Aire d'un polygone par coordonnées : formule du lacet

Calculez l'aire, le périmètre et le centroïde d'un polygone simple à partir de ses sommets, avec chaque terme de la formule du lacet détaillé.

Mis à jour 28 septembre 2026Exemples vérifiés : 7

Aire

m²

Décimales maximales : 6 ; Au plus proche, égalités en s’éloignant de zéro

Signed area: 16.5m²
Périmètre: 24.307995m
Centroid x: 3.88888889m
Centroid y: 3.66666667m
Number of vertices: 5
Vertex order: Counterclockwise

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).

The 5-sided outline

Shoelace terms Lignes : 5

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

Comment le calcul est effectué S

Shoelace sum
$\sum_{i=1}^{5} (x_i y_{i+1} - x_{i+1} y_i) = -17 - 1 + 20 - 12 + 43 = 33$
Index n + 1 wraps round to vertex 1.
Aire
$A = \frac{|33|}{2} = 16.5\,\text{m}^2$
The sum is positive, so the vertices run counterclockwise.
Périmètre
$P = \sum \sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2} = 24.307995\,\text{m}$
Centroid
$C_x = \frac{\sum (x_i + x_{i+1})(x_i y_{i+1} - x_{i+1} y_i)}{6A_\pm} = 3.888889,\quad C_y = 3.666667$
A± is the signed area; using it keeps the centroid right for either vertex order.

À propos de Aire d'un polygone par coordonnées : formule du lacet

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.

Exemples détaillés

Wikipedia's pentagon

Vertices in order (x, y per line): 1, 6 3, 1 7, 2 4, 4 8, 5
Coordinates are in: Mètres (m)

Aire: 16.5 m²
Number of vertices: 5
Vertex order: Counterclockwise
Périmètre: 24.307995 m
Centroid x: 3.88888889 m
Centroid y: 3.66666667 m

Source de vérification : 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

Rectangle 4 × 3

Vertices in order (x, y per line): 0, 0 4, 0 4, 3 0, 3
Coordinates are in: Mètres (m)

Aire: 12 m²
Périmètre: 14 m
Centroid x: 2 m
Centroid y: 1.5 m
Vertex order: Counterclockwise

Source de vérification : 4 × 3 rectangle; centroid at the centre

Right triangle, centroid at the mean of the vertices

Vertices in order (x, y per line): (0, 0) (4, 0) (0, 3)
Coordinates are in: Mètres (m)

Aire: 6 m²
Périmètre: 12 m
Centroid x: 1.33333333 m
Centroid y: 1 m

Source de vérification : ½ × 4 × 3; 3-4-5 perimeter; a triangle's centroid is the vertex mean (4/3, 1)

Clockwise square (edge case: negative signed area)

Vertices in order (x, y per line): 0 0; 0 2; 2 2; 2 0
Coordinates are in: Mètres (m)

Aire: 4 m²
Signed area: -4 m²
Vertex order: Clockwise

Source de vérification : Shoelace sum for clockwise order is −2 × area

Questions

How does the shoelace formula work?

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.

Does the order of the points matter?

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.

How do you find the area of an irregular plot of land from its corners?

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.

How do you find the centroid of a polygon?

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.

Quelle est la précision de « Aire d'un polygone par coordonnées : formule du lacet » ?

La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 7. Par exemple, « Wikipedia's pentagon » est vérifié à l’aide de 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.

D’où vient cette méthode ?

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”.

À propos de ce calculateur

\begin{gathered} A = \frac{1}{2}\sum_{i=1}^{n}(x_i y_{i+1} - x_{i+1} y_i) \\[6pt] C_x = \frac{1}{6A}\sum (x_i + x_{i+1})(x_i y_{i+1} - x_{i+1} y_i) \end{gathered}

Sources

Vérifié avec les références

Ce calculateur comprend 7 exemples résolus dont les réponses proviennent de sources indépendantes. Ils font partie de la suite de tests et peuvent aussi être exécutés ici.