Listed anticlockwise, which is what the sign of 12 tells you. The shape is convex.
Area
12
exact, from the shoelace sum
Signed area
12
its sign is the winding direction
Winding
anticlockwise
the order the vertices were given
Perimeter
14
sum of the edge lengths
Centroid
(2, 1.5)
of the filled shape, not the vertices
Convex
yes
no interior reflex angle
The signed area is 12, and its sign says the vertices are listed anticlockwise. Taking the absolute value at the end is what turns a signed quantity into an area.
The formula pairs each vertex with the next and cross-multiplies — which is where the name comes from, since the pairings look like laces when the coordinates are written in two columns.
It assumes the outline does not cross itself. For a self-intersecting polygon it still returns a number, and that number is not the enclosed area — the overlapping parts cancel instead of adding.
The centroid reported is the centroid of the filled shape, which is not the average of the vertices unless the polygon is very regular.
The formula assumes the outline does not cross itself. For a self-intersecting one it still returns a number, and that number is not the enclosed area.
What this tool shows
The shoelace formula pairs each corner with the next and cross-multiplies. Its result is a signed area — the sign is the winding direction — and taking the absolute value at the end is what turns it into an area.
The area of any simple polygon
The signed area and what its sign means
Perimeter from the same coordinates
The centroid of the filled shape
Whether the outline is convex
Where the formula stops working
Any number of corners Exact area Winding direction Centroid and convexity
Updated 7 September 2026 · Works in any browser, no installation
Pair each corner with the next, cross-multiply, add, and halve. For a rectangle with corners (0,0), (4,0), (4,3) and (0,3) that gives 12. The sum is signed — positive here because the corners were listed anticlockwise — and the absolute value at the end is what makes it an area.
At a glance
Formula shown
Area = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| taken round the outline with the last vertex paired back to the first. Without the absolute value the result is the signed area, positive for an anticlockwise ordering.
Scenario support
Measuring a plot of land from surveyed corners; finding the area of a shape drawn on a grid; computing the area of a region in a graphics or GIS program.
Educational estimate
Planning support from the values you enter — not professional advice.
How the shoelace works
Write the coordinates in two columns, repeat the first vertex at the bottom, and multiply diagonally. The crossing pattern of those multiplications is where the name comes from — written out, they look like laces.
Each term x₁y₂ − x₂y₁ is twice the signed area of the triangle formed by the origin and that edge. Summing them adds those triangles up, and the ones outside the polygon cancel because the edges pass them in opposite directions.
That cancellation is the elegant part. The origin does not have to be inside the shape, or anywhere near it — the sum comes out the same wherever it is.
It works for any number of vertices, convex or not, without triangulating first.
The sign is the winding
Before the absolute value, the sum carries a sign, and the sign is information rather than an annoyance.
Positive means the vertices were listed anticlockwise. Negative means clockwise. The magnitude is the same either way.
Graphics systems use exactly this to decide which face of a surface you are looking at, so that the back of an object can be skipped rather than drawn. GIS formats use it to distinguish an outer boundary from a hole cut in it.
A signed area of zero means the vertices are collinear, or the outline retraces itself exactly. Nothing is enclosed. That is a real answer rather than an error, and it is why the sum is computed exactly here — in floating point a genuine zero can arrive as 1e−16 and the page would confidently report a winding direction that does not exist.
Where it breaks
One assumption, stated plainly: the outline must not cross itself.
For a self-intersecting polygon — a bowtie, say — the formula still returns a number, and that number is not the enclosed area. The two lobes are traversed in opposite directions, so their contributions subtract instead of adding. A symmetric bowtie returns zero.
It is worth being clear that this is not the formula failing. It is computing the signed area correctly, and for a self-intersecting outline the signed area genuinely is not the enclosed region.
The other requirement is vertex order. The corners have to be given in the order they appear round the outline. Listing them in some other order describes a different polygon, and the formula will happily measure that one instead.
Lattice points and Pick’s theorem
When every vertex sits on integer coordinates, the area is always a whole number of halves. Never a third, never anything else.
It falls straight out of the formula: every term is a product of integers, so the sum is an integer, and the area is half of it.
Pick’s theorem is the surprising consequence. For such a polygon, the area equals i + b/2 − 1, where i is the number of grid points strictly inside and b the number on the boundary. The area can be found by counting dots, with no multiplication at all.
It is one of the more charming results in elementary geometry, and it works only because the half-integer fact above holds exactly rather than approximately.
Where it gets used
Surveying and land measurement. Corners are recorded as coordinates and the area follows directly, without dividing the plot into triangles first.
GIS and mapping. Every polygon area in a mapping system is a shoelace sum, and the winding sign distinguishes an outer ring from a hole.
Computer graphics. Back-face culling uses the sign of the projected area to skip the far side of an object, which roughly halves the work per frame.
Physics and engineering. Cross-sectional areas of irregular profiles, and the centroid needed to find where a beam bends.
Sources and methodology
The formula and its limits are standard; these are the references.
Method. The sum is computed on exact rationals rather than in floating point, which matters for a reason beyond precision: the sign of the result decides the winding direction, and a quantity that should be exactly zero cannot be allowed to come out as a tiny positive number. On integer coordinates the area is always a whole number of halves, so an exact answer is achievable and an approximate one would be visibly wrong. The suite re-derives every test polygon by fan triangulation from the first vertex, and checks that reversing the vertex order flips the winding while leaving the area alone. That engine is verified on every change against 119 hand-written assertions, including that reversing the vertex order flips the sign and preserves the magnitude across four hundred generated polygons, and that the shoelace sum matches an independent fan triangulation. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
TriangleSolve any triangle from three measurements — and when two sides and a non-included angle describe TWO triangles, this one shows both instead of picking one.
CircleRadius, diameter, circumference and area from any one of them — and why a 16-inch pizza is four times an 8-inch one rather than twice.
DistanceThe distance between two points in the plane or in space, kept exact as a surd wherever the square root does not come out — √50 stays √50, and is also shown as 5√2.
MidpointThe midpoint of a segment, plus the reverse problem nobody else solves: given one endpoint and the midpoint, where is the other endpoint?
Plane EquationThe equation of a plane through three points, with its normal vector and the signed distance from a fourth — so you get which side, not only how far.
Similar TrianglesScale factor and the missing sides — plus the area factor, which is its square. Two pairs with different ratios are reported as not similar rather than averaged.
For a triangle specifically — solved from sides and angles as well as coordinates — the Triangle Calculator covers more cases.
Educational use disclaimer
This is an educational tool. The shoelace formula assumes the outline does not cross itself; for a self-intersecting polygon it returns a number that is not the enclosed area.
Published the polygon area page computing the shoelace sum on exact rationals, because the sign of that sum decides the winding direction and a quantity that should be exactly zero cannot be left to floating point.
Warns plainly that the formula assumes the outline does not cross itself — for a bowtie it still returns a number, and that number is not the enclosed area because the lobes are traversed in opposite directions and subtract.
Notes that on integer coordinates the area is always a whole number of halves, which is the fact Pick’s theorem rests on.
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