Centroid Calculator

The centroid represents the arithmetic mean position of all the points in a figure. For a simple triangle defined by three vertices (x1, y1), (x2, y2), and (x3, y3), the centroid coordinates (xc3, yc3) are found by averaging the respective X and Y coordinates: xc3 = (x1 + x2 + x3) / 3 and yc3 = (y1 + y2 + y3) / 3. For more complex polygons with n vertices, the calculation utilizes the shoelace formula framework to compute both the area and the weighted coordinate moments, ensuring absolute mathematical precision across multi-sided figures.

Let us find the centroid of a right-angled triangle with vertices at coordinates (0, 0), (0, 3), and (3, 3). First, identify the coordinates: (x1, y1) = (0, 0), (x2, y2) = (0, 3), and (x3, y3) = (3, 3). Next, apply the triangle centroid formulas. For the X-coordinate: xc3 = (0 + 0 + 3) / 3 = 3 / 3 = 1. For the Y-coordinate: yc3 = (0 + 3 + 3) / 3 = 6 / 3 = 2. Thus, the exact centroid of this triangle is located at the coordinate point (1, 2).

Always enter your polygon vertices in a consistent sequential order—either clockwise or counterclockwise—around the perimeter to ensure accurate area and centroid computations. Double-check your positive and negative signs when inputting coordinates into the Cartesian grid. If you are calculating the center of a non-standard shape, ensure all vertex values are measured from the exact same origin point (0, 0).