Shoelace Formula Polygon Area Calculator
Enter polygon vertex coordinates to find area and perimeter with the shoelace (Gauss) formula, and reshape it on an interactive SVG grid
Calculate the area and perimeter of any polygon from its vertex coordinates (X, Y) using the shoelace formula, also known as Gauss's area formula. An interactive SVG grid lets you drag the vertices around and watch the area update in real time.
도구를 불러오는 중…
🔒 Everything runs 100% in your browser. Your files and input are never uploaded to any server.
How to use
- Enter the X and Y coordinate for each vertex in the coordinate table, or drag the points to adjust them.
- You can also paste a block of coordinate data as text to import many vertices at once.
- Read the polygon's total area and perimeter, recalculated live as you edit.
- Scroll down for the detailed work, including the cross-multiplication sums and the running perimeter calculation.
FAQ
- Does the shoelace formula work on concave polygons?
- Yes. As long as the polygon is simple — meaning its edges never cross themselves — the shoelace formula returns the exact area whether the shape is convex or concave.
- Does the order of the coordinates matter?
- Yes. The vertices must be listed in order around the polygon's boundary, either clockwise or counterclockwise. If the order is scrambled, the edges cross each other and the formula returns a meaningless area.
- Does it calculate correctly for a self-intersecting polygon, like a bowtie shape?
- No. The shoelace formula only holds for a "simple polygon" — one whose edges never cross each other. If you plug in coordinates for a shape whose sides intersect themselves, like a bowtie, the overlapping regions can cancel out or get double-counted, producing a number that doesn't match the shape's visible area.