Online Tool Store
Polygon Area From Coordinates
Enter polygon vertices to compute Shoelace area, perimeter, centroid, signed-area winding direction, and an auto-scaled 2D plot with reverse and clear controls.
This works right here on your device — nothing gets sent out.
Vertex Coordinates (x, y)
Enter coordinates in clockwise or counter-clockwise order (one (x, y) coordinate pair per line):
Polygon Surface Area
Calculated via Shoelace Gauss formula
Perimeter Length
5 Vertices
2D Coordinate Plane Visualization
How it works
- Enter your polygon vertex (x, y) coordinates sequentially in clockwise or counter-clockwise order (one coordinate pair per line), or load an example.
- The calculator applies Gauss's Shoelace formula to compute exact enclosed surface area.
- Perimeter length is calculated using the Euclidean distance formula across all connecting boundary edges.
- View the auto-scaled plot, centroid, vertex count, and signed-area winding direction; reverse or clear the input when needed.
The formula
Area = ½ | ∑ [i=1 to n] (xᵢ · yᵢ₊₁ − xᵢ₊₁ · yᵢ) | (where xₙ₊₁ = x₁, yₙ₊₁ = y₁)
Perimeter = ∑ [i=1 to n] √((xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²)
FAQ
What is the Shoelace formula (Gauss area formula)?
The Shoelace formula is an algorithm to determine the area of a simple polygon whose vertices are described by ordered 2D Cartesian coordinates (x₁, y₁), (x₂, y₂), ... (xₙ, yₙ) by summing the cross products of adjacent coordinates: Area = ½ |(x₁y₂ + x₂y₃ + ... + xₙy₁) − (y₁x₂ + y₂x₃ + ... + yₙx₁)|.
Must coordinates be entered in a specific sequence?
Yes. Vertices must be listed in consecutive perimeter order (either clockwise or counter-clockwise). If vertices are listed out of order, the shape will self-intersect, yielding an incorrect area.
Does the Shoelace formula work for convex and concave polygons?
Yes! The Shoelace formula works for any simple 2D polygon (both convex and concave), provided the boundary does not cross or self-intersect itself.
How is the geometric centroid calculated?
The geometric center of mass (centroid Cx, Cy) is computed by weighting the cross-product areas of consecutive trapezoidal segments formed by the polygon vertices.
How we compare
| Feature | Online Tool Store | CAD software | Manual formula |
|---|---|---|---|
| Interactive 2D Cartesian SVG plot | ✓ | ✓ | ✗ |
| Perimeter, centroid & winding direction | ✓ | ✓ | ✗ |
| Instant browser access with zero software installation | ✓ | ✗ (Heavy install) | ✓ |
| Reverse order and clear controls | ✓ | ✗ | ✗ |