Shoelace Formula Area Calculator

Work out shoelace formula area instantly with clear inputs, formula shown and shareable results.

About Shoelace Formula Area Calculator

The shoelace (surveyor's) formula sums the cross products of consecutive vertices: A = ½|Σ(xₖyₖ₊₁ − xₖ₊₁yₖ)|. It works for any simple polygon, convex or not, as long as the vertices are given in order around the boundary.

Formula

Shoelace formula

A = ½|Σₖ (xₖ·yₖ₊₁ − xₖ₊₁·yₖ)|, wrapping the last vertex to the first

Worked example

With the default values below, Shoelace Formula Area Calculator returns Enclosed area of 12.0000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Vertices in order (x, y pairs): 0,0 4,0 4,3 0,3
Treat the outline as closedone of Yes, join last to first, No, report open path length: Yes, join last to first

Results

Enclosed area: 12.0000
Boundary length: 14.0000
Number of vertices: 4
Average edge length: 3.5000

Step by step

1Values used
2Shoelace formula
3Enclosed area= 12.0000
4Boundary length= 14.0000
5Number of vertices= 4
6Average edge length= 3.5000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Does 0,0 4,0 4,3 0,3 give 12?

Yes — that is a 4 by 3 rectangle, and the shoelace sum returns exactly 12.

What if the edges cross?

The formula assumes a simple, non-self-intersecting outline. Crossing edges cause the signed pieces to cancel and the answer will be wrong.