Irregular Polygon
The Shoelace formula computes polygon area from vertex coordinates—no need to split into triangles. Works for convex and concave simple polygons. Vertices ordered clockwise or counterclockwise.
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The Shoelace formula works for convex and concave simple polygons. Surveyor's formula is the same as the Shoelace formula. Centroid = geometric center from weighted vertex formula.
Ready to run the numbers?
Why: Irregular polygons appear in land surveying, GIS, and CAD. The Shoelace formula (also Gauss area formula) handles any simple polygon without triangulation.
How: Shoelace: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. Vertices must be ordered clockwise or counterclockwise. Perimeter = sum of edge lengths.
Run the calculator when you are ready.
Irregular Polygon — Shoelace Formula
Area, perimeter, and centroid from vertex coordinates. Add or remove vertices (3–20). Step-by-step Shoelace breakdown.
📐 Examples — Click to Load
Vertices (3–20) — Add or remove (x, y) pairs
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
Shoelace formula: A = ½|Σ(xᵢyᵢ₊₁−xᵢ₊₁yᵢ)| from vertex coordinates.
— Formula
Works for convex and concave—no triangulation needed.
— Property
📋 Key Takeaways
- • The Shoelace formula computes polygon area from vertex coordinates — no need to split into triangles
- • Vertices must be ordered clockwise or counterclockwise around the boundary; the absolute value handles both
- • Works for convex and concave simple polygons (no self-intersections)
- • The centroid is the geometric center — the balance point of the shape
- • Perimeter is the sum of distances between consecutive vertices
💡 Did You Know?
📖 Shoelace Formula Explained
Area of a Polygon
For vertices (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ) in order:
where (xₙ₊₁, yₙ₊₁) = (x₁, y₁) to close the polygon
Example: Triangle (0,0), (4,0), (2,3) → A = ½|0+12+0| = 6 sq units
🎯 Expert Tips
💡 Vertex Order
List vertices in order around the polygon. Clockwise or counterclockwise both work — the absolute value gives positive area.
💡 Land Surveying
For irregular lots, surveyors record boundary coordinates. The Shoelace formula gives exact area from those points.
💡 Self-Intersections
The formula assumes simple polygons (no crossing edges). For self-intersecting shapes, it gives signed area — interpret with care.
💡 Coordinate Systems
Use consistent units (meters, feet). The result is in square units of your coordinate system.
⚖️ Calculator Comparison
| Feature | This Calculator | Basic | Manual |
|---|---|---|---|
| 3–20 vertices, add/remove | ✅ | ❌ | ⚠️ |
| Shoelace step-by-step | ✅ | ❌ | ✅ |
| Charts (Bar, Doughnut, Radar) | ✅ | ❌ | ❌ |
| Centroid & perimeter | ✅ | ⚠️ | ⚠️ |
| Copy & share | ✅ | ❌ | ❌ |
| 7 real-world examples | ✅ | ❌ | ❌ |
📊 Polygon Quick Facts
❓ Frequently Asked Questions
What is the Shoelace formula?
A method to compute polygon area from vertex coordinates: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. Also called the Surveyor's or Gauss's area formula.
Does vertex order matter?
Yes — list vertices in order around the boundary (clockwise or counterclockwise). The absolute value ensures positive area either way.
Does it work for concave polygons?
Yes. The Shoelace formula works for any simple polygon (convex or concave) with no self-intersecting edges.
What is the centroid?
The geometric center — the balance point. Calculated from the same cross terms used in the Shoelace formula.
Why "Shoelace"?
The crisscross multiplication pattern resembles lacing a shoe when written in a table.
How accurate is it?
Mathematically exact. Any error comes from rounding or imprecise input coordinates.
Can I use it for land area?
Yes. Surveyors use it for irregular lots. Use consistent units (e.g., meters) for coordinates.
What about self-intersecting polygons?
The formula gives signed area. For crossing shapes, interpret the result carefully — it may not match intuitive "area".
📚 Official & Educational Sources
⚠️ Disclaimer: This calculator provides mathematically precise results for simple polygons. Real-world measurements may vary. For surveying and construction, verify coordinates and units. Self-intersecting polygons require careful interpretation.
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