A numerical walkthrough
Vertices (0,0),(6,0),(3,9) have centroid (3,3).
Math calculator
Triangle Centroid Calculator
Average three vertices to locate the intersection of a triangle’s medians. A checkable formula accompanies centroid coordinates instead of leaving an unexplained number.
Triangle Centroid inputs
Known quantities
The idea behind the result
The centroid is ((x₁+x₂+x₃)/3,(y₁+y₂+y₃)/3) and divides every median in a 2:1 ratio from vertex to midpoint.
The calculation treats vertex 1 x, vertex 1 y, vertex 2 x, vertex 2 y, vertex 3 x and vertex 3 y according to their labels, not as interchangeable values in triangle centroid.
Working through Triangle Centroid on paper
Average the three x-coordinates and separately average the three y-coordinates. To continue from Triangle Centroid, try triangle area.
Details to check in Triangle Centroid
Collinear points still have an average coordinate but do not define a nondegenerate triangle; interpret that case carefully. If the Triangle Centroid assumptions do not fit, consider side midpoint.
Save extra digits internally if centroid coordinates will become an input to another calculation.
When the result is useful
It is the balance point of a uniform triangular lamina and a common reference in meshes and coordinate geometry.
How the inputs shape the output
A field name is part of the formula. Match the problem's quantities to vertex 1 x, vertex 1 y, vertex 2 x, vertex 2 y, vertex 3 x and vertex 3 y, then check that they share the scale assumed by the triangle centroid relationship.
Moving one vertex by a vector moves the centroid by one third of that vector. This is a stronger check than judging the answer only by how many decimal places it shows.
Match the reported precision to vertex 1 x, vertex 1 y, vertex 2 x, vertex 2 y, vertex 3 x and vertex 3 y, not to the number of digits the browser can display. Preserve an exact form when it communicates the triangle centroid structure more clearly than a decimal.
The midpoint averages two points; the centroid averages all three triangle vertices. The formula panel makes the chosen definition explicit.
The minimum audit trail is short: vertex 1 x, vertex 1 y, vertex 2 x, vertex 2 y, vertex 3 x and vertex 3 y, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
Questions about Triangle Centroid
Is the centroid always inside?
Yes for a nondegenerate triangle.
Does vertex order matter?
No.
What ratio does it make on a median?
2:1 from the vertex.
Is it the same as the incenter?
Only for special symmetric triangles.