What is the centroid of a triangle?
The centroid (often labelled G) is the point where the three medians of a triangle intersect. A median connects a vertex to the midpoint of the opposite side. The centroid is also the triangle's center of mass or balance point: a uniform triangular plate balances perfectly on a pin placed at G. It always lies inside the triangle, and it divides each median in a 2:1 ratio measured from the vertex.
How to use this calculator
Enter the (x, y) coordinates of the three corners — Vertex A, Vertex B and Vertex C — in any order. The calculator returns the centroid coordinates instantly. Coordinates can be negative, decimal, or zero, and the order of the vertices does not affect the result.
The formula explained
The centroid is simply the average of the three vertices:
$$\left( C_x, C_y \right) = \left( \frac{\text{x}_1 + \text{x}_2 + \text{x}_3}{3},\ \frac{\text{y}_1 + \text{y}_2 + \text{y}_3}{3} \right)$$You add the three x-values and divide by 3 to get the centroid's x-coordinate, then do the same for the y-values. No square roots or trigonometry are required.
Worked example
Take a triangle with vertices A(0, 0), B(6, 0) and C(3, 6).
$$G_x = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3$$$$G_y = \frac{0 + 0 + 6}{3} = \frac{6}{3} = 2$$So the centroid is at \((3, 2)\).
FAQ
Is the centroid the same as the circumcenter or incenter? No. The centroid is the average of the vertices. The circumcenter (center of the circumscribed circle) and incenter (center of the inscribed circle) are generally different points unless the triangle is equilateral.
Can the centroid lie outside the triangle? Never — the centroid of any triangle always lies inside it.
Does the order I enter the vertices matter? No. Because addition is commutative, swapping the vertices gives the same centroid.
