What Is the Centroid of a Triangle?
The centroid of a triangle is the point where its three medians intersect. A median is the line segment connecting a vertex to the midpoint of the opposite side. Every triangle has exactly three medians, and they always meet at a single point: the centroid.
The centroid has a special physical meaning: it is the triangle's centre of mass. If you were to cut a triangle out of a uniform flat material (such as cardboard or sheet metal), it would balance perfectly on a pin placed at the centroid. Engineers and designers use this property when calculating the behaviour of triangular structural elements.
The Centroid Formula
For a triangle with vertices at A(x1, y1), B(x2, y2) and C(x3, y3), the centroid G has coordinates:
Cx = (x1 + x2 + x3) / 3
Cy = (y1 + y2 + y3) / 3
In other words, you average the three x-coordinates to find the centroid x, and average the three y-coordinates to find the centroid y. This formula works for any triangle regardless of its shape or orientation.
Worked Example
Consider a triangle with vertices at A(0, 0), B(6, 0) and C(3, 6). These are the default values in the calculator above.
| Step | Calculation | Result |
|---|
| Sum of x-coordinates | 0 + 6 + 3 | 9 |
| Centroid x (Cx) | 9 / 3 | 3.00 |
| Sum of y-coordinates | 0 + 0 + 6 | 6 |
| Centroid y (Cy) | 6 / 3 | 2.00 |
| Centroid | | (3.00, 2.00) |
The centroid of this triangle is at the point (3, 2), which you can verify lies inside the triangle.
Key Properties of the Centroid
- The centroid always lies inside the triangle, regardless of whether the triangle is acute, right-angled, or obtuse.
- The centroid divides each median in the ratio 2:1 from the vertex. This means it sits two-thirds of the way along each median from the vertex end.
- The three medians divide the triangle into six smaller triangles of equal area.
- For an equilateral triangle, the centroid, circumcentre, incentre, and orthocentre are all the same point.
- The centroid of a composite shape is found by a weighted average of the centroids of its component parts.
Centroid vs Other Triangle Centres
| Centre | Definition | Always inside? |
|---|
| Centroid (G) | Intersection of the three medians; centre of mass | Yes |
| Circumcentre (O) | Centre of the circumscribed circle; equidistant from all vertices | Only for acute triangles |
| Incentre (I) | Centre of the inscribed circle; equidistant from all sides | Yes |
| Orthocentre (H) | Intersection of the three altitudes | Only for acute triangles |
Applications
The centroid appears in many practical contexts:
- Structural engineering: The centroid of a cross-section determines where bending stress is zero (the neutral axis).
- Physics: The centre of mass of a uniform triangular plate equals its centroid.
- Computer graphics: The centroid is used to position labels, calculate collision detection, and determine the orientation of triangular meshes.
- Navigation and surveying: The centroid of a triangulated area gives a representative central point.
Related Calculators
Sources and method: Standard Euclidean geometry. Centroid formula: G = ((x1+x2+x3)/3, (y1+y2+y3)/3). Reference: Coxeter, H.S.M. and Greitzer, S.L., Geometry Revisited, Mathematical Association of America, 1967. Weisstein, Eric W., "Triangle Centroid," MathWorld, Wolfram Research.
This calculator computes the centroid of a triangle from Cartesian coordinates using the standard averaging formula. Coordinates can be any real numbers including negatives and decimals. Results are exact to the precision of your inputs.
