Polygon Area Calculator
Calculate the area of any polygon using the Shoelace formula given vertex coordinates.
Enter vertices in order (clockwise or counter-clockwise). Minimum 3 vertices, maximum 100.
The Shoelace formula (also known as the Surveyor's formula or Gauss's area formula) calculates the area of a simple polygon whose vertices are given by their Cartesian coordinates.
Formula: For a polygon with n vertices (x₁, y₁), (x₂, y₂), ..., (xₙ, yₙ), the area is:
The formula gets its name because the calculation resembles the criss-cross pattern of lacing a shoe. You multiply coordinates diagonally, alternating between adding and subtracting.
Requirements: The polygon must be simple (non-self-intersecting), and vertices should be listed in order (either clockwise or counter-clockwise).
Example: For a triangle with vertices (0,0), (4,0), (2,3), the area is ½|0×0 - 4×0 + 4×3 - 2×0 + 2×0 - 0×3| = ½|12| = 6 square units.
Applications: This formula is widely used in computer graphics, geographic information systems (GIS), surveying, and computational geometry for calculating land areas, polygon rendering, and collision detection.
What is the Shoelace formula?
The Shoelace formula (also called Gauss's area formula or the surveyor's formula) calculates the area of a simple polygon when you know the coordinates of its vertices. The name comes from the criss-cross pattern of multiplications, resembling the pattern of lacing a shoe.
Do the vertices need to be in a specific order?
Yes, vertices must be listed in order around the polygon, either clockwise or counter-clockwise. If vertices are out of order, the formula will compute incorrect results. The formula automatically handles both winding directions by taking the absolute value.
Can the Shoelace formula work with any polygon?
The formula works for any simple (non-self-intersecting) polygon, regardless of whether it's convex or concave. It doesn't work for self-intersecting polygons (like a figure-8) because different regions may cancel out in the calculation.
Where is the Shoelace formula used in practice?
It's widely used in computer graphics for polygon rendering, in GIS (Geographic Information Systems) for calculating land areas, in surveying, game development for collision detection, and in computational geometry algorithms for determining polygon properties.