Education
Centroid Calculator
Last updated: June 19, 2026
A centroid calculator is a geometry and physics utility that determines the geometric center (centroid) or center of mass of a shape or a set of coordinates. For discrete points, the calculator computes the arithmetic mean of the coordinates. For continuous two-dimensional regions, it applies integral calculus to divide the first moments of area by the total area. It supports common polygons, triangles, and custom coordinate inputs. Engineering and physics students use this tool to calculate balance points and analyze structural distributions.
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.
Quick Answer
Find the centroid of 2D geometric shapes or sets of points. Enter coordinates to calculate the exact center of area or center of mass.
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.
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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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