Area, perimeter and centroid of any simple polygon from its vertex coordinates, using the shoelace (surveyor's) formula with every term shown.
By AdminDrafted with AI writing toolsUpdated Checked examples: 7
The 5 points trace a Counterclockwise outline enclosing 16.5 m², with a perimeter of 24.307995 m and its centroid (balance point) at (3.8889, 3.6667).
| Vertex | x | y | xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ |
|---|---|---|---|
| 1 | 1 | 6 | −17 |
| 2 | 3 | 1 | −1 |
| 3 | 7 | 2 | 20 |
| 4 | 4 | 4 | −12 |
| 5 | 8 | 5 | 43 |
Index n + 1 wraps round to vertex 1.
The sum is positive; for a simple polygon this means counterclockwise vertex order.
A± is the signed area; using it keeps the centroid right for either vertex order.
List a polygon's corners in order around the outline and the shoelace formula gives its area: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|, with the sum wrapping from the last vertex back to the first. The sign of the sum shows the direction of travel, positive for counterclockwise. The same terms give the centroid, and the edge lengths add up to the perimeter.
The default five points, (1, 6), (3, 1), (7, 2), (4, 4) and (8, 5), are the worked example in Wikipedia's article on the formula: the terms sum to 33, so the area is 16.5 m². Surveyors use the method, also called the surveyor's area formula, to find a plot's area from the corner coordinates on a site plan.
The outline must not cross itself. Up to 200 vertices, crossing edges trigger a warning because overlapping parts can cancel or count twice; for larger lists this check is skipped. Coordinates must be planar, so convert latitude and longitude to a projected grid such as UTM first.
List the corners in order around the outline, then apply the shoelace formula A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|, pairing the last corner with the first. For (0, 0), (4, 0), (0, 3), the terms are 0, 12 and 0, so A = 6. It works for any polygon whose edges do not cross.
Write one term per edge, multiplying each x by the next y and subtracting the next x times this y. For the outline (1, 6), (3, 1), (7, 2), (4, 4), (8, 5):
The terms sum to 33, so the area is 33/2 = 16.5 square units. The sum is positive because these corners run counterclockwise.
Without the absolute value, the sum's sign shows the direction of travel: positive for counterclockwise, negative for clockwise, on axes with y pointing up.
Listing the triangle as (0, 0), (0, 3), (4, 0) runs clockwise and gives −6, the same area of 6. Screen and image coordinates often have y increasing downward, which flips what looks clockwise.
| Outline | Vertices in boundary order | Area | Centroid |
|---|---|---|---|
| Rectangle | (0,0), (4,0), (4,3), (0,3) | 12 | (2, 1.5) |
| Right triangle | (0,0), (4,0), (0,3) | 6 | (4/3, 1) |
| L-shaped outline | (0,0), (6,0), (6,2), (2,2), (2,5), (0,5) | 18 | (7/3, 11/6) |
| Default five-point outline | (1,6), (3,1), (7,2), (4,4), (8,5) | 16.5 | (35/9, 11/3) |
The L shape splits into a 6 × 2 rectangle and a 2 × 3 rectangle above its left end, 12 + 6 = 18. Its shoelace terms, 0, 12, 8, 6, 10 and 0, sum to 36, which halves to the same 18.
Weight each shoelace term by the coordinates of its edge: Cx = Σ(xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)/(6A), and Cy the same way with y, where A is the signed area. The L shape's centroid is (7/3, 11/6) ≈ (2.333, 1.833).
The plain average of the L's six corners is (8/3, 7/3), a different point. The corner average equals the centroid for a triangle, but not for polygons in general, and a concave shape's centroid can fall outside it.
Paul Bourke's area and centroid derivation gives both formulas.
(0,0); (4,0); (4,3); (0,3).For standard shapes from their dimensions, use the area and perimeter calculator. For the distance and midpoint between two corners, use the coordinate geometry calculator.
Checked against: Area 16.5 from the Wikipedia “Shoelace formula” worked example
Checked against: 4 × 3 rectangle; centroid at the centre
Checked against: ½ × 4 × 3; 3-4-5 perimeter; a triangle's centroid is the vertex mean (4/3, 1)
Checked against: Shoelace sum for clockwise order is −2 × area
Multiply each x by the next vertex's y, subtract the next x times this y, add the results all the way round and halve the absolute value. For (0, 0), (4, 0), (4, 3), (0, 3) the terms are 0, 12, 12 and 0, so the area is 24/2 = 12. The name comes from the criss-cross pattern the products make when the coordinates are written in two columns.
Yes. The points must follow the boundary, clockwise or counterclockwise, without jumping across. Direction only flips the sign: counterclockwise gives a positive sum, clockwise a negative one, and the area is the absolute value. The corners of a 2 × 2 square taken as (0, 0), (2, 2), (2, 0), (0, 2) trace a crossed bow-tie whose shoelace area is 0 instead of 4.
Record each corner as coordinates on a flat grid, such as eastings and northings in metres from a site plan, list them in order round the boundary and apply the shoelace formula. An L-shaped plot with corners (0, 0), (6, 0), (6, 2), (2, 2), (2, 5) and (0, 5) in feet covers 18 ft². Latitude and longitude must be projected first, because a degree of longitude shrinks towards the poles.
Weight each shoelace term by the sum of the two x coordinates involved: Cx = Σ(xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)/(6A), with A the signed area, and likewise for Cy. For a triangle this equals the average of the three vertices; for the L-shaped plot above it is (7/3, 11/6) ≈ (2.333, 1.833). The centroid of a non-convex shape can lie outside it.
This calculator includes 7 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.