What this calculator does
The shoelace formula computes the area of any simple polygon from its vertices alone, no matter how irregular. The name comes from the criss-cross pattern of the multiplications when the coordinates are written out in two columns.
The sign of the result carries information. A positive signed area means the vertices were listed anticlockwise and a negative one means clockwise, which is why the absolute value is taken for the area itself.
The formula
Each pair of consecutive vertices contributes a cross product term. Summing them and halving gives the signed area, and related sums give the centroid.
| Term | Meaning |
|---|---|
| Shoelace formula | The cross-product sum that gives polygon area from vertices. |
| Signed area | The result before taking the absolute value, whose sign gives orientation. |
| Simple polygon | One whose edges do not cross, which the formula requires. |
The inputs explained
| Field | What to enter |
|---|---|
| Vertices in order, x,y pairs separated by ; (e.g. 0,0; 6,0; 6,4; 0,4) | Vertices in order around the polygon, as x,y pairs separated by semicolons. The order matters and the edges must not cross. |
When to use it
Finding the area of an irregular plot
Survey coordinates go straight into the formula.
Computing a centroid
The same working yields the centre of area.
Checking vertex ordering
The orientation result confirms whether points were listed clockwise.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Does the formula handle any shape?
A triangle, a rectangle and a square.
| Vertices | Area | Centroid |
|---|---|---|
| 3 vertices: 0,0; 4,0; 4,3 | 6.000 | (2.667, 1.000) |
| 4 vertices: 0,0; 6,0; 6,4; 0,4 | 24.000 | (3.000, 2.000) |
| 4 vertices: 0,0; 5,0; 5,5; 0,5 | 25.000 | (2.500, 2.500) |
Questions
Why is it called the shoelace formula?
Because writing the coordinates in two columns and drawing lines between the terms being multiplied produces a criss-cross pattern like laced shoes. The name describes the bookkeeping rather than the mathematics.
Does the vertex order matter?
Very much. The vertices must go around the perimeter in sequence, either clockwise or anticlockwise. Listing them in a jumbled order describes a self-intersecting shape and gives a meaningless area.
What does the orientation tell me?
Whether the points were listed anticlockwise, giving a positive signed area, or clockwise, giving a negative one. It matters in computer graphics, where orientation determines which face of a polygon is visible.
Does it work for concave polygons?
Yes, provided the edges do not cross themselves. Concavity is no obstacle at all, which is one of the formula's real strengths over breaking a shape into triangles by hand.
For the incentre of a triangle, see the triangle incenter calculator. For triangle area from sides, see the triangle area calculator.