Polygon Area Calculator
Paste a list of vertex coordinates to get the area of any simple polygon by the shoelace formula, plus perimeter, centroid, winding order and a convexity check.
Shoelace working
- Signed area ½·Σ(xᵢ·yᵢ₊₁ - xᵢ₊₁·yᵢ) = 18
- Absolute area = 18
- Winding order: counter-clockwise (positive signed area)
- Shape: convex — every turn goes the same way
- Bounding box: x from -1 to 4, y from 0 to 5
Plot
Side lengths
| Edge | Length |
|---|---|
| Vertex 1 → 2 | 4 |
| Vertex 2 → 3 | 3 |
| Vertex 3 → 4 | 3.605551 |
| Vertex 4 → 5 | 3.605551 |
| Vertex 5 → 1 | 2.236068 |
What is the Polygon Area Calculator?
The ByteTools Polygon Area Calculator applies the shoelace formula, A = ½|Σ(xᵢ·yᵢ₊₁ - xᵢ₊₁·yᵢ)|, to any list of vertex coordinates you paste in.
- Shoelace formula for any simple polygon, verified on (0,0), (4,0), (4,3) giving an area of 6
- Perimeter, area centroid, bounding box and per-edge lengths
- Winding order from the signed area — clockwise or counter-clockwise
- Convex versus concave detection from the cross-product signs
- Warns on self-intersecting outlines and on zero-area collinear input
- Runs entirely in your browser — coordinates never leave your device
How to use the Polygon Area Calculator
- 1
Paste your vertices one per line as x then y, separated by a comma or a space.
- 2
Read the area, perimeter, vertex count and centroid in the summary tiles.
- 3
Check the shoelace working panel for the signed area, winding order and convexity.
- 4
Scroll the edge table for individual side lengths, or study the plot.
- 5
Click Copy results to take the whole summary.
About the Polygon Area Calculator
The ByteTools Polygon Area Calculator applies the shoelace formula, A = ½|Σ(xᵢ·yᵢ₊₁ - xᵢ₊₁·yᵢ)|, to any list of vertex coordinates you paste in. It handles triangles, quadrilaterals and irregular polygons with hundreds of vertices, in whatever order you traced them.
Along with the area you get the perimeter from the segment lengths, the centroid of the enclosed region, the winding order read from the sign of the signed area, a convexity test from the cross-product signs, the bounding box, and a per-edge length table. A scaled plot draws the outline with the centroid marked.
Everything is computed in your browser with JavaScript — coordinates are never uploaded — so the tool is private, instant and works offline. Self-intersecting outlines are detected and flagged, because the shoelace formula assumes a simple polygon.
Frequently asked questions
What is the shoelace formula?
It computes a polygon's area from its vertex coordinates: sum xᵢ·yᵢ₊₁ - xᵢ₊₁·yᵢ around the outline, then halve the absolute value. The name comes from the crisscross pattern of the multiplications when you write the coordinates in two columns.
Do the vertices need to be in order?
Yes. The shoelace formula traces the boundary in sequence, so the points must follow the outline, either clockwise or counter-clockwise. A scrambled order produces a self-intersecting path and a meaningless area — the tool detects crossing edges and warns you.
Does the polygon need to be closed?
No. The calculator closes it for you by joining the last vertex back to the first, so do not repeat the starting point. If you do repeat it, the extra zero-length edge is harmless and the area stays the same.
What does clockwise versus counter-clockwise mean here?
The signed area is positive when the vertices run counter-clockwise and negative when they run clockwise. Winding order does not change the size of the area, but it matters in computer graphics, GIS and CNC toolpaths where orientation determines which side is inside.
Can I use this for a plot of land?
Yes, as long as you have the corner coordinates in a flat coordinate system such as UTM or local survey grid, and the boundary is made of straight lines. Latitude and longitude in degrees will not work directly, because a degree of longitude is not the same distance as a degree of latitude.
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