Education
Centroid Calculator
Pick triangle mode or point-set mode. Enter the vertices or the list of points; the calculator returns the centroid coordinate (x-bar, y-bar) along with a step note.
Mode
Vertex 1
Vertex 2
Vertex 3
Formula
For a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), the centroid is at ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3). For a general point set, take the mean of all x values and all y values separately.
This is the centroid of a discrete set of points (equally weighted). For an arbitrary 2D region or polygon, the centroid formula uses signed areas.
Triangle centroid
(3, 1.333333)
x̄ = 3; ȳ = 1.333333
The centroid is the average position of the points. For a triangle it is also the intersection of the three medians, and it sits two-thirds of the way along each median from the vertex.
Examples
Triangle (0,0), (6,0), (3,4)
Centroid (3, 4/3) ≈ (3, 1.333)
Triangle (1,1), (4,1), (1,5)
Centroid (2, 7/3) ≈ (2, 2.333)
Points (1,2), (3,4), (5,6), (7,8)
Centroid (4, 5)
Single point (3,3)
Centroid (3, 3)
How it works
The centroid is the mean of the coordinates. Average the x values; average the y values; the resulting pair is the centroid. For a triangle, this lands at the intersection of the three medians.
Triangle · ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3)
Point set · (Σ x / n, Σ y / n)
Related geometry calculators
- Midpoint formula calculator for the midpoint of just two points (a special case of the centroid).
- Triangle area calculator for the area enclosed by three vertices (Heron, base/height, or SAS).
- Distance formula calculator for the distance between any two points (handy when checking medians).
- Area calculator for the area of 2D shapes (the centroid of a polygon depends on its area).
- All education calculators.
Frequently asked questions
The centroid is the average position of a set of points (or the geometric center of a shape). For a discrete point set, it is the simple mean of x values and y values. For a triangle, it is also the intersection of the medians.
Centroid = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3). The same average over the three vertices works for any triangle regardless of shape.
For a general polygon (not just a triangle), the centroid uses a signed-area formula that weights each edge by its midpoint. That is not implemented here; this calculator handles triangles and discrete point sets, which are the most common use cases.
Switch the mode to 'Point set' and type one point per line, x and y comma-separated. The calculator parses any number of points and returns the average.
For equal-mass points, they are the same. The centroid is the mean position; the center of mass would weight each point by its mass. This page assumes equal weighting (a geometric centroid).
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