Polygon area from coordinates
Enter the corners in order, get the area.
What is the shoelace formula?
A way to find the area of a polygon from the coordinates of its corners: A = ½ |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, going around the vertices in order and wrapping from the last back to the first.
Example
The triangle (0, 0), (4, 0), (4, 3): 0×0 − 4×0 = 0, 4×3 − 4×0 = 12, 4×0 − 0×3 = 0. Half of |12| is 6 square units.
Does the order of the points matter?
Yes. List the corners in order around the shape, either clockwise or counterclockwise. Jumping across the shape creates crossing edges and a wrong area.
Can I use it for land?
Yes, if you have corner coordinates in feet or meters on a flat grid. The calculator converts square feet to acres (43,560 sq ft) and square meters to hectares (10,000 m²).
Does it work for concave shapes?
Yes — for any simple polygon, convex or not, as long as the edges do not cross each other.
Why is it called the shoelace formula?
Write the coordinates in two columns and repeat the first point at the bottom. The products x·y' and x'·y then criss-cross like shoe laces, and each difference x·y' − x'·y is the 2 × 2 determinant of two neighboring corners (Wolfram MathWorld, “Polygon Area”).
What do clockwise and counterclockwise mean for the result?
Without the absolute value the sum is the signed area: positive when the corners run counterclockwise, negative when clockwise. The triangle (0, 0), (4, 0), (4, 3) runs counterclockwise and gives +6. The calculator shows the positive area and the direction; listing the L-shaped lot below backwards flips the sign, 3300 to −3300.
What happens if the edges cross?
The formula is stated only for polygons that do not cross themselves. For the bow-tie (0, 0), (4, 4), (4, 0), (0, 4) the two triangles are walked in opposite directions and cancel, so the result is 0, although together they cover 8 square units. Reorder the points so the outline never crosses itself.
The formula and what it assumes
For corners (x₁, y₁) to (xₙ, yₙ) in order around the outline, A = ½ |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, with the last corner wrapping back to the first (Wolfram MathWorld, “Polygon Area”). It needs a flat polygon whose edges do not cross; convex or concave makes no difference. The perimeter is the sum of the edge lengths √(Δx² + Δy²). Coordinates must be lengths on a flat grid in one unit, such as feet from a survey corner. Latitude and longitude are angles, so their products give square degrees. The calculator converts ft² to acres (43,560 per acre) and m² to hectares (10,000 per hectare).
Worked example: an L-shaped lot
A lot has corners (0, 0), (60, 0), (60, 40), (30, 40), (30, 70), (0, 70), in feet. Edge by edge, the terms xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ are 0, 2400, 1200, 900, 2100, 0. They add up to 6600, and half of that is 3,300 ft². Splitting the L into a 60 × 40 rectangle and a 30 × 30 square gives 2,400 + 900 = 3,300 ft², the same. The perimeter is 260 ft and the area is 0.0758 acres. The inward corner at (30, 40) needs no special handling.
Common mistakes
The usual error is the wrong corner order. Swapping two corners of the lot above changes the area from 3,300 to 1,500 ft² because the outline now crosses itself; always walk around the boundary. Mixing units, say feet for x and inches for y, gives a number that matches nothing. Latitude and longitude typed in directly give degrees squared; convert to a local grid first.
Sources: Wolfram MathWorld, “Polygon Area” and “Triangle Area” (formula, orientation, non-intersecting condition); NIST SP 811, Appendix B (acre and hectare in square meters). For land purchases rely on a licensed survey, not on sketched coordinates. Related tools: polygon calculator, distance formula calculator and acreage calculator. How we check our calculators.