शिरोबिंदूंच्या निर्देशकांवरून साध्या बहुभुजाचे क्षेत्रफळ, परिमिती आणि क्षेत्रकेंद्र शूलेस सूत्राने काढा. प्रत्येक पदाचा हिशेब पाहा.
अद्यतनित तपासलेली उदाहरणे: ७
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).
| 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 |
Index n + 1 wraps round to vertex 1.
The sum is positive, so the vertices run counterclockwise.
A± is the signed area; using it keeps the centroid right for either vertex order.
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.
पडताळणीचा स्रोत: 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
पडताळणीचा स्रोत: 4 × 3 rectangle; centroid at the centre
पडताळणीचा स्रोत: ½ × 4 × 3; 3-4-5 perimeter; a triangle's centroid is the vertex mean (4/3, 1)
पडताळणीचा स्रोत: Shoelace sum for clockwise order is −2 × area
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.
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.
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.
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.
अचूकता तुमच्या इनपुटवर आणि पद्धतीच्या गृहीतकांवर अवलंबून असते. दशांश गणना 50 सार्थ अंक वापरते, पण अंदाज, संख्यात्मक पद्धती आणि मूळ डेटा कमी अचूक असू शकतात; दाखवलेल्या मूल्यांचे पूर्णांकन केल्याने या मर्यादा दूर होत नाहीत. स्वतंत्र स्रोतांतील सोडवलेल्या उदाहरणांशी पडताळणी: ७. उदाहरणार्थ, “Wikipedia's pentagon” ची पडताळणी 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 याच्याशी केली आहे.
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”.
या कॅल्क्युलेटरमध्ये स्वतंत्र स्रोतांमधील उत्तरांसह ७ सोडवलेली उदाहरणे आहेत. ती चाचणी संचात चालतात आणि तुम्ही ती येथेही चालवू शकता.
दोन बिंदूंमधील अंतर, मध्यबिंदू व उतार काढा. रेषेचे समीकरण उतार-छेद, प्रमाणित व बिंदू-उतार रूपात आणि दोन रेषांचा छेदनबिंदू पाहा.
आयत, त्रिकोण, समलंब चौकोन, वर्तुळ, लंबवर्तुळ, सम बहुभुज आणि वलय अशा 15 सपाट आकृत्यांचे क्षेत्रफळ, परिमिती व प्रमाणबद्ध चित्रे पाहा.
बाजूंची संख्या किंवा एका कोनावरून सम बहुभुजाच्या अंतर्गत कोनांची बेरीज, प्रत्येक अंतर्गत व बाह्य कोन आणि कर्णांची संख्या काढा.
तीन बिंदूंमधून जाणाऱ्या वर्तुळाचे केंद्र, त्रिज्या, क्षेत्रफळ व समीकरण काढा. (x − h)² + (y − k)² = r² आणि सामान्य रूप पाहा.