What this calculator does
The centroid of a triangle is the point where its three medians meet, and it is also the triangle's centre of mass: the single point where a flat, evenly weighted triangular shape would balance perfectly on a pin. The centroid formula finds it directly from the coordinates of the three corners, without needing to draw a single median.
The centroid equation is simply the average of the three vertices: add up the three x-coordinates and divide by three for the centroid's x-coordinate, then do the same with the three y-coordinates. It works for any triangle, however irregular, as long as the three points are not sitting on one straight line.
The formula
The centroid formula averages each coordinate separately: centroid x = (x₁ + x₂ + x₃) / 3, and centroid y = (y₁ + y₂ + y₃) / 3. This calculator also reports the triangle's area from the same three vertices, using the shoelace formula, mainly to confirm the points genuinely form a triangle rather than falling on a single line.
| Term | Meaning |
|---|---|
| Vertex | One of the three corner points of the triangle, given as an (x, y) coordinate. |
| Centroid | The average position of the three vertices; also the triangle's centre of mass. |
| Median | A line from one vertex to the midpoint of the opposite side; all three medians cross at the centroid. |
The inputs explained
| Field | What to enter |
|---|---|
| Vertex A: x | The x-coordinate of the first vertex. |
| Vertex A: y | The y-coordinate of the first vertex. |
| Vertex B: x | The x-coordinate of the second vertex. |
| Vertex B: y | The y-coordinate of the second vertex. |
| Vertex C: x | The x-coordinate of the third vertex. |
| Vertex C: y | The y-coordinate of the third vertex. |
When to use it
A geometry problem giving three coordinates
A textbook question that states three vertex coordinates and asks for the centroid is solved directly here, without needing to plot the medians by hand.
Finding the balance point of a triangular shape
In design, engineering or fabrication, the centroid marks where a flat triangular panel would balance, which matters for anything from mounting points to weight distribution.
Checking a hand calculation
After averaging the coordinates by hand, entering the same three vertices here confirms the arithmetic before it feeds into a larger problem.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the centroid moves as one vertex shifts
A base fixed between (0,0) and (6,0), with the third vertex moving further from the base.
| Third vertex y-coordinate | Centroid | Triangle area |
|---|---|---|
| 2 | (3.000, 0.6667) | 6.000 |
| 4 | (3.000, 1.333) | 12.000 |
| 6 | (3.000, 2.000) | 18.000 |
| 8 | (3.000, 2.667) | 24.000 |
| 10 | (3.000, 3.333) | 30.000 |
| 12 | (3.000, 4.000) | 36.000 |
Centroid of a right-angled triangle at the origin
A right-angled triangle with one vertex at the origin and the other leg length increasing.
| Third vertex x-coordinate | Centroid | Triangle area |
|---|---|---|
| 3 | (1.000, 4.000) | 9.000 |
| 6 | (2.000, 4.000) | 18.000 |
| 9 | (3.000, 4.000) | 27.000 |
| 12 | (4.000, 4.000) | 36.000 |
| 15 | (5.000, 4.000) | 45.000 |
Questions
What is the centroid formula for a triangle?
Centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3): average the three x-coordinates for the centroid's x-coordinate, and average the three y-coordinates for its y-coordinate.
Is the centroid the same as the centre of a triangle?
It is one of several points called a triangle's "centre", alongside the circumcentre, incentre and orthocentre. The centroid specifically is the centre of mass, the average of the vertices, and the point where the three medians intersect.
Why does the centroid formula just average the coordinates?
Each median splits the opposite side in half and meets the other medians at a point exactly two thirds of the way from each vertex to the midpoint of the opposite side. Working through that ratio algebraically for all three medians shows they all land on the simple average of the three vertices.
Does the centroid always sit inside the triangle?
Yes, for any valid triangle the centroid always lies strictly inside it, unlike the circumcentre or orthocentre, which can fall outside the triangle for very obtuse shapes.
For the incentre and the largest circle that fits inside a triangle, see the triangle incenter calculator. For the area and angles of a general triangle from its three vertices, see the scalene triangle calculator.