Shoelace is a surveyor’s hack
The “shoelace” formula is the same trick land surveyors used with notebooks: multiply down, multiply up, subtract, and you’ve got area—no calculus needed.
Paper math
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.
5 Fun Facts about Irregular Polygons
Clockwise flips the sign
Reverse your vertex order and the signed area becomes negative. 3D engines use that sign to decide which face of a polygon is “front.”
Orientation test
Holes vs. islands
GIS files decide whether a ring is a “hole” or “island” by its winding: outer rings counterclockwise, inner rings clockwise—your area sign is the clue.
Mapping lore
Triangulate anything
Every simple polygon can be chopped into n−2 triangles. Shoelace is just all those tiny wedge areas summed in one swipe.
Hidden triangles
Self-crossing breaks it
A bow-tie polygon has edges that cross; the shoelace sums cancel and area turns weird. That’s why intersection checks matter before you trust the number.
Quality check
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.