Centroid Calculator
Free Centroid Calculator for 2d geometry. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Formula
Centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
For a triangle, the centroid is the average of the three vertex coordinates. For a quadrilateral, the centroid is the area-weighted average of the centroids of two constituent triangles formed by a diagonal.
Worked Examples
Example 1: Triangle Centroid
Problem: Find the centroid of a triangle with vertices A(2, 4), B(8, 2), and C(5, 10).
Solution: Centroid x = (2 + 8 + 5) / 3 = 15 / 3 = 5.0\nCentroid y = (4 + 2 + 10) / 3 = 16 / 3 = 5.333\nArea = |2(2-10) + 8(10-4) + 5(4-2)| / 2\n = |(-16) + 48 + 10| / 2 = 42 / 2 = 21\nPerimeter: AB=6.32, BC=8.54, CA=6.71 → Total=21.58
Result: Centroid = (5.0, 5.333) | Area = 21 sq units | Perimeter = 21.58
Example 2: Quadrilateral Centroid
Problem: Find the centroid of a quadrilateral with vertices A(0,0), B(6,0), C(8,5), D(2,7).
Solution: Triangle 1 (A,B,C): Area = 15, Centroid = (4.667, 1.667)\nTriangle 2 (A,C,D): Area = 18, Centroid = (3.333, 4.0)\nWeighted Cx = (15×4.667 + 18×3.333)/(15+18) = 3.939\nWeighted Cy = (15×1.667 + 18×4.0)/(15+18) = 2.939\nTotal area = 33 sq units
Result: Centroid = (3.94, 2.94) | Area = 33 sq units
Frequently Asked Questions
What is the centroid of a shape?
The centroid is the geometric center of a shape, often described as the point where the shape would balance perfectly if placed on a pin. For a triangle, the centroid is the intersection of the three medians, which are lines drawn from each vertex to the midpoint of the opposite side. The centroid divides each median in a 2:1 ratio, with the longer segment being closer to the vertex. Mathematically, for a triangle with vertices (x1,y1), (x2,y2), and (x3,y3), the centroid is at ((x1+x2+x3)/3, (y1+y2+y3)/3). The centroid is one of four major triangle centers, along with the circumcenter, incenter, and orthocenter. For uniform density shapes, the centroid coincides with the center of mass. In engineering, centroids are essential for calculating bending stresses, deflections, and moments of inertia in structural members.
How is the centroid of a quadrilateral calculated?
Unlike triangles, there is no simple averaging formula for the centroid of a general quadrilateral. The correct method involves dividing the quadrilateral into two triangles by drawing a diagonal, computing the centroid of each triangle, then taking the weighted average of these two centroids where the weights are the areas of the respective triangles. The formula is: Cx = (A1 * Cx1 + A2 * Cx2) / (A1 + A2), and similarly for Cy, where A1 and A2 are the areas of the two triangles and (Cx1, Cy1) and (Cx2, Cy2) are their centroids. Simply averaging the four vertices gives the centroid only for parallelograms and is an approximation for other quadrilaterals. For convex quadrilaterals, either diagonal can be used to split into triangles and the result will be the same. For concave (non-convex) quadrilaterals, care must be taken to choose a diagonal that stays inside the shape.
What is the difference between centroid, center of mass, and center of gravity?
These three concepts are closely related but technically distinct. The centroid is a purely geometric property determined by the shape's outline, calculated without considering mass or material properties. The center of mass is the average position of mass in a body, accounting for density variations. The center of gravity is the point where gravitational force effectively acts on a body. For a uniform density object in a uniform gravitational field, all three coincide at the same point. They differ when the density varies across the object or when the gravitational field is non-uniform. For example, a wooden board with a metal weight attached on one end would have its center of mass shifted toward the metal, while its centroid remains at the geometric center. In most engineering and mathematics applications at human scales, the gravitational field is effectively uniform, so center of mass and center of gravity are treated as identical.
How accurate are the results from Centroid Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Does Centroid Calculator work offline?
Once the page is loaded, the calculation logic runs entirely in your browser. If you have already opened the page, most calculators will continue to work even if your internet connection is lost, since no server requests are needed for computation.
What formula does Centroid Calculator use?
The formula used is described in the Formula section on this page. It is based on widely accepted standards in the relevant field. If you need a specific reference or citation, the References section provides links to authoritative sources.