Centroid Calculator

Find the centroid (geometric center) of a triangle from its three vertex coordinates, plus the triangle's area.

Quick Facts

Formula
Centroid = average of the 3 vertices
x-bar = (x1+x2+x3)/3, y-bar = (y1+y2+y3)/3 — the point where the medians meet.

Your Results

Calculated
Centroid x-bar
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Average of x1, x2, x3
Centroid y-bar
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Average of y1, y2, y3
Triangle area
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Shoelace formula
Avg. distance to vertices
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Centroid to each vertex, averaged

Ready

Enter the three vertex coordinates, then press Calculate.

How the Centroid Calculator works

The centroid of a triangle is its geometric center — the point where the three medians (the lines from each vertex to the midpoint of the opposite side) intersect. It is also the triangle's balance point: if the triangle were cut from a sheet of uniform material, it would balance perfectly on a pin placed at the centroid.

Formula and method

Given three vertices (x1, y1), (x2, y2), and (x3, y3), the centroid coordinates are simply the average of the vertex coordinates:

  • x-bar = (x1 + x2 + x3) / 3
  • y-bar = (y1 + y2 + y3) / 3

The calculator also reports the triangle's area using the shoelace formula, Area = |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)| / 2, and the average straight-line distance from the centroid to each of the three vertices, computed with the standard distance formula √((x − x1)² + (y − y1)²) for each vertex and then averaged.

Common sources of error

  • Mixed-up coordinates: double-check that each x pairs with the correct y for that vertex — swapping a coordinate between two vertices changes the shape entirely.
  • Collinear points: if the three points lie on a straight line, the "triangle" has zero area; the centroid formula still returns a valid average point, but it no longer represents a triangle's balance point.
  • Rounding early: enter full-precision coordinates rather than rounded ones, especially if you plan to feed the centroid into a further calculation.

Checking your result

The centroid should always land between the minimum and maximum x-value of the vertices, and likewise between the minimum and maximum y-value — it can never fall outside that bounding box. For a valid (non-degenerate) triangle, the centroid also always lies strictly inside the triangle itself, roughly two-thirds of the way along each median from its vertex.

Applications

Centroids are used in engineering and physics to find the center of mass of uniform flat shapes, in computer graphics to position labels or pivot points on polygons, and in geometry to prove properties about medians and triangle balance. For shapes with more than three vertices, the same averaging idea extends to an area-weighted centroid formula for general polygons.

Frequently Asked Questions

What is the formula for the centroid of a triangle?
The centroid (x-bar, y-bar) of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) is the average of the three vertices: x-bar = (x1+x2+x3)/3 and y-bar = (y1+y2+y3)/3. This is also the point where the triangle's three medians intersect.
How is triangle area calculated from the vertices?
Given vertices (x1,y1), (x2,y2), and (x3,y3), the shoelace formula gives the area as half the absolute value of x1(y2-y3) + x2(y3-y1) + x3(y1-y2). An area of zero means the three points are collinear and do not form a triangle.
Is the centroid the same as the circumcenter or incenter?
No. The centroid is the average of the vertex coordinates and the intersection of the medians. The circumcenter (equidistant from all vertices) and incenter (center of the inscribed circle) are computed differently and only coincide with the centroid in an equilateral triangle.
Does the centroid have to lie inside the triangle?
Yes, for any non-degenerate triangle the centroid always lies inside it, roughly two-thirds of the way from each vertex to the midpoint of the opposite side. This is unlike the circumcenter, which can fall outside an obtuse triangle.

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