Calcula área, perímetro y centroide de un polígono simple con las coordenadas de sus vértices y cada término de la fórmula del cordón explicado.
Actualizado Ejemplos verificados: 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, so the vertices run counterclockwise.
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; crossing edges trigger a warning, because the overlapping parts then cancel or count twice. Coordinates must be planar, so convert latitude and longitude to a projected grid such as UTM first.
Fuente de comprobación: Area 16.5 from the Wikipedia “Shoelace formula” worked example; Python 3.8 fractions: signed sum +33 (counterclockwise), centroid (35/9, 11/3); math.sqrt edge sum for the perimeter
Fuente de comprobación: 4 × 3 rectangle; centroid at the centre
Fuente de comprobación: ½ × 4 × 3; 3-4-5 perimeter; a triangle's centroid is the vertex mean (4/3, 1)
Fuente de comprobación: 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.
La precisión depende de tus datos y de los supuestos del método. El cálculo decimal usa 50 cifras significativas, pero las estimaciones, los métodos numéricos y los datos de origen pueden ser menos precisos; el redondeo mostrado no elimina esos límites. Ejemplos resueltos comprobados con fuentes independientes: 7. Por ejemplo, «Wikipedia's pentagon» se comprueba con Area 16.5 from the Wikipedia “Shoelace formula” worked example; Python 3.8 fractions: signed sum +33 (counterclockwise), centroid (35/9, 11/3); math.sqrt edge sum for the perimeter.
Wikipedia, “Shoelace formula” — worked example with vertices (1, 6), (3, 1), (7, 2), (4, 4), (8, 5); Bourke, P. (1988) “Calculating the area and centroid of a polygon”.
Esta calculadora incluye 7 ejemplos resueltos con respuestas de fuentes independientes. Forman parte del conjunto de pruebas y también puedes ejecutarlos aquí.
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