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 calculator helps you find the area of an irregular polygon from a list of coordinates. It is built for shapes that do not have a simple formula, such as odd floor plans, custom land plots, or sketch-like outlines. Instead of measuring triangles by hand, you can drop in your vertices and get the polygon area, perimeter, and centroid in one clear result.
The core idea is the shoelace formula (also called Gauss’s area formula). Imagine writing the x and y coordinates in two columns and “lacing” them diagonally; multiplying down, multiplying up, then subtracting gives the signed area. If your points run counterclockwise, the signed area is positive; clockwise points make it negative. The calculator also shows the absolute area so you always see a positive measurement for real-world use.
How to use the tool:
- Enter points in order around the boundary. You can click on the canvas, type coordinates, or paste a list.
- Make sure the polygon does not self-intersect. A clean boundary is required for a simple polygon area.
- Click Calculate to see the area, perimeter, and centroid.
- If your pasted points are out of order, use Auto-order (radial) to sort them into a loop.
The calculator also computes the perimeter by summing each edge length, and the centroid (the balance point of the shape). These are useful when you need total fencing length, the center of mass for a cutout, or a reference point for CAD work. If you are using the tool as a coordinate area calculator or a polygon perimeter calculator, the results are consistent as long as your points are in the correct order.
Real-world examples include estimating the area of an irregular garden, verifying a property survey outline, calculating the footprint of a room with angled walls, or computing a custom shape for laser cutting. Designers and engineers often use the centroid to place labels, drill holes, or align components. GIS users can also benefit from the signed area to confirm orientation when working with map data.
Reference formulas:
- 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.