Centroid Calculator (Triangle)
Enter the three vertices of a triangle to find its centroid.
The centroid of a triangle is the point where its three medians meet, and it is simply the average of the three vertices. Add the x-coordinates and divide by three for the centroid's x; do the same with the y-coordinates. It is the triangle's balance point.
How to Use This Calculator
Enter the x and y coordinates of each of the triangle's three corners. The calculator returns the centroid as an (x, y) point and shows how it averaged the coordinates.
The centroid is the triangle's center of mass — the point where it would balance on a pin if it were a uniform flat shape. It always lies inside the triangle.
How the Calculation Works
The centroid is where the three medians of a triangle intersect — each median runs from a vertex to the midpoint of the opposite side. A useful fact is that this intersection is exactly the average of the three vertex coordinates.
So you find it by adding the three x-values and dividing by three, then adding the three y-values and dividing by three. No angle or side length is needed — only the coordinates.
centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
Worked Example
Take vertices (0, 0), (6, 0) and (0, 9). The centroid's x is (0 + 6 + 0) ÷ 3 = 2, and its y is (0 + 0 + 9) ÷ 3 = 3.
So the centroid is (2, 3), which sits inside the triangle, two-thirds of the way along each median from its vertex.
Assumptions and Limitations
- Finds the centroid of a triangle from three vertices
- The centroid is the average of the coordinates — no side lengths needed
- Works in the standard 2D coordinate plane
- For polygons with more sides, the centroid formula is different
When to Use a Centroid Calculator
Geometry and engineering students use this to find a triangle's centroid — its balance point — from the three vertices. It comes up in coursework, in physics and statics problems, and in computer graphics. Enter the three corner coordinates to get the centroid, with the averaging shown.
Related Calculators
Diagonal Calculator
Find the diagonal of a rectangle, square or box.
Open calculator →Octagon Calculator
Area, perimeter and width of a regular octagon from its side.
Open calculator →Reflection Calculator (Coordinate)
Reflect a point over an axis or line, with the rule shown.
Open calculator →Dilation Calculator (Coordinate)
Scale a point from a center by a factor, with the rule shown.
Open calculator →Frequently Asked Questions
What is the centroid of a triangle?
It is the point where the three medians meet — the triangle's balance point or center of mass. It equals the average of the three vertex coordinates and always lies inside the triangle.
How do you find the centroid from vertices?
Average the coordinates: add the three x-values and divide by three for the centroid's x, and do the same for y. For (0,0), (6,0), (0,9) the centroid is (2, 3).
Is the centroid the same as the center?
For a triangle, the centroid is one of several centers (others include the circumcenter and incenter). It is specifically the center of mass and the intersection of the medians.
Does the centroid always lie inside the triangle?
Yes. Because it is the average of the three corners, the centroid is always strictly inside the triangle, unlike some other triangle centers which can fall outside.
What is a median of a triangle?
A median is a line from a vertex to the midpoint of the opposite side. A triangle has three, and they all cross at the centroid, which divides each median in a 2:1 ratio.
Looking for a different calculator?
CalculatorPlus has free tools across finance, construction, math, health and more — each one showing the formula and a worked example.