Math
Centroid Calculator
Find the centroid (geometric center) of a triangle, polygon (shoelace), point set, weighted points, or common shapes like semicircles and quarter circles.
Centroid Calculator
Enter the three vertices. The centroid is the average of the coordinates.
Vertex 1
Vertex 2
Vertex 3
Centroid
(2, 1)
Triangle centroid from three vertices.
Cx
2
Cy
1
Area
9
Frequently Asked Questions about the Centroid Calculator
What is a centroid?
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.
How is the centroid of a triangle calculated?
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.
What is the shoelace formula?
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.
Where is the centroid of a semicircle?
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.
Can I add weights to the points?
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.