How this instrument works
The centroid of a triangle is the point where its three medians — the lines from each corner to the midpoint of the opposite side — all cross. Remarkably, that single crossing point is found with plain averaging: take the mean of the three vertices' x-coordinates for the centroid's x-coordinate, and separately the mean of the three y-coordinates for its y-coordinate. No line-intersection algebra is actually needed, even though that's how the point is originally defined.
Physically, the centroid is a triangle's center of mass — a flat plate cut in the triangle's exact shape, of uniform thickness and density, would balance perfectly on a pin placed right at that point. This is a genuinely different point from a triangle's circumcenter (equidistant from all three corners) or its orthocenter (where the altitudes meet); the three coincide only in an equilateral triangle, but sit at three distinct locations in every other triangle shape.
A classic ratio worth knowing: the centroid always sits exactly two-thirds of the way along each median from its corner to the midpoint of the opposite side, and one-third of the way from that midpoint. This 2:1 split is a fixed property of every triangle, regardless of its shape, and is provable directly from the coordinate-averaging formula itself.
- Enter the first corner's coordinates into the Vertex 1: x and Vertex 1: y fields.
- Enter the second corner's coordinates into the Vertex 2: x and Vertex 2: y fields.
- Enter the third corner's coordinates into the Vertex 3: x and Vertex 3: y fields.
- Read Centroid: x and Centroid: y: the sheet averages each coordinate independently.
Worked example — vertices (0,0), (6,0), (3,6)
A triangle has corners at (0,0), (6,0), and (3,6). The centroid sits at ((0+6+3)⁄3, (0+0+6)⁄3) = (3, 2) — a plain average of the three x-values and, separately, the three y-values, landing well inside the triangle's interior.
A physical model of this same triangle, cut from cardboard of uniform thickness, would balance exactly on a pin at (3, 2) — not at the geometric middle of any single side, but at the point every median passes through, two-thirds of the way from each corner toward the midpoint of the opposite side.
Questions
What is a triangle's centroid?
It's the point where a triangle's three medians (the lines from each corner to the midpoint of the opposite side) all intersect — and physically, it's the triangle's center of mass, the point a uniform triangular plate would balance on.
How do you find the centroid from the three vertices?
Average the x-coordinates of the three vertices for the centroid's x-coordinate, and separately average the three y-coordinates for its y-coordinate. This plain averaging always gives the exact point where the three medians cross, with no line-intersection algebra required.
Is the centroid the same as the circumcenter?
No — the circumcenter is equidistant from all three corners (the center of the circle passing through them), while the centroid is the balance point found by averaging the vertices. The two coincide only in an equilateral triangle; in every other triangle shape, they sit at different locations.
What is the 2:1 ratio along a median?
The centroid always divides each median in a 2:1 ratio, sitting two-thirds of the way from a corner to the midpoint of the opposite side, and one-third of the way back from that midpoint. This fixed ratio holds for every triangle, regardless of its shape.
Does the centroid always sit inside the triangle?
Yes — unlike the orthocenter or circumcenter, which can fall outside an obtuse triangle, the centroid is always inside the triangle, for every possible triangle shape, since it's a weighted average of three points that always sits within their bounding region.