Free Irregular Polygon Area Calculator

Last updated October 10, 2026 · Report an error

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The Shoelace formula (also called Gauss's area formula) computes the area of any simple — meaning non-self-intersecting — polygon directly from the (x, y) coordinates of its vertices, no matter how irregular or lopsided the shape is. Unlike triangle- or rectangle-specific area formulas, it needs no special decomposition into simpler shapes; it works identically whether the polygon is a perfect regular hexagon or a jagged, asymmetric ten-sided outline. This makes it the standard tool for measuring real, irregular boundaries: land surveyors and GIS software use it to compute the area of an oddly-shaped property parcel from its corner coordinates (often gathered by GPS), computer graphics and CAD systems use it to calculate the area of arbitrary 2D shapes and cross-sections, and it's a foundational technique in computational geometry for anything involving polygon area, from collision detection to mesh processing.

How it works

Choose the number of vertices (3 to 10) and enter each vertex's (x, y) coordinates in order as you'd trace them walking around the polygon's perimeter — either clockwise or counterclockwise, but consistently one or the other. The calculator then sums a cross-term for each consecutive pair of vertices, xᵢyᵢ₊₁ − xᵢ₊₁yᵢ (wrapping back around from the last vertex to the first), adds all of those terms together, and takes half the absolute value of that sum. Geometrically, each cross-term corresponds to twice the signed area of a thin triangular wedge from the origin to that edge, and summing them lets the wedges outside the polygon cancel out the wedges that overlap, leaving exactly the polygon's true enclosed area.

Enter number of vertices.
Enter vertex 1 — x.
Enter vertex 1 — y.
Enter vertex 2 — x.
Enter vertex 2 — y.
Enter vertex 3 — x.
Enter vertex 3 — y.
Enter vertex 4 — x.
Enter vertex 4 — y.
Enter vertex 5 — x.
Enter vertex 5 — y.
Enter vertex 6 — x.
Enter vertex 6 — y.
Enter vertex 7 — x.
Enter vertex 7 — y.
Enter vertex 8 — x.
Enter vertex 8 — y.
Enter vertex 9 — x.
Enter vertex 9 — y.
Enter vertex 10 — x.
Enter vertex 10 — y.
Click Calculate to see your results.

Examples

Unit square (0,0), (1,0), (1,1), (0,1)

The Shoelace sum works out to 1, and half its absolute value gives an area of 1 — matching the obvious 1×1 square area computed the ordinary way.

Right triangle (0,0), (4,0), (0,3)

The Shoelace sum is (0×0−4×0) + (4×3−0×0) + (0×0−0×3) = 0 + 12 + 0 = 12, and half its absolute value gives an area of 6 — matching the standard right-triangle formula ½ × base × height = ½ × 4 × 3 = 6.

Irregular quadrilateral (0,0), (5,0), (6,4), (1,3)

Summing the cross-terms gives (0×0−5×0)+(5×4−6×0)+(6×3−1×4)+(1×0−0×3) = 0+20+14+0 = 34, so the area is 34/2 = 17 square units — a shape with no obvious base-times-height shortcut, solved directly from its four corner coordinates.

Common mistakes to avoid

Entering vertices out of order (not tracing consistently around the polygon's perimeter) — this produces a self-intersecting shape and an incorrect area.
Mixing clockwise and counterclockwise segments partway through the vertex list, which effectively describes a different, tangled shape.
Applying the formula to a genuinely self-intersecting outline (like a bowtie shape) and trusting the raw result without checking whether it matches the visually expected area.
Forgetting to close the loop mentally — the formula always wraps the last vertex back to the first one, so omitting or duplicating that final connection isn't necessary and can throw off manual verification.

Frequently asked questions

It only flips the sign of the raw sum before the final step — counterclockwise ordering by convention gives a positive sum and clockwise gives a negative one — but since the calculator takes the absolute value at the end, the reported area is always positive either way.

The formula assumes consecutive vertices in your list are connected by an edge of the polygon. Entering them out of order effectively describes a different, usually self-intersecting shape (like a bowtie), and the Shoelace formula no longer gives the true area of the polygon you actually meant to describe.

Yes — the Shoelace formula works for any simple polygon, convex or concave, as long as its edges don't cross themselves. It doesn't require the shape to "bulge outward" everywhere the way some simpler area formulas implicitly assume.

The Shoelace formula's standard form isn't reliable for self-intersecting polygons — overlapping regions can cancel out or double-count in ways that don't correspond to the shape's visual area, so results should be treated with caution unless the outline never crosses itself.

This tool supports 3 to 10 vertices, covering everything from a simple triangle up to a ten-sided (decagon) shape; the underlying formula itself has no upper limit and scales to any number of vertices.

Yes — the formula works with any real-valued coordinates, positive or negative, since it only depends on the differences and cross-products between coordinate values, not their absolute position relative to the origin.