Irregular Polygon Area Calculator
Enter the (x, y) coordinates of each vertex, one per line, in order around the polygon. The calculator applies the Shoelace formula to find the exact area, along with the perimeter, centroid, and bounding box. You can work in any unit system: the coordinates are unitless, so enter metres to get square metres, feet to get square feet, and so on. The step-by-step panel shows every cross-product term of the Shoelace formula with your actual numbers.
Formula
Worked example
House pentagon with vertices (0,0), (10,0), (10,8), (5,12), (0,8): forward sum = 0*0 + 10*8 + 10*12 + 5*8 + 0*0 = 240, backward sum = 0*10 + 0*10 + 8*5 + 12*0 + 8*0 = 40, Area = 0.5 * |240 - 40| = 100 square units.
What is the Shoelace formula and why does it work?
The Shoelace formula (also called the Gauss area formula or Surveyor's formula) computes the area of any simple polygon from its vertex coordinates. You list the (x, y) coordinates in order around the perimeter, multiply each x by the next y, subtract each y times the next x, sum those cross terms, and halve the absolute value. The name comes from the visual pattern of diagonals that crisscross like shoelace eyelets. The formula works because it sums signed trapezoid areas from each edge down to the x-axis; the interior trapezoids add up while the exterior ones cancel. It handles concave (inward-dipping) shapes exactly, and it generalizes naturally from 3 vertices to hundreds.
How to enter coordinates and what units to use
Type each vertex on its own line in "x,y" format. You may also use a space or semicolon as the separator. List the vertices in consecutive order, either clockwise or counter-clockwise. The unit system selector just labels the output: choose Metric (metres) and the area will be stated in m2, choose Imperial (feet) for ft2, or Unitless for dimensionless coordinates such as pixels. If you are measuring a real plot of land with GPS readings, first convert latitude and longitude to a projected coordinate system such as UTM (Universal Transverse Mercator) or State Plane; using raw degrees introduces significant distortion because one degree of longitude is shorter near the poles than near the equator.
Perimeter, centroid, and bounding box
In addition to the area, the calculator reports the perimeter (the sum of all side lengths computed via the Pythagorean theorem), the centroid (the geometric center of mass of the polygon), and the bounding box (the smallest axis-aligned rectangle that encloses the shape). The centroid is useful for structural analysis, label placement in GIS, and finding the balance point of a flat plate. For convex polygons the centroid always lies inside the shape; for concave polygons it can fall outside the boundary, which is mathematically correct but can be surprising. The bounding box width and height let you quickly estimate the overall footprint of the shape.
Real-world applications in surveying and architecture
Irregular polygon area calculation is essential in land surveying, where property boundaries rarely follow perfect rectangles. A surveyor records the GPS coordinates of each corner stake, projects them to a flat coordinate system, and feeds the resulting (x, y) pairs into the Shoelace formula. Architects use the same approach to compute the gross floor area of non-rectangular rooms or to estimate material quantities (flooring, roofing, concrete). In computer graphics, the Shoelace formula is the basis of polygon rasterisation and collision detection. Game developers use it to compute the area of procedurally generated terrain cells, and GIS software applies it to every polygon feature in a spatial database.
Common irregular polygon types and their vertex counts
| Polygon type | Min vertices | Common use case |
|---|---|---|
| Triangle | 3 | Truss segments, triangular land plots |
| Quadrilateral (general) | 4 | Rectangular rooms with non-right angles |
| L-shaped floor plan | 6 | Building footprints, lot surveys |
| Arrow / chevron | 7 | Directional signage, map icons |
| Star polygon | 10+ | Decorative design, logo shapes |
| Land parcel (typical) | 4-20 | Property surveys, cadastral mapping |
| Complex building footprint | 10-50 | Architectural floor area calculation |
Examples of irregular polygon shapes used in surveying, architecture, and computer graphics.
Frequently asked questions
Does the order of vertices matter?
Yes, but only in the sense that they must be listed consecutively around the polygon boundary. Clockwise and counter-clockwise both give the correct area because the formula takes the absolute value. What you must avoid is jumping across the shape: if vertex 3 is on the far side from vertex 2, the formula will compute the area of a butterfly-shaped self-intersecting figure, not the intended polygon.
Can I use this for a concave (inward-dipping) polygon?
Yes. The Shoelace formula works for any simple polygon, convex or concave. A concave polygon has at least one interior angle greater than 180 degrees, and the formula still gives the correct enclosed area. The only shapes it cannot handle are self-intersecting polygons (where edges cross each other), which require subdivision into non-self-intersecting pieces first.
What is the minimum number of vertices?
Three vertices define the smallest possible polygon: a triangle. Two vertices define only a line segment, which has zero area, so the calculator requires at least three points to return a result.
How do I calculate the area of a land parcel from GPS coordinates?
Collect the GPS coordinates (latitude, longitude) of each corner, then convert them to a local projected coordinate system such as UTM. Record the easting (x) and northing (y) values in metres, enter them here in order, and the result is the parcel area in square metres. Divide by 10,000 for hectares or by 4,047 for acres. Do not use raw decimal degrees as coordinates because their scale varies significantly with latitude.
What is the centroid and can it fall outside the polygon?
The centroid is the arithmetic mean of all points inside the polygon, equivalent to the balance point if the shape were cut from a sheet of uniform material. For convex polygons it always lies inside the boundary. For concave polygons it can fall outside: imagine a very deep U-shape where the center of gravity is in the gap at the top. This is mathematically correct and not an error in the calculation.
Is this the same as the Surveyor's formula?
Yes. The Shoelace formula, Gauss area formula, and Surveyor's formula all refer to the same algorithm. The name "Surveyor's formula" reflects that land surveyors have used this technique for centuries to compute plot areas from field measurements.
How does the formula handle large coordinates such as UTM values?
The formula is numerically stable for any floating-point values, but very large coordinates (e.g. UTM eastings around 500,000) can cause cancellation errors in the cross-product subtraction. A simple fix is to subtract a reference point (such as the centroid or the first vertex) from all coordinates before computing, so the numbers stay small. The area is translation-invariant, so subtracting a constant offset from all x-values and another from all y-values does not change the result.