Area del poligono da coordinate: formula di Gauss

Calcola area, perimetro e centroide di un poligono semplice dalle coordinate dei vertici con la formula di Gauss, mostrando ogni termine.

Aggiornato 28 settembre 2026Esempi verificati: 7

Area

m²

Massimo di cifre decimali: 6; Al più vicino; a parità lontano da zero

Signed area: 16.5m²
Perimetro: 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 Righe: 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

Come si calcola 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.
Area
$A = \frac{|33|}{2} = 16.5\,\text{m}^2$
The sum is positive, so the vertices run counterclockwise.
Perimetro
$P = \sum \sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2} = 24.307995\,\text{m}$
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.

Informazioni su Area del poligono da coordinate: formula di Gauss

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.

Esempi svolti

Wikipedia's pentagon

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

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

Fonte di verifica: 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: Metri (m)

Area: 12 m²
Perimetro: 14 m
Centroid x: 2 m
Centroid y: 1.5 m
Vertex order: Counterclockwise

Fonte di verifica: 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: Metri (m)

Area: 6 m²
Perimetro: 12 m
Centroid x: 1.33333333 m
Centroid y: 1 m

Fonte di verifica: 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: Metri (m)

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

Fonte di verifica: Shoelace sum for clockwise order is −2 × area

Domande

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.

Quanto è preciso «Area del poligono da coordinate: formula di Gauss»?

La precisione dipende dai dati inseriti e dalle ipotesi del metodo. Il calcolo decimale usa 50 cifre significative, ma stime, metodi numerici e dati di origine possono essere meno precisi; l’arrotondamento visualizzato non elimina questi limiti. Esempi svolti verificati con fonti indipendenti: 7. Per esempio, «Wikipedia's pentagon» viene verificato 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.

Da dove proviene il metodo?

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”.

Informazioni su questa calcolatrice

\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}

Fonti

Verificato con le fonti

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