Rectangle
For (0,0), (6,0), (6,4), (0,4), the centroid is (3,2) and the area is 24 square units.
Polygon Centroid Calculator from Coordinates
Ordered polygon coordinates
Accepted: 3, 4, 3 4, or (3, 4). Scientific notation is supported. Follow the boundary in either direction.
Ready to calculate the five-vertex example.
Centroid result
Centroid x, Cx—
Centroid y, Cy—
Polygon area—
Vertex order—
The blue cross marks the area centroid after a valid calculation.
Normalized cross sum ΣΔ—
Normalized x numerator—
Normalized y numerator—
Calculation notes
Calculate a valid polygon to see the substituted centroid relationships.
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How to Find a Polygon Centroid from Coordinates
The centroid of a uniform polygon is its area-weighted geometric center. Unlike the simple average of the vertices, it accounts for all of the enclosed surface. This makes it useful as a planar balance point, a label position, a CAD reference, or a center for shape transformations.
- List at least three unique vertices in consecutive order around the boundary.
- Paste one x,y pair per line. Repeating the first point at the end is optional.
- Select the coordinate unit and the display precision.
- Choose Calculate Centroid and resolve any boundary warning.
- Read Cx and Cy in the same linear unit as the coordinates. Area is shown in the corresponding square unit.
Formula: let Δᵢ = xᵢyᵢ₊₁ − xᵢ₊₁yᵢ and close the polygon with vertex n+1 = vertex 1. Then A = ½ΣΔᵢ, Cx = Σ(xᵢ+xᵢ₊₁)Δᵢ ÷ (3ΣΔᵢ), and Cy = Σ(yᵢ+yᵢ₊₁)Δᵢ ÷ (3ΣΔᵢ).
The sign of A depends on winding direction; reversing all vertices changes the signs of the sums but not the centroid or absolute area. Internally, this calculator translates and scales the coordinates before applying the formula. That reduces avoidable floating-point overflow and loss of precision for very large or very small coordinate magnitudes.
Assumptions and Validation
Simple planar polygon
The boundary must not cross itself. Convex and concave shapes are supported; holes and multiple rings are not.
Uniform area
The result is the geometric centroid of the filled region, equivalent to a center of mass only when density and thickness are uniform.
Numerical limits
Up to 5,000 vertices and finite coordinates with absolute value no greater than 10150 are accepted. Nearly zero-area outlines are rejected.
Last reviewed: July 30, 2026. Formula, winding invariance, concave behavior, duplicate handling, crossing detection, and representative examples were checked for calculator-specific accuracy.
Polygon Centroid Examples
Right triangle
For (0,0), (6,0), (0,3), the centroid is (2,1), matching the average of the triangle's three vertices.
Concave polygon
For the loaded arrow-like example, the area is 18 m² and the centroid is (3, 1.555556) m.
Polygon Centroid: Frequently Asked Questions
What is the centroid of a polygon?
It is the area-weighted geometric center of a uniform, flat polygon—the balance point of an ideal lamina with the same shape.
Do the polygon coordinates need to be in order?
Yes. List vertices consecutively around the boundary, clockwise or counterclockwise. Unordered points can make crossing edges or describe a different polygon.
Does the centroid formula work for concave polygons?
Yes. Convex and concave simple polygons work when the vertices follow a single boundary without self-intersections.
Can the centroid lie outside a polygon?
A convex polygon's centroid lies inside it. A sufficiently concave polygon can have its area centroid outside the filled region.
Should I repeat the first coordinate at the end?
It is optional. The calculator closes the last vertex to the first automatically and ignores one repeated closing coordinate.
Can I use latitude and longitude coordinates?
Not directly for an accurate geographic centroid. Project longitude and latitude to an appropriate planar coordinate reference system first.
Is a polygon centroid the same as the average of its vertices?
Usually not. A vertex mean depends on how many boundary points were sampled; the polygon centroid weights the complete enclosed area.
Are my coordinates uploaded?
No. All input handling and calculations remain in this browser page.