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Irregular Polygon Area Calculator

DonnéesPromoteurMathématiques

SAISIR

Processus automatique

Vertices

Coordonnées *

Enter at least 3 vertices, one per line, as comma-separated x, y pairs. Order them around the polygon (clockwise or counter-clockwise).

Units & Display

Unité

Label used for area, perimeter and centroid values. Pure numbers — no conversion is applied.

Decimals

Number of decimal places shown in the output.

Show Vertex Labels

Show Centroid Marker

SORTIR

Côté client

Propriété	Valeur
Vertex Count
Zone
Perimeter
Centroid (x, y)
Bounding Box
Vertex Order
Convexe
Simple (no self-intersection)

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Guide

Irregular Polygon Area Calculator

Type a list of vertex coordinates and instantly get the area, perimeter, and centroid of any flat polygon — convex or concave, axis-aligned or skewed, three vertices or thirty. The calculator uses the classic Shoelace formula so the answer is exact (no triangulation, no approximations), and it draws a live SVG preview so you can confirm the shape matches what you intended before trusting the numbers.

It’s a quick way to verify a hand calculation, double-check a CAD value, work out the footprint of a non-rectangular floor plan, or estimate how much paint, fabric, or material an oddly-shaped surface needs. Everything runs in your browser — no upload, no account, no waiting.

Comment utiliser

Type one vertex per line in the Coordonnées box as x, y. Spaces, semicolons, or parentheses are also fine: (2.5, 7) works the same as 2.5, 7.
List the vertices in the order they appear around the polygon — clockwise or counter-clockwise, it doesn’t matter. Just don’t skip vertices.
Choisissez un Unité if your coordinates represent metres, feet, or anything else. The label gets appended to the results; no unit conversion is applied.
Modifier Decimals to control the precision shown in the table.
Watch the live SVG preview to confirm the shape looks right. Use the Try shortcuts if you want a starter example to play with.
Cliquez Télécharger SVG to save the preview, or Copier les résultats to grab a plain-text summary for a report or notebook.

Caractéristiques

Exact Shoelace area – Uses Gauss’s area formula directly on your coordinates. No rounding errors from triangulating the shape.
Perimeter & centroid – Reports the total edge length and the geometric centroid (true center of mass for a uniform thin plate of that shape).
Live SVG preview – Polygon, vertex dots, optional vertex labels, and an optional centroid marker, all redrawn as you type.
Convexity & self-intersection detection – Tells you whether the polygon is convex and whether any edges cross. A crossed polygon is highlighted in red so you can fix the vertex order.
Orientation reported – Shows whether your vertices run clockwise or counter-clockwise (handy for CAD and graphics pipelines that care about winding).
Bounding box – Width × height of the axis-aligned rectangle that just contains the polygon.
Flexible input parsing – Accepts comma, space, or semicolon separators, and parentheses or brackets around each pair.
Download & copy – One-click SVG download for documentation, and a clipboard-ready plain-text summary.

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 FAQ

What is the Shoelace formula?

The Shoelace formula (also called Gauss's area formula or the surveyor's formula) computes the area of a polygon directly from its vertex coordinates. For vertices (x1,y1), (x2,y2), ..., (xn,yn) listed in order, the area equals half the absolute value of the sum of x_i * y_{i+1} minus x_{i+1} * y_i, with the last vertex wrapping back to the first. It is exact for any simple polygon — convex or concave — and avoids triangulating the shape.
Does the order of the vertices matter?

The order matters because it defines which edges connect to which. The vertices must be listed sequentially around the boundary, but you can start anywhere and go either clockwise or counter-clockwise — the signed area changes sign, but the absolute area is the same. If two vertices are out of order the polygon edges will cross themselves and the area becomes ambiguous.
What is the difference between a simple and a complex polygon?

A simple polygon is one whose edges only meet at shared endpoints — no two edges cross each other. A complex (or self-intersecting) polygon has edges that cross, which makes its enclosed area ambiguous: the Shoelace formula still returns a number, but it is a signed combination of overlapping regions, not a meaningful physical area. The calculator flags self-intersecting input so you can reorder the vertices.
What does it mean for a polygon to be convex?

A polygon is convex when every interior angle is at most 180 degrees, equivalently when the line segment between any two points inside the polygon stays entirely inside. Algorithmically, a polygon is convex when the cross product of every pair of consecutive edges keeps the same sign. Convex polygons are easier to triangulate, intersect, and test point membership against, which is why many computer-graphics and physics algorithms require convex shapes.
How is the centroid different from the geometric center?

For a polygon, the centroid is the average position of all the points enclosed by the boundary — it is also the center of mass of a flat plate of uniform density cut in that shape. The geometric center can mean the centroid, but it is sometimes loosely used to mean the average of the vertices or the center of the bounding box, which are generally different points except for very symmetric shapes. This tool reports the true centroid using the area-weighted polygon formula.

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