaded.

Sources

Bureau of Land Management, Manual of Surveying Instructions (2009) — public domain; the authority for section subdivision, aliquot parts, government lots and the chain-to-acre relationships
Washington State DOT, Highway Surveying Manual M 22-97, Chapter 11 — published state manual; the coordinate geometry conventions used for latitudes, departures and areas

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