Surveyors and GIS software use the shoelace (Gauss) formula to compute the area of a plot from its corner coordinates. It works for any polygon whose sides do not cross, convex or not.
How to use the Irregular Polygon Area Calculator
- Enter the x-coordinates of the vertices in order around the polygon.
- Enter the y-coordinates in the same order.
- Read the area, perimeter and centroid.
Results update as you type, and everything is calculated in your browser.
How it works
List the vertices in order around the shape (either direction). The cross products of consecutive vertices are added, halved and made positive. The perimeter adds the side lengths.
Inputs explained
| Field | What to enter |
|---|---|
| x-coordinates of the vertices (in order) | A list of numbers separated by commas, spaces or new lines |
| y-coordinates of the vertices (same order) | A list of numbers separated by commas, spaces or new lines |
What each result means
- Polygon area = shoelace([x-coordinates of the vertices (in order)], [y-coordinates of the vertices (same order)])
- Perimeter = polyperim([x-coordinates of the vertices (in order)], [y-coordinates of the vertices (same order)])
- Number of vertices = min(count([x-coordinates of the vertices (in order)]), count([y-coordinates of the vertices (same order)]))
- Centroid – x = polycx([x-coordinates of the vertices (in order)], [y-coordinates of the vertices (same order)])
- Centroid – y = polycy([x-coordinates of the vertices (in order)], [y-coordinates of the vertices (same order)])
Formula
Area = ½ |x₁y₂ − y₁x₂ + x₂y₃ − y₂x₃ + … + xₙy₁ − yₙx₁|
Worked example
Inputs — x-coordinates of the vertices (in order): 0, 4, 6, 3, -1; y-coordinates of the vertices (same order): 0, 0, 3, 6, 3.
Results — Polygon area = 27 square units; Perimeter = 20.01046962; Number of vertices = 5; Centroid – x = 2.425926; Centroid – y = 2.555556.
Key terms
- Shoelace formula
- Gauss's area formula for polygons from vertex coordinates.
Frequently asked questions
Why is it called the shoelace formula?
Writing the coordinates in two columns and cross-multiplying looks like lacing a shoe.
Does the order of the vertices matter?
They must go around the polygon in order; clockwise or counter-clockwise both work.
What if the sides cross?
The formula then gives a net area, not the true area of a self-intersecting shape.
Can I use latitude and longitude?
Only for small areas after projecting to a flat grid in metres.