Enter your polygon's (x, y) vertex coordinates — one pair per line — and the Shoelace Formula Calculator computes the area of any polygon from those coordinates. Works for triangles, quadrilaterals, and complex irregular polygons. Paste coordinates in the Vertex Coordinates field, choose your decimal precision, and get the polygon area and perimeter back.
Polygon Area
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Perimeter
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Number of Vertices
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Shoelace Sum (|Σ|)
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The Shoelace Formula (also called the Surveyor's Formula or Gauss's area formula) calculates the area of a polygon given the coordinates of its vertices. The formula is: A = ½ |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|, where the sum runs over all consecutive vertex pairs and wraps back to the first vertex. It works for any simple (non-self-intersecting) polygon, regardless of shape.
The name comes from the visual pattern when you write out the cross-multiplication steps. If you list x and y coordinates in two columns and draw diagonal lines connecting the terms you multiply, the crisscrossing lines resemble the lacing pattern of a shoe. The alternating additions and subtractions of the diagonal products give the formula its distinctive 'laced' appearance.
The Shoelace Formula works for any simple polygon — convex or concave — as long as the edges don't cross each other. It supports triangles, quadrilaterals, pentagons, and polygons with many vertices. For self-intersecting (complex) polygons, the formula may still run, but the result reflects the net signed area, which may not match the intuitive total enclosed area.
The vertices must be entered in order around the polygon — either all clockwise or all counter-clockwise. The calculator takes the absolute value of the signed area, so both orderings give the same positive area result. What matters is that points follow the boundary of the polygon consecutively; entering them out of order will produce an incorrect area.
Before taking the absolute value, the signed shoelace sum is positive for counter-clockwise vertex order and negative for clockwise order. This sign (the winding direction) is useful in computational geometry for determining orientation, but for the purposes of area calculation, only the magnitude matters.
If using GPS or map coordinates (latitude/longitude), you can treat latitude as Y and longitude as X, entering them in decimal degree format. Note that the resulting 'area' will be in square degrees, not square meters, so a conversion factor based on your location is needed for real-world measurements. For small areas, a local projection coordinate system gives more accurate results.
For an irregular polygon, collect the (x, y) coordinates of all its vertices in order. Then apply the Shoelace Formula: multiply each xᵢ by yᵢ₊₁, subtract xᵢ₊₁ times yᵢ for each consecutive pair, sum all terms (wrapping the last vertex back to the first), and divide the absolute value of that sum by 2. This calculator automates every step — just paste your coordinates.
Using the Shoelace Formula with those four vertices in order, the area is 20 square units. You can verify this by entering the coordinates into the calculator above: 1,6 on line 1; 5,6 on line 2; 8,1 on line 3; 4,1 on line 4.