The formula in one line
Take the corners in order round the figure. For each side, multiply the easting of the first point by the northing of the second, subtract the easting of the second times the northing of the first, and keep a running total. When you have been all the way round, halve the total and drop the sign. That is the area. It works for any simple polygon, convex or not, with three corners or three hundred, and it does not care where the origin is.
The sign you dropped is not useless. A positive total means the corners were listed counter-clockwise and a negative one means clockwise, which is why the page can tell you which way round your list runs. Software that silently takes the absolute value loses that, and a list that reverses direction halfway through is one of the ways a corner list goes wrong without looking wrong.
Why double meridian distances exist
DMD is the same area computed in a form you can run down a column of a field book without ever writing a coordinate. The double meridian distance of the first course is its own departure. Every course after that is the previous DMD plus the previous departure plus its own. Multiply each DMD by that course latitude, add the lot, halve the absolute value, and there is the area again.
It looks like a strange piece of bookkeeping until you see what it is doing: each term is twice the area of the trapezoid between one course and the meridian through the first point, signed by whether the course runs north or south. Adding them cancels everything outside the parcel. It survives because it runs on latitudes and departures, which a traverse sheet already has in front of it, so the area comes out of the same columns as the closure with no extra computation.
The third method, and why it is here
Shoelace and DMD are algebraically the same identity written two ways. That makes them a poor check on each other. If you had the sign convention wrong, or forgot the factor of a half, both would be wrong together and their perfect agreement would be reassuring and meaningless.
So the page adds a fan triangulation. Pick the first corner, run a line from it to every other corner, and you have cut the polygon into triangles. Add up their signed areas using the cross product, take the absolute value, and you have the area by a route with no shared algebra. When all three land on the same number, a factor of two or a swapped axis is off the table. The five-corner default lands all three on the same 203,999.512 square feet, and the largest disagreement between them is floating point noise several decimal places below anything measurable.
Acres, and how precise they really are
An acre is 43,560 square feet, which is an awkward number with a tidy history: a chain is 66 feet, an acre is ten square chains, and ten times 66 squared is 43,560. A hectare is 10,000 square metres and needs no history.
The interesting question is how many decimal places an acreage deserves. The page prints a line for that. Move any one corner by a tenth of a foot and the area changes by roughly half the span between the two neighbours of that corner, times the tenth. On the default parcel the worst case is 29 square feet out of 203,999 — 0.014 percent, or about seven ten-thousandths of an acre on a parcel of 4.6832. So the fourth decimal place of an acreage is already at the mercy of a tenth of a foot at one corner, and a disagreement in the third is a disagreement about where a monument sits, not about the arithmetic.
There is a second reason the sensitivity is low that catches people out. Sliding a corner along the boundary line does not change the area at all. Only the component perpendicular to the line between its two neighbours counts. A corner that is out by a foot in the wrong direction can cost nothing.
The crossed outline
Type two points in the wrong order and the outline develops a bow tie. The shoelace formula does not object. It treats the area on one side of the crossing as positive and the other as negative and returns the difference, which is a real number and a meaningless area. Nothing about the result looks unusual — no error, no warning, just a smaller figure than it should be. The page tests every pair of non-adjacent sides for an intersection and says so if it finds one, because this is the failure that otherwise sails through unnoticed.
What an area computation is not
It is a property of a list of numbers. What a parcel of land contains is a legal question answered by a licensed surveyor from record research, from monuments recovered in the ground and from the rights of the people on either side of the line. A deed that calls for 5.00 acres and a computation that returns 4.97 do not settle anything between them. Take both to the surveyor responsible for the work.
Questions people ask
How does the shoelace formula work?
Go round the corners in order. For each side, multiply the easting of the first point by the northing of the second, subtract the easting of the second times the northing of the first, and keep a running total. Halve the total and take the absolute value. The halving and the absolute value are both essential and both easy to forget: without the half you get twice the area, and without the absolute value a clockwise list returns a negative. The sign itself is worth reading, because it tells you which way round the corners were listed.
What is a double meridian distance?
It is the area of a traverse computed straight from the latitude and departure columns instead of from coordinates. The DMD of the first course is its own departure; each course after that is the previous DMD plus the previous departure plus its own. Multiply each DMD by its own latitude, sum, halve, drop the sign. Each term is twice the trapezoid between one course and the meridian through the first corner, and the parts outside the parcel cancel. It gives the same answer as the shoelace formula because it is the same identity in the form a field book can run.
How many square feet are in an acre?
43,560. The number comes from the chain: a surveying chain is 66 feet, an acre is ten square chains, and ten times 66 squared is 43,560. A hectare, by contrast, is simply 10,000 square metres. One acre is about 0.4047 hectares and one hectare is about 2.471 acres.
How accurate is an acreage computed from coordinates?
The arithmetic is exact; the corners are not. Move one corner a tenth of a foot and the area shifts by roughly half the span between its two neighbours times that tenth, which on a suburban parcel is a few tens of square feet out of hundreds of thousands. That puts the honest limit somewhere around the third decimal place of an acre, and only if the corners themselves are that well located. An argument about the fourth decimal place is an argument about monuments, not about the formula.
Why is my computed area smaller than it should be?
The most common cause is two points listed out of sequence, which puts a crossing in the outline. The shoelace formula counts the area on one side of the crossing as negative and returns the difference without complaining, so the answer comes out low with no sign that anything is wrong. This page tests every pair of non-adjacent sides for an intersection and reports it. The other common cause is a repeated or missing corner.
Can I use this to settle how big my lot is?
No. It computes the area enclosed by the coordinates you typed. What a parcel legally contains depends on where the boundary actually runs, which is determined by a licensed surveyor from the record, from the monuments found on the ground and from the rights of adjoining owners. A deed acreage and a computed acreage disagreeing is normal and does not by itself mean either is wrong.