Centroid Calculator

The centroid is the geometric center of a shape: the average position of all its points. For a uniform-density object it is also the center of mass. For a triangle it is one third of the way from each side to the opposite vertex, and the three medians intersect at it.

Average the three vertex coordinates: Cx = (x1 + x2 + x3) / 3, Cy = (y1 + y2 + y3) / 3. The result is always inside the triangle. This calculator does the same for a free-form polygon using the shoelace formula, which weights each edge contribution by the signed area.

It computes the signed area of a simple polygon as half the sum of cross products of consecutive vertex pairs. The centroid uses the same cross-product terms, weighted by the average x or y coordinate of each edge. The polygon must not self-intersect for the result to be meaningful.

For a semicircle of radius r sitting on its diameter, the centroid is at height 4r / (3 pi) above the flat edge, about 0.4244 r. A quarter circle in the first quadrant has its centroid at (4r / (3 pi), 4r / (3 pi)). These are standard closed-form results from integration.

Yes. The weighted-points mode multiplies each point by its weight before averaging, giving you a center of mass when the weights represent mass. Negative weights work (useful for holes in composite shapes) as long as the total weight does not sum to zero.