The Shoelace Formula: Finding the Area of an Irregular Lot From Corner Coordinates
October 21, 2026
Land parcels, garden beds, and floor plans are often not rectangles. If you know the coordinates of each corner, measured from a convenient origin, you can compute the area exactly with the shoelace formula (also called Gauss's area formula). For vertices (x₁, y₁), (x₂, y₂), ..., (xₙ, yₙ) listed in order around the boundary, area = ½ |Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|, where the last vertex connects back to the first.
The name comes from the crisscross pattern of multiplication that resembles lacing a shoe.
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Open the Area Calculator →A Worked Example
Consider a four-sided lot with corners at (0,0), (40,0), (50,30), and (10,40), with coordinates in feet. Compute each term xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ: from (0,0) to (40,0): 0×0 − 40×0 = 0. From (40,0) to (50,30): 40×30 − 50×0 = 1,200. From (50,30) to (10,40): 50×40 − 10×30 = 2,000 − 300 = 1,700. From (10,40) back to (0,0): 10×0 − 0×40 = 0. The sum is 0 + 1,200 + 1,700 + 0 = 2,900. Area = ½ × 2,900 = 1,450 square feet.
As a check, split the lot into two triangles using the diagonal from (0,0) to (50,30). Triangle 1 with vertices (0,0), (40,0), (50,30) has area ½ × 40 × 30 = 600. Triangle 2 with vertices (0,0), (50,30), (10,40) has area ½ |50×40 − 10×30| = ½ × 1,700 = 850. The total is 1,450.
| Edge | From | To | x1*y2 - x2*y1 |
|---|---|---|---|
| 1 | (0,0) | (40,0) | 0 |
| 2 | (40,0) | (50,30) | 1,200 |
| 3 | (50,30) | (10,40) | 1,700 |
| 4 | (10,40) | (0,0) | 0 |
| Sum | 2,900 (area = 1,450 sq ft) |
Rules for Using the Formula
List vertices in order around the shape, either clockwise or counterclockwise. The sign of the sum tells you the direction (positive for counterclockwise), and taking the absolute value gives the area. Do not list vertices in a crisscross order, since that creates a self-intersecting polygon, and the formula will not give the true area. The formula works for convex and concave polygons without holes. For shapes with holes, subtract the area of the hole.
Use consistent units. If coordinates are in meters, the area is in square meters.
Converting Units
Convert square feet to acres by dividing by 43,560: 1,450 square feet is 0.033 acres. A one-acre lot is 43,560 square feet, roughly 209 feet on a side if square. Square meters convert to hectares by dividing by 10,000. In land surveys, area is often calculated from bearings and distances, which are first converted into coordinates, giving the same result.
Calculate to a reasonable precision. Surveyed boundaries may be accurate to a few centimeters or tenths of a foot.
Getting Coordinates in Practice
Measure from a fixed baseline: pick one corner as the origin, run a tape along one edge as the x-axis, and measure perpendicular offsets to other corners. GPS apps and mapping tools provide coordinates in latitude and longitude, which must be projected onto a flat plane first, because degrees of longitude vary in length with latitude. For small areas, convert using local approximations or use a mapping software area tool.
Uses Beyond Land
The shoelace formula appears in computer graphics for polygon area, in geographic information systems, and in cutting patterns for fabric or flooring. For the cost of materials, calculate the area, then add the waste factor appropriate for the material.
Frequently Asked Questions
What is the shoelace formula?
A method to compute polygon area from vertex coordinates by summing cross-products.
Does the order of vertices matter?
Yes. List them in order around the boundary, either clockwise or counterclockwise.
Can it handle concave shapes?
Yes, as long as the polygon does not cross itself.
How do I convert square feet to acres?
Divide by 43,560.
How can I verify my result?
Split the shape into triangles and add their areas, or use a mapping tool.