Irregular Polygon Area
Calculate the area of any irregular polygon from vertex coordinates using the shoelace formula. Free geometry solver for land surveys, CAD, and shape analysis.
About This Calculator
Use this irregular polygon area calculator to find the area of any polygon from its vertex coordinates. Unlike regular polygons, irregular polygons have sides of different lengths and angles that are not equal -- making standard formulas inapplicable. This calculator uses the shoelace formula (Gauss's area formula) to compute the area directly from vertex coordinates.
The shoelace formula works by dividing the polygon into triangles from the origin and summing their signed areas. Enter vertices as space-separated x,y coordinate pairs in order around the polygon. The formula is: A = 1/2 x |Sigma(x_i x y_i+1) - Sigma(y_i x x_i+1)|. The result is always positive, representing the absolute area.
Regional Notes: Coordinate-based area calculations are entirely unit-agnostic and work worldwide. If your coordinates are in meters, the area is in square meters; if in feet, square feet; if in kilometers, square kilometers. This makes the tool useful for land surveying, CAD, and geographic applications across all countries.
Frequently Asked Questions
How do you find the area of an irregular polygon?
The area of an irregular polygon is found using the shoelace formula (also called Gauss's area formula). List the vertices in order (clockwise or counterclockwise), then for each vertex multiply its x-coordinate by the next vertex's y-coordinate, subtract the product of its y-coordinate with the next vertex's x-coordinate, sum all these differences, and take half the absolute value.
What coordinate format does this calculator accept?
Enter vertex coordinates as space-separated x,y pairs. For example, the default sample '0,0 4,0 4,3 0,3' represents a rectangle with vertices at (0,0), (4,0), (4,3), and (0,3) -- giving an area of 12 square units. Coordinates can be integers or decimals, and vertices must be listed in order around the polygon.
Can I use this calculator for polygons with more than 4 vertices?
Yes, this calculator works for any polygon with 3 or more vertices. Simply list all vertices in order separated by spaces. For example, a pentagon could be '0,0 2,0 3,2 1,3 -1,1'. The shoelace formula works for any number of vertices.
What is the shoelace formula for polygon area?
The shoelace formula calculates polygon area as A = (1/2) x |Sigma(x_i x y_i+1) - Sigma(y_i x x_i+1)|, where vertices are (x_i, y_i) in order and the last vertex connects back to the first. It is called the shoelace formula because of the crisscross multiplication pattern resembling shoelaces.
Is the Irregular Polygon Area Calculator free to use?
Yes, all calculators on Calculy are completely free to use. No registration, login, or payment is required. You can bookmark any calculation with shareable URLs that preserve your inputs.
Can I calculate the area of a self-intersecting polygon?
The shoelace formula works correctly for simple (non-self-intersecting) polygons. For self-intersecting (complex) polygons, the formula may give unexpected results because it computes the signed area. Make sure your vertices are listed in order around the perimeter without crossing edges.