Irregular Polygon Area Calculator
Calculate the area and perimeter of any irregular polygon from its (x, y) vertex coordinates using the Shoelace formula. Enter one vertex per line as an x,y pair — supports any number of vertices (3+).
Input
Enter at least 3 vertices, one per line, as comma-separated x,y pairs. Order them around the polygon (clockwise or counter-clockwise).
Output
| Metric | Value |
|---|---|
| No data yet | |
Guides
The Irregular Polygon Area Calculator finds the area and perimeter of any polygon — regular or irregular, convex or concave — directly from its corner coordinates. Instead of forcing you to break an odd shape into triangles by hand, you simply list each vertex as an x, y pair and the calculator does the geometry for you. It works for triangles, quadrilaterals, pentagons, L-shapes, land plots, floor plans, and any other flat polygon with three or more corners.
How to use it
- Enter your vertices one per line in the input box, as comma-separated
x, ypairs — for example0,0,4,0,4,3,0,3for a 4 × 3 rectangle. - List the points in order as you walk around the polygon's outline, either clockwise or counter-clockwise. The order matters: the calculator connects each vertex to the next, and the last vertex back to the first.
- Read the results — vertex count, area, and perimeter — updated automatically as you type.
You need at least three vertices to form a polygon. Coordinates can be whole numbers or decimals, positive or negative.
The Shoelace formula
The area is computed with the Shoelace formula (also called the surveyor's formula or Gauss's area formula). For vertices (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ) listed in order, the area is:
area = ½ × | Σ (xᵢ · yᵢ₊₁ − xᵢ₊₁ · yᵢ) |
where the sum runs over every edge and the last vertex wraps back to the first. The name comes from the way the cross-multiplied terms lace together like a shoelace when you write the coordinates in two columns. Taking the absolute value means the answer is the same whether you entered the points clockwise or counter-clockwise.
The perimeter is the straightforward sum of the straight-line (Euclidean) distances between each pair of consecutive vertices, including the closing edge from the last point back to the first. Each edge length is √((x₂ − x₁)² + (y₂ − y₁)²).
Worked example
Take the right triangle with corners at 0,0, 4,0, and 0,3. The Shoelace sum is (0·0 − 4·0) + (4·3 − 0·0) + (0·0 − 0·3) = 12, so the area is ½ × 12 = 6. The three edges measure 4, 5 (the hypotenuse), and 3, giving a perimeter of 12.
Frequently asked questions
Does the polygon have to be convex?
No. The Shoelace formula handles concave (non-convex) polygons and complex outlines like L-shapes just as accurately, as long as the edges don't cross themselves.
What units does it use?
The calculator treats coordinates as pure numbers, so results carry no units. If your coordinates are in metres, the area is in square metres and the perimeter in metres; the same holds for feet, pixels, or any consistent unit.
What if my points are collinear?
If all vertices lie on a single straight line, the enclosed area is zero — the calculator reports an area of 0, which is correct for a degenerate polygon.
Is my data sent anywhere?
No. All calculations run locally in your browser. Your coordinates never leave your device.