Centroid of a Triangle
Find the centroid of any triangle from its three vertex coordinates. Free online geometry calculator computes the geometric center using the average of x and y coordinates with instant results for students, teachers, and professionals.
About This Calculator
The centroid of a triangle is the point where all three medians intersect. It is also known as the geometric center or the center of mass of a uniform triangular lamina. This calculator computes the centroid from the coordinates of the three vertices using the formula Cx = (x1 + x2 + x3) / 3 and Cy = (y1 + y2 + y3) / 3.
This tool is ideal for students learning coordinate geometry, teachers preparing classroom examples, engineers working on structural analysis, and anyone needing to find the centroid quickly without manual calculations. Simply enter the x and y coordinates for each vertex and click calculate to get the centroid coordinates rounded to four decimal places.
The centroid has several important properties: it always lies inside the triangle, it divides each median in a 2:1 ratio (with the longer segment from the vertex), and it is the point that minimizes the sum of squared distances to the vertices. Unlike the circumcenter or orthocenter, the centroid exists for every triangle and is always inside the triangle regardless of whether the triangle is acute, right, or obtuse.
Frequently Asked Questions
How do you find the centroid of a triangle?
The centroid of a triangle is found by averaging the x-coordinates and y-coordinates of the three vertices. The formula is Cx = (x1 + x2 + x3) / 3 and Cy = (y1 + y2 + y3) / 3. Simply enter the coordinates of each vertex into the calculator and it computes the geometric center instantly.
Is the centroid the same as the center of mass?
Yes, for a uniform density triangular lamina, the centroid coincides with the center of mass and the center of gravity. The centroid is the point where the three medians intersect and it is always located inside the triangle regardless of the triangle type.
What is the centroid formula for a triangle?
The centroid formula for a triangle with vertices A(x1,y1), B(x2,y2), and C(x3,y3) is: Cx = (x1 + x2 + x3) / 3 and Cy = (y1 + y2 + y3) / 3. The centroid divides each median in a 2:1 ratio, with the longer segment being from the vertex to the centroid.
Can the centroid be outside the triangle?
No, the centroid of a triangle is always inside the triangle. Unlike the orthocenter or circumcenter which can lie outside for obtuse triangles, the centroid always lies inside the triangle for all triangle types -- acute, right, and obtuse.
What are the properties of the triangle centroid?
The centroid has several key properties: it is the intersection point of the three medians, it divides each median in a 2:1 ratio, it is the center of mass of a uniform triangular lamina, and it always lies inside the triangle. The centroid is also the point that minimizes the sum of squared distances to the vertices.
Is this centroid calculator free to use?
Yes, all calculators on Calculy including this centroid of a triangle calculator are completely free to use. No registration or payment is required. You can share your results by copying the URL which preserves your input values.