Área de polígono por coordenadas: fórmula do cadarço

Calcule área, perímetro e centroide de um polígono simples pelas coordenadas dos vértices, com cada termo da fórmula do cadarço detalhado.

Atualizado 28 de setembro de 2026Exemplos verificados: 7

Área

m²

Máximo de casas decimais: 6; Ao mais próximo; empates afastando-se de zero

Signed area: 16.5m²
Perímetro: 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 Linhas: 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

Como é calculado 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.
Área
$A = \frac{|33|}{2} = 16.5\,\text{m}^2$
The sum is positive, so the vertices run counterclockwise.
Perímetro
$P = \sum \sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2} = 24.307995\,\text{m}$
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.

Sobre Área de polígono por coordenadas: fórmula do cadarço

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.

Exemplos resolvidos

Wikipedia's pentagon

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

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

Fonte de verificação: 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: Metros (m)

Área: 12 m²
Perímetro: 14 m
Centroid x: 2 m
Centroid y: 1.5 m
Vertex order: Counterclockwise

Fonte de verificação: 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: Metros (m)

Área: 6 m²
Perímetro: 12 m
Centroid x: 1.33333333 m
Centroid y: 1 m

Fonte de verificação: 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: Metros (m)

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

Fonte de verificação: Shoelace sum for clockwise order is −2 × area

Perguntas

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.

Qual é a precisão de “Área de polígono por coordenadas: fórmula do cadarço”?

A precisão depende dos dados inseridos e das hipóteses do método. O cálculo decimal usa 50 algarismos significativos, mas estimativas, métodos numéricos e dados de origem podem ter menor precisão; o arredondamento exibido não elimina essas limitações. Exemplos resolvidos verificados com fontes independentes: 7. Por exemplo, “Wikipedia's pentagon” é verificado com 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.

De onde vem o método?

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”.

Sobre esta calculadora

\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}

Fontes

Verificado com as referências

Esta calculadora inclui 7 exemplos resolvidos com respostas de fontes independentes. Eles fazem parte do conjunto de testes e você também pode executá-los aqui.