Irregular Polygon Area Calculator

Enter your polygon's vertex coordinates in order (clockwise or counterclockwise) to compute area, perimeter, and centroid using the Shoelace formula — works for any simple convex or concave polygon.

Quick Facts

Shoelace formula
A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|
Also called the surveyor's formula; works for any simple polygon.
Vertex order matters
List points sequentially
Go clockwise or counterclockwise around the perimeter, not randomly.
Concave shapes work too
Convex or concave, both fine
Only requirement: edges must not cross themselves.

Your Results

Calculated
Area
-
Shoelace formula, in square units
Perimeter
-
Sum of side lengths
Centroid (x, y)
-
Geometric center of the polygon
Vertices used
-
Number of points parsed

Ready

Enter vertex coordinates in order, then press Calculate.

Formula and Method for Irregular Polygon Area

An irregular polygon has sides and angles of different lengths and sizes, so there is no single side-length formula like there is for a square or a regular polygon. Instead, this calculator uses the Shoelace formula (also called the surveyor's formula or Gauss's area formula), which computes area directly from the (x, y) coordinates of the polygon's vertices, listed in order around its perimeter.

How the calculation works

List each vertex as an (x, y) pair, one per line, walking around the polygon in order — either clockwise or counterclockwise, without skipping around. For n vertices (x₁, y₁) through (xₙ, yₙ), with the list wrapping back to the first point, the calculator computes Area = ½ |Σᵢ (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. It also sums the straight-line distance between each consecutive pair of vertices to get the perimeter, and uses the signed version of the same cross-product terms to locate the centroid (geometric center) at Cx = (1/6A)Σ(xᵢ+xᵢ₊₁)(xᵢyᵢ₊₁−xᵢ₊₁yᵢ) and Cy = (1/6A)Σ(yᵢ+yᵢ₊₁)(xᵢyᵢ₊₁−xᵢ₊₁yᵢ), where A is the signed area.

Common mistakes

  • Vertices out of order: the Shoelace formula assumes points are listed sequentially around the boundary. Entering them in a random order (not tracing the perimeter) produces a self-intersecting shape and a wrong — often much smaller — area.
  • Mixing units: every coordinate must use the same unit (all feet, all meters, etc.); the result inherits that unit squared for area and un-squared for perimeter.
  • Repeating the first point: don't re-enter the starting vertex at the end of the list — the calculator automatically closes the polygon back to point 1.

Real-world applications

  • Land surveying and real-estate parcels use the Shoelace formula (often from GPS or total-station coordinates) to compute lot area for deeds and tax assessments.
  • CAD, GIS, and mapping software rely on the same formula to report the area of any digitized boundary or floor plan.
  • Construction and landscaping use it to estimate material (sod, paving, roofing) for oddly shaped lots that don't reduce to a rectangle or circle.
  • Engineering and physics use the centroid location to find the center of mass of a flat, uniform-density lamina shaped like the polygon.

Frequently Asked Questions

What is the Shoelace formula?
The Shoelace formula (also called the surveyor's formula or Gauss's area formula) computes a polygon's area directly from its vertex coordinates: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|, summed over all vertices in order and wrapping back to the first point. It works for any simple polygon — convex or concave — with three or more vertices.
Does the order I list the vertices in matter?
Yes. Vertices must be entered sequentially as you'd trace the polygon's outline, either clockwise or counterclockwise — not in a random order. Listing them out of sequence connects the wrong edges and produces a self-intersecting shape with an incorrect area.
Can this calculator handle concave (non-convex) polygons?
Yes. The Shoelace formula gives the correct area for any simple polygon, meaning one whose edges don't cross each other, whether it's convex or has inward-pointing (concave) vertices. It cannot be used directly on self-intersecting shapes.
How many vertices can I enter?
Enter at least 3 vertices to form a polygon; there's no fixed upper limit — add one "x, y" pair per line for as many corners as your shape has.

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