GeoProof — Shoelace Formula explorer

5 corners

Area = ½ · | Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) |

The Shoelace Formula

The shoelace (or surveyor's) formula gives the area of any simple polygon directly from its corner coordinates (x₁,y₁), …, (xₙ,yₙ), taken in order:

Area = ½ · | Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) |

(the indices wrap around, so after the last corner you return to the first).

Why "shoelace"?

If you list the coordinates in a column and multiply diagonally — down-right then down-left — the crossing pattern looks like lacing a shoe. Each x·y product is one "cross."

What it's really doing

Each term is twice the signed area of the triangle from the origin to one edge. Going around the polygon, the outside parts cancel and only the inside is left — so the sum is twice the polygon's area. The sign tracks direction (counter-clockwise is positive), which is why we take the absolute value.

Handy because

It needs no heights or angles — just coordinates — so it's the go-to method in surveying, computer graphics, and GIS for the area of a plotted region.