Centroid of a Triangle Calculator
This centroid of a triangle calculator takes the coordinates of the three vertices. Example: A(1|2), B(7|2) and C(4|8) have the centroid S(4|4) and an area of 18.
Example: x-coordinate of A 1 · y-coordinate of A 2 · x-coordinate of B 7 · y-coordinate of B 2 · x-coordinate of C 4 · y-coordinate of C 8 → Result: Centroid S(4 | 4), area A = 18. Source: Wikipedia – Centroid. Updated: .
How it is calculated
The centroid of a triangle is the point where its three medians meet; a flat triangular plate would balance exactly there. It always lies inside the triangle.
Formula and steps
For the vertices A(x₁|y₁), B(x₂|y₂) and C(x₃|y₃) the centroid is S = ((x₁ + x₂ + x₃) / 3 | (y₁ + y₂ + y₃) / 3). So you only need to average the coordinates. A median joins a vertex to the midpoint of the opposite side, and the centroid divides it in the ratio 2 : 1 with the longer part at the vertex. The area comes from the shoelace formula.
Worked example
For A(1|2), B(7|2) and C(4|8) we get xₛ = (1 + 7 + 4) / 3 = 4 and yₛ = (2 + 2 + 8) / 3 = 4, so S(4|4). The area is |1 · (2 − 8) + 7 · (8 − 2) + 4 · (2 − 2)| / 2 = |−6 + 42 + 0| / 2 = 18. Side AB has the midpoint (4|2); the median from C to it is 6 long, of which 4 lie between C and S and 2 between S and the midpoint.
- The centroid coincides with the balance point of the three vertices treated as equal masses, but not with the circumcentre.
- In an equilateral triangle the centroid, circumcentre and incentre are the same point.
Frequently asked questions
How do you find the centroid of a triangle?
Average the x-coordinates and the y-coordinates of the three vertices separately: S = ((x₁ + x₂ + x₃) / 3 | (y₁ + y₂ + y₃) / 3).
In what ratio does the centroid divide the medians?
In the ratio 2 : 1. The distance to the vertex is twice the distance to the midpoint of the opposite side; in the example 4 to 2.
Is the centroid always inside the triangle?
Yes, in every triangle, obtuse ones included. That is not true for the orthocentre or circumcentre, which can lie outside.
Sources and legal basis
As of: · Editorial policy
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