Paste the corner points of any polygon, one x, y pair per line, and get its area with the shoelace formula. The calculator also gives the perimeter, centroid and vertex order, shows a full shoelace table and plots the shape so you can spot a mistyped point.
Last updated:
One point per line, such as 3, 7 or (3, 7). Go around the edge clockwise or counterclockwise; don't repeat the first point.
Enter your values to see the result.
Advertisement
Advertisement
How to use the Polygon Area from Coordinates
Enter the vertices in order around the shape, one x, y pair per line. Don't repeat the first point at the end.
Pick a unit if your coordinates are in meters, feet or another length unit.
Read the area at the top, with the perimeter, centroid and shape type below.
Check the plot: if edges cross, a point is out of order.
Follow the shoelace table and steps, then copy the results or download the table as CSV.
How the shoelace formula works
Write the vertices P₁ to Pₙ in order around the polygon and imagine returning to P₁ at the end. For each edge from Pᵢ to Pᵢ₊₁, compute the cross product xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ. Half the sum of these cross products is the signed area: positive when the points run counterclockwise, negative when they run clockwise. The absolute value is the area.
Each cross product is twice the signed area of the triangle formed by the origin and one edge. Triangles on the far side of the shape are added and those on the near side are subtracted, so everything outside the polygon cancels and only the enclosed region remains.
Worked example
Take the quadrilateral with vertices (1, 1), (5, 2), (4, 6) and (0, 4).
Edge
xᵢ·yᵢ₊₁
xᵢ₊₁·yᵢ
Difference
(1, 1) → (5, 2)
1 × 2 = 2
5 × 1 = 5
−3
(5, 2) → (4, 6)
5 × 6 = 30
4 × 2 = 8
22
(4, 6) → (0, 4)
4 × 4 = 16
0 × 6 = 0
16
(0, 4) → (1, 1)
0 × 1 = 0
1 × 4 = 4
−4
Sum
48
17
31
Area = ½ × |31| = 15.5 square units. The sum is positive, so the points were listed counterclockwise.
A = I + B ÷ 2 − 1, where I = interior points and B = boundary points
Tips for accurate results
Survey sketches and floor plans work well: set one corner as (0, 0) and measure the other corners from it in the same unit.
For a shape with a hole, find the outer area and the hole area separately and subtract.
Very large coordinates, such as state-plane survey values, are fine, but moving the origin near the shape reduces rounding.
Frequently asked questions
What is the shoelace formula?
It is a way to find the area of a polygon from the coordinates of its vertices. List the points in order, add up xᵢ·yᵢ₊₁ for every vertex, subtract the sum of xᵢ₊₁·yᵢ, and take half the absolute value: A = ½ |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. The criss-cross pattern of the products looks like lacing a shoe, which gives the formula its name.
Does the order of the points matter?
Yes. The points must follow the boundary of the shape, either clockwise or counterclockwise. The direction only changes the sign of the signed area, which the calculator reports, but jumping across the shape out of order makes the edges cross and gives a wrong area.
Do I need to repeat the first point at the end?
No. The formula wraps around from the last vertex back to the first automatically. If you do repeat the first point, the calculator notices and drops the duplicate.
Why is my area wrong when the polygon crosses itself?
For a self-intersecting shape such as a star drawn with five crossing lines, the shoelace formula gives a net area where some regions cancel out or count twice. Reorder the points so they trace the outer boundary to get the true enclosed area.
How do I find the area of a triangle from coordinates?
Use the same formula with three points: A = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. For (0, 0), (4, 0) and (0, 3) the area is ½ × |0 + 12 + 0| = 6.
How is the centroid of a polygon calculated?
The centroid uses the signed area A and the same cross products cᵢ = xᵢyᵢ₊₁ − xᵢ₊₁yᵢ: Cx = Σ(xᵢ + xᵢ₊₁)cᵢ ÷ 6A and Cy = Σ(yᵢ + yᵢ₊₁)cᵢ ÷ 6A. It is the balance point of a flat plate with that shape, not the average of the vertices.
Can I use GPS latitude and longitude?
Not directly. Latitude and longitude are angles, so the shoelace formula would return square degrees. Convert the points to a flat grid in meters or feet first, or use a map-based land area tool for GPS points.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.