Area by Coordinates (Shoelace)

This formula finds the area of a polygon directly from its vertex coordinates, without needing side lengths or angles. It works by adding the signed cross-products of each consecutive pair of points; geometrically, those terms measure the area contributions from the edges, and the absolute value removes the sign caused by the order of the vertices. The result is the same whether the polygon is drawn in a coordinate grid, on a map, or from survey data, as long as the vertices are listed around the boundary in order.

Use this when you know the coordinates of a closed polygon and need its area quickly, such as in land surveying, GIS, computer graphics, or geometry problems. It is especially useful for irregular shapes where breaking the region into triangles would be cumbersome.

• Not listing the vertices in the correct boundary order, which can give the wrong result
• Forgetting to repeat the first vertex at the end of the calculation
• Leaving out the absolute value and getting a negative area