Irregular Polygon Area Calculator — Shoelace, Perimeter, Centroid
Diagram & Vertices
Tip: Click to add a vertex. Drag points to adjust. The polygon is closed automatically for calculations.
Results
Vertices (x, y)
| # | x | y | Actions |
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Paste or Import
Accepted formats: “x,y” or “x y” per line. Scientific notation OK.
How the Irregular Polygon Calculator Works
This tool uses the classic shoelace formula (a.k.a. Gauss’s area formula) for a simple polygon with vertices ordered around the boundary. It reports the signed area to indicate orientation (positive for counterclockwise, negative for clockwise) and also the absolute area. The perimeter is the sum of edge lengths, and the centroid uses the polygon centroid formula based on the same cross products.
- Signed area \( A_s = \tfrac{1}{2}\sum_{i=1}^{n}(x_i y_{i+1} - x_{i+1} y_i) \), with \( (x_{n+1},y_{n+1})=(x_1,y_1) \)
- Perimeter \( P=\sum_{i=1}^{n}\sqrt{(x_{i+1}-x_i)^2+(y_{i+1}-y_i)^2} \)
- Centroid \( C_x=\frac{1}{6A_s}\sum (x_i+x_{i+1})(x_i y_{i+1}-x_{i+1} y_i) \), \( C_y=\frac{1}{6A_s}\sum (y_i+y_{i+1})(x_i y_{i+1}-x_{i+1} y_i) \)
All computation and rendering occur entirely in your browser.
Irregular Polygon Calculator: FAQs
How do I enter a polygon?
Click on the diagram to add vertices, drag them to adjust, or type coordinates in the table. You can paste CSV or “x y” pairs; use Auto-order (radial) if your pasted points aren’t in boundary order.
What formula is used for area?
The shoelace (Gauss’s) formula. The tool shows both signed area (for orientation) and absolute area.
Does it support concave shapes?
Yes. Concave polygons work fine. The tool also warns if the shape self-intersects; in that case, the simple area becomes ambiguous.
Is my data private?
Yes. Everything runs locally; nothing is uploaded.