Polygon Area from Coordinates Calculator

How it works

The shoelace formula (also called Gauss's area formula or the surveyor's formula) finds the area of any polygon whose corners you know as coordinates. Walk round the shape and, for each edge, multiply the x of one end by the y of the other, cross-wise, like lacing a shoe. Adding up the differences gives twice the signed area: positive if you walked counter-clockwise, negative if clockwise. The area is half the absolute value.

Each cross-wise term is twice the signed area of the triangle formed by the origin and that edge; the triangles outside the shape cancel out, leaving exactly the enclosed region. The same edge terms, weighted by the sum of the two endpoints' coordinates, give the centroid, the point where a flat cut-out of the shape would balance.

The perimeter is simply the sum of the edge lengths, each by the distance formula. The calculator reports which way round you listed the corners so that a clockwise survey traverse is not mistaken for an error; it also drops a final point that repeats the first, as closed-ring exports from GIS and CAD tools often include one.

Formula

cᵢ = xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ          (edge i; the last edge returns to vertex 1)
area      = ½ · |Σ cᵢ|
perimeter = Σ √((xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²)
Cx = Σ (xᵢ + xᵢ₊₁)·cᵢ ÷ (6·A_signed)      Cy = Σ (yᵢ + yᵢ₊₁)·cᵢ ÷ (6·A_signed)

Example

A four-cornered lot at (0, 0), (6, 0), (8, 4) and (2, 5) ft. The cross terms are 0×0 − 6×0 = 0, 6×4 − 8×0 = 24, 8×5 − 2×4 = 32 and 2×0 − 0×5 = 0, totalling 56, so the area is 56 ÷ 2 = 28 sq ft. The sides are 6, √20 = 4.4721, √37 = 6.0828 and √29 = 5.3852 ft, a perimeter of 21.9401 ft. The centroid is at (656 ÷ 168, 384 ÷ 168) = (3.9048, 2.2857) ft, and the corners were listed counter-clockwise.

The triangle (0, 0), (4, 0), (0, 3) has area ½ × 4 × 3 = 6, perimeter 3 + 4 + 5 = 12 and centroid (1.3333, 1), the average of its three corners.

Assumptions and limitations

• The polygon must be simple, with no crossing edges, and the corners must be listed in order around the boundary. A self-intersecting figure or an out-of-order list gives a meaningless area.
• Coordinates are in a flat plane. Survey points in latitude and longitude must be projected to a grid first; the formula is not for a sphere.
• All coordinates share one unit, the one you choose, and the area is in that unit squared.
• The centroid is the centre of area of the region. It is not the average of the corner points, except for a triangle.

Frequently asked questions