Calculated Result
The centroid, geometry metrics, point table, and graph appear here after submission.
Polygon Plot
Parsed Vertex Table
| # | Original X | Original Y | Used X | Used Y |
|---|
Metric Summary Table
| Metric | Value |
|---|
Calculator Inputs
Example Data Table
This sample polygon is preloaded in the calculator. The metrics below match the default example values.
| Vertex | X | Y |
|---|---|---|
| V1 | 0 | 0 |
| V2 | 6 | 0 |
| V3 | 8 | 4 |
| V4 | 4 | 7 |
| V5 | 1 | 5 |
| Example Metric | Value |
|---|---|
| Centroid X | 3.8139 |
| Centroid Y | 2.9957 |
| Area | 38.5000 square units |
| Perimeter | 24.1767 units |
| Orientation | Counterclockwise |
| Bounding Box | 0 to 8, 0 to 7 |
Formula Used
Signed Area
A = (1 / 2) × Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)
Centroid Coordinates
Cₓ = [1 / (6A)] × Σ(xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)
Cᵧ = [1 / (6A)] × Σ(yᵢ + yᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)
Perimeter
P = Σ √[(xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²]
Additional Diagnostics
Compactness = 4π|A| / P²
Equivalent Radius = √(|A| / π)
The centroid equations require a nonzero polygon area. If the vertices are collinear or the polygon collapses, the centroid of area is undefined.
How to Use This Calculator
- Enter one vertex per line using the format x,y.
- Keep the vertices in boundary order around the polygon.
- Adjust scale and shifts if your coordinates need transformation.
- Choose a unit label and decimal precision.
- Click Calculate Centroid to display results above the form.
- Review centroid, area, perimeter, orientation, and bounding box values.
- Inspect the plot to verify vertex order and shape closure.
- Use the CSV and PDF buttons to export the summary and parsed points.
FAQs
1. What is a polygon centroid?
A polygon centroid is the geometric center of its area. For a uniform flat shape, it is the balance point where the polygon could rest evenly.
2. Does vertex order matter?
Yes. Vertices should follow the boundary in order, either clockwise or counterclockwise. Random order can create crossed edges and incorrect area or centroid values.
3. Can this calculator handle concave polygons?
Yes. Concave polygons work as long as the boundary does not self-intersect and the points are entered in correct edge sequence around the shape.
4. Why is my signed area negative?
A negative signed area means the vertices were entered clockwise. The centroid location remains valid, but orientation changes the sign convention used in the shoelace formula.
5. What if the polygon is self-intersecting?
Self-intersecting shapes can produce misleading results because the area formula assumes a simple polygon. Reorder vertices or split the shape into simple polygons first.
6. Do I need to repeat the first vertex?
No. When auto-close is enabled, the calculator automatically connects the final vertex back to the first one for perimeter, area, and plot calculations.
7. Why are scale and shift options included?
They help when your source data must be resized or translated before analysis. This is useful for normalized coordinates, map offsets, or drawing conversions.
8. Which units appear in the output?
The unit label applies to lengths and area text. Centroid coordinates use the same coordinate units, while area is shown in square units of that label.