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Triangle Centroid Calculator

Find the centroid from the vertices.

How it is calculated

The centroid is where the three medians of a triangle meet, and its coordinates are simply the averages of the vertex coordinates: Cx = (x1 + x2 + x3) / 3 and Cy = (y1 + y2 + y3) / 3. Coordinates can be negative, zero or decimal, and the order of the three vertices does not matter. The results come with four decimals in the same unit as the inputs. The tool does not check that the points form a real triangle: three collinear points, or three identical points, still return their average. Leaving any of the six fields empty gives no result.

Worked example

Take the vertices (1, 2), (7, 4) and (4, 9). Add the x-values: 1 + 7 + 4 = 12, and divide by 3 to get 4. Add the y-values: 2 + 4 + 9 = 15, and divide by 3 to get 5. The calculator shows the centroid at (4.0000, 5.0000). A flat plate cut in this triangle would balance on a pin at that point.

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Common mistakes

  • Averaging only two vertices, or taking the midpoint of one side. That gives a side midpoint, not the centroid.
  • Confusing the centroid with the circumcentre or the orthocentre. They coincide only for an equilateral triangle.
  • Entering the x and y of one vertex into the wrong fields, which mixes up the pairs. Keep each (x, y) pair together.
  • Mixing units between coordinates, such as x in metres and y in centimetres.
  • Expecting the tool to warn about collinear or repeated points. It will still return their average.

About this calculator

The centroid of a triangle — the point where its three medians (lines from each vertex to the midpoint of the opposite side) intersect — is calculated simply by averaging the x-coordinates and y-coordinates of the three vertices. This point always lies inside the triangle and divides each median in a 2:1 ratio from vertex to midpoint.

The centroid represents the triangle's geometric center of mass — if the triangle were a flat, uniform-density plate, it would balance perfectly on a pin placed at the centroid — which makes it essential in engineering statics and structural analysis for locating a shape's center of gravity, in computer graphics for mesh and polygon calculations, and in physics problems involving rotational inertia and balance.

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