# مساحة المضلع من الإحداثيات: صيغة رباط الحذاء

> احسب مساحة مضلع بسيط ومحيطه ومركز مساحته من إحداثيات رؤوسه بصيغة رباط الحذاء، مع عرض حساب كل حد.

النسخة التفاعلية: https://www.calcopenly.com/ar/geometry/polygon-area-from-coordinates
الموضوع: حاسبات الهندسة

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.

## المدخلات

- **Vertices in order (x, y per line)**: Go around the outline in either direction. Brackets and semicolons are fine; a repeated first point at the end is ignored.
- **Coordinates are in** (الخيارات: No unit, مليمتر (⁨mm⁩), سنتيمتر (⁨cm⁩), متر (⁨m⁩), كيلومتر (⁨km⁩), بوصة (⁨in⁩), قدم (⁨ft⁩), ياردة (⁨yd⁩), ميل (⁨mi⁩))

## النتائج

- المساحة — النتيجة الرئيسية
- Signed area
- المحيط
- Centroid x
- Centroid y
- Number of vertices
- Vertex order

## الصيغة

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

## أمثلة محلولة

### Wikipedia's pentagon

- Vertices in order (x, y per line): 1, 6 / 3, 1 / 7, 2 / 4, 4 / 8, 5
- Coordinates are in: متر (⁨m⁩)
- **المساحة: 16.5 m²**
- **Number of vertices: 5**
- **Vertex order: Counterclockwise**
- **المحيط: 24.307995 m**
- **Centroid x: 3.88888889 m**
- **Centroid y: 3.66666667 m**
- مصدر التحقق: ⁨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: متر (⁨m⁩)
- **المساحة: 12 m²**
- **المحيط: 14 m**
- **Centroid x: 2 m**
- **Centroid y: 1.5 m**
- **Vertex order: Counterclockwise**
- مصدر التحقق: ⁨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: متر (⁨m⁩)
- **المساحة: 6 m²**
- **المحيط: 12 m**
- **Centroid x: 1.33333333 m**
- **Centroid y: 1 m**
- مصدر التحقق: ⁨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: متر (⁨m⁩)
- **المساحة: 4 m²**
- **Signed area: -4 m²**
- **Vertex order: Clockwise**
- مصدر التحقق: ⁨Shoelace sum for clockwise order is −2 × area⁩

### L-shaped plot in feet

- Vertices in order (x, y per line): 0,0 / 6,0 / 6,2 / 2,2 / 2,5 / 0,5
- Coordinates are in: قدم (⁨ft⁩)
- **المساحة: 18 ft²**
- **المحيط: 22 ft**
- **Centroid x: 2.33333333 ft**
- **Centroid y: 1.83333333 ft**
- مصدر التحقق: ⁨6×2 + 2×3 = 18 ft²; Python 3.8 fractions: centroid (7/3, 11/6)⁩

### Closed ring with the first point repeated (edge case)

- Vertices in order (x, y per line): 0 0 / 4 0 / 4 3 / 0 3 / 0 0
- Coordinates are in: متر (⁨m⁩)
- **المساحة: 12 m²**
- **Number of vertices: 4**
- مصدر التحقق: ⁨The repeated closing vertex adds a zero-length edge; same rectangle as above⁩

## الأسئلة

### 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.

### ما مدى دقة «⁨مساحة المضلع من الإحداثيات: صيغة رباط الحذاء⁩»؟

تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 7. مثلًا، يجري التحقق من «⁨Wikipedia's pentagon⁩» بالرجوع إلى ⁨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.

### ما مصدر هذه الطريقة؟

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”.

## المصادر

- [Wikipedia, “Shoelace formula” — worked example with vertices (1, 6), (3, 1), (7, 2), (4, 4), (8, 5)](https://en.wikipedia.org/wiki/Shoelace_formula)
- [Bourke, P. (1988) “Calculating the area and centroid of a polygon”](https://paulbourke.net/geometry/polygonmesh/)