محددات سے کثیرالاضلاع کا رقبہ: شو لیس فارمولا

راسوں کے محددات سے سادہ کثیرالاضلاع کا رقبہ، محیط اور مرکزِ رقبہ شو لیس فارمولے سے نکالیں۔ ہر جز کا حساب دیکھیں۔

تازہ کاری 28 ستمبر، 2026جانچی گئی مثالیں: 7

رقبہ

m²

زیادہ سے زیادہ اعشاری مقامات: 6؛ قریب ترین؛ برابر فاصلے پر صفر سے دور

Signed area: 16.5m²
محیط: 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 قطاریں: 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

حساب کا طریقہ 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.
رقبہ
$A = \frac{|33|}{2} = 16.5\,\text{m}^2$
The sum is positive, so the vertices run counterclockwise.
محیط
$P = \sum \sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2} = 24.307995\,\text{m}$
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.

محددات سے کثیرالاضلاع کا رقبہ: شو لیس فارمولا کے بارے میں

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.

حل شدہ مثالیں

Wikipedia's pentagon

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

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

جانچ کا ماخذ: ⁨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: میٹر (⁨m⁩)

رقبہ: 12 m²
محیط: 14 m
Centroid x: 2 m
Centroid y: 1.5 m
Vertex order: Counterclockwise

جانچ کا ماخذ: ⁨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: میٹر (⁨m⁩)

رقبہ: 6 m²
محیط: 12 m
Centroid x: 1.33333333 m
Centroid y: 1 m

جانچ کا ماخذ: ⁨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: میٹر (⁨m⁩)

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

جانچ کا ماخذ: ⁨Shoelace sum for clockwise order is −2 × area⁩

سوالات

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.

“⁨محددات سے کثیرالاضلاع کا رقبہ: شو لیس فارمولا⁩” کتنا درست ہے؟

درستی آپ کی درج کردہ قدروں اور طریقے کے مفروضوں پر منحصر ہے۔ اعشاری حساب 50 بامعنی ہندسے استعمال کرتا ہے، مگر تخمینے، عددی طریقے اور ماخذ کا ڈیٹا کم درست ہو سکتے ہیں؛ دکھائی گئی قدروں کو راؤنڈ کرنے سے یہ حدود ختم نہیں ہوتیں۔ آزاد ذرائع کی حل شدہ مثالوں سے جانچ: 7۔ مثلاً، “⁨Wikipedia's pentagon⁩” کو ⁨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 سے جانچا جاتا ہے۔

اس طریقے کا ماخذ کیا ہے؟

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”.

اس کیلکولیٹر کے بارے میں

\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}

ماخذ

حوالوں سے جانچ شدہ

اس کیلکولیٹر میں آزاد ذرائع سے حاصل کردہ جوابات والی 7 حل شدہ مثالیں شامل ہیں۔ یہ ٹیسٹ مجموعے میں چلتی ہیں اور آپ انہیں یہاں بھی چلا سکتے ہیں۔