The formula
area = ½ · |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|
Why it works
The shoelace formula — also called Gauss’s area formula — walks the polygon adding cross products of consecutive vertices, and it works for any simple polygon: convex, concave, three sides or thirty. The idea behind it is that each pair of consecutive vertices defines a triangle with the origin, and those triangles add and subtract until exactly the enclosed area is left. You do not have to break the plot into triangles by hand, and the sides need not be square to anything: all you need are the coordinates, in order.
How to solve it by hand
- Note the coordinates of the corners following the boundary, skipping none
- Multiply each corner’s x by the next corner’s y and add them all
- Subtract the sum of the cross products the other way round
- Divide by two and take the absolute value
What is worth knowing
The corners have to go in order along the boundary: skip one or shuffle them and the polygon crosses itself, and the answer stops meaning anything. The sign before the absolute value tells you which way you went round — positive anticlockwise, negative clockwise — and it is a quick way to check that no corner is out of place. If the coordinates come from a GPS in degrees, convert them to metres first: a degree of latitude is about 111 kilometres, but a degree of longitude depends on the latitude, and using them raw gives nonsense areas.