Centroid Calculator

The centroid of a triangle is its geometric center, the single point where the three medians intersect. A median connects one vertex to the midpoint of the opposite side. Every triangle has exactly one centroid, and that point always lies inside the triangle. This calculator finds its Cartesian coordinates from the x and y coordinates of the triangle's three vertices. For vertices (x1, y1), (x2, y2), and (x3, y3), the centroid coordinates are the arithmetic means of the corresponding coordinates. Add the three x values and divide by three to obtain the centroid's x coordinate. Add the three y values and divide by three to obtain its y coordinate. This simple average works for acute, right, obtuse, scalene, isosceles, and equilateral triangles. The centroid is also called the barycenter or center of area for a uniform triangular lamina. If a flat triangular sheet has constant density and thickness, it balances at the centroid. The point divides every median in a two-to-one ratio: the segment from a vertex to the centroid is twice as long as the segment from the centroid to the opposite side's midpoint. That property makes the centroid useful in geometric constructions and proofs. Coordinate centroids appear in computer graphics, engineering, architecture, surveying, mechanics, and geographic analysis. Designers use them to position labels, rotations, and supports. Engineers use them when beginning center-of-mass and moment calculations. In a physical application, remember that the coordinate average represents center of mass only when density is uniform. Unequal point masses require a weighted average instead. Coordinates may be positive, negative, zero, whole numbers, or decimals, but all six values must use the same coordinate system and scale. Enter vertices in any order because averaging produces the same centroid regardless of ordering. If all three points lie on one line, the arithmetic mean is still defined, although the points describe a degenerate triangle with zero area. Results are rounded only for display, preserving useful precision for practical calculations.