Irregular Polygon Area (Shoelace) Calculator
Find the area of any simple polygon from a list of its vertex coordinates using the shoelace (surveyor’s) formula.
How it works
A = ½ · | Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) | · vertices taken in order, wrapping the last back to the first
The shoelace formula finds the area of any simple (non-self-intersecting) polygon directly from its vertex coordinates, no matter how irregular. You multiply each x by the next y, subtract each next-x times y, sum around the polygon, and halve the absolute value — the criss-cross pattern of multiplications is what gives it the name. List the vertices consistently in order around the boundary; the sign of the raw sum tells you whether that order was clockwise or counter-clockwise.
Worked example
For the square 0,0 → 4,0 → 4,3 → 0,3: the cross terms sum to (0·0−4·0) + (4·3−4·0) + (4·3−0·3) + (0·0−0·3) = 0 + 12 + 12 + 0 = 24, so A = |24|/2 = 12 square units.
Sources: Shoelace formula — Wikipedia
Frequently asked questions
Do I need to repeat the first vertex at the end?
No. The formula automatically wraps the last vertex back to the first, so you list each corner exactly once in order around the polygon.
Does the order of the vertices matter?
They must be listed sequentially around the boundary, either all clockwise or all counter-clockwise. Listing them out of order or crossing edges gives a wrong area because the polygon self-intersects.
Does it work for non-convex shapes?
Yes. The shoelace formula is exact for any simple polygon, convex or concave, as long as the edges do not cross each other.
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This calculator is for educational purposes. Double-check important results before acting on them.