Enter the X and Y coordinates of each vertex of your irregular polygon — minimum 3 points, up to 12 — and the Irregular Polygon Area Calculator applies the shoelace formula to compute the exact polygon area. You also get the perimeter and number of vertices as supporting results. List vertices in order (clockwise or counter-clockwise) for accurate output.
Polygon Area
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Perimeter
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Vertices Used
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Use the shoelace formula (also known as the Gauss area formula). List all vertices in order — either clockwise or counter-clockwise — as (x, y) coordinate pairs. The formula is: Area = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, summed over all consecutive vertex pairs, wrapping the last vertex back to the first. This calculator does all of that automatically once you enter your coordinates.
The name comes from the visual pattern of the calculation. When you write the x and y coordinates in two columns and draw diagonal lines between them to show which pairs multiply together, the crossing lines look like the lacing pattern on a shoe. The formal name is the Gauss area formula or the surveyor's formula, but 'shoelace formula' is the most widely used nickname.
Yes — vertices must be entered in either consistently clockwise or consistently counter-clockwise order as they appear around the polygon. Entering them in a random or criss-crossing order will produce an incorrect area, as the shoelace formula assumes the polygon boundary is traced without jumping around.
Absolutely. The shoelace formula works perfectly with negative, zero, and positive coordinate values. There is no requirement for vertices to be in a particular quadrant. This makes it suitable for real-world survey data, CAD coordinates, or any coordinate system where the origin may fall inside or outside the polygon.
A parallelogram is generally an irregular polygon. While its opposite sides are equal and parallel, its angles are not necessarily 90° and its sides are not necessarily all equal. Only a square (a special case of rectangle and rhombus) qualifies as a regular quadrilateral.
Yes, an isosceles triangle is typically an irregular polygon. It has two equal sides and two equal base angles, but all three sides are not equal and all three angles are not equal, so it does not meet the definition of a regular polygon. Only an equilateral triangle — with all sides and angles equal — is a regular polygon.
You need at least 3 vertices to form a polygon (a triangle). With fewer than 3 points, you only have a line segment or a single point, and no enclosed area can be calculated. This calculator requires a minimum of 3 vertices and supports up to 12.
The shoelace formula gives a result for self-intersecting polygons, but the result represents a net signed area rather than the true total enclosed area. Regions where the polygon crosses itself may cancel out. For accurate area calculations, ensure your polygon is simple — meaning its edges do not cross each other.