The shoelace formula for any polygon's area
How a list of corner coordinates becomes an exact area, why the method is named after a shoe, and the one case where it fails.
From corners to area
The shoelace formula turns a polygon into a running sum over its edges. For each corner you multiply its x by the next corner's y, subtract the next corner's x times this corner's y, and add the results all the way around. Halving the absolute value of that total gives the enclosed area. It needs nothing more than the ordered list of vertices, which is why it suits surveying, mapping and any shape defined by points rather than a neat formula.
Why it is called the shoelace
If you write the coordinates in two columns and draw lines between the terms you multiply, the crossing diagonals look like the lacing of a shoe. Each downward diagonal is a positive product and each upward diagonal a negative one, and the difference of the two running sums is what you halve. The nickname is a memory aid for the criss-cross pattern rather than anything about footwear. It makes the bookkeeping easy to do by hand for a few points.
Winding direction and sign
Before taking the absolute value, the raw sum carries a sign that reveals the winding direction. Listing the corners counter-clockwise yields a positive value and clockwise a negative one, but the magnitude is identical either way. This tool reports the absolute area, so you never have to worry about which way you traced the outline. The signed version is handy in graphics for telling front-facing from back-facing polygons.
The one case it cannot handle
The formula assumes a simple polygon whose edges meet only at shared corners and never cross. If the boundary intersects itself, forming a figure-eight or bow-tie, the sums partly cancel and the result is not the visible area. Ordering the points correctly around the perimeter avoids this. For a genuinely self-intersecting region, split it into simple pieces and add their areas separately.