Centroid of a Triangle Calculator
Enter the coordinates of all three vertices to instantly find the centroid (geometric center) of your triangle. You also get the three median lengths, the distance from each vertex to the centroid, the triangle area (via Shoelace formula), the perimeter, and the triangle type. The step-by-step panel shows every calculation with your actual numbers.
Formula
Worked example
For A(0,0), B(6,0), C(3,4): x_G = (0+6+3)/3 = 3, y_G = (0+0+4)/3 = 1.333. Side BC = sqrt((3-6)^2+(4-0)^2) = 5. Median from A = 0.5*sqrt(2*5^2+2*6^2-5^2) = 0.5*sqrt(50+72-25) = 0.5*sqrt(97) = 4.924. Area = |0*(0-4)+6*(4-0)+3*(0-0)|/2 = |0+24+0|/2 = 12.
What is the centroid of a triangle?
The centroid is the point where the three medians of a triangle intersect. A median is a line segment drawn from any vertex to the midpoint of the opposite side. Because every triangle has three vertices, it has three medians, and all three cross at exactly one point: the centroid, usually labelled G. The centroid is also called the geometric center or center of mass of the triangle, because a thin flat triangle of uniform density balances perfectly on a pin placed at G.
How to calculate the centroid coordinates
Place the three vertices at coordinates A(x1, y1), B(x2, y2), and C(x3, y3). The centroid x-coordinate is the arithmetic mean of the three x-values: x_G = (x1 + x2 + x3) / 3. The centroid y-coordinate is the mean of the three y-values: y_G = (y1 + y2 + y3) / 3. This follows directly from the fact that the centroid lies on every median at the point 2/3 of the way from each vertex to the opposite midpoint. For example, with A(0,0), B(6,0), C(3,4), the centroid is ((0+6+3)/3, (0+0+4)/3) = (3, 1.33).
Median lengths and the 2:1 ratio
A key property of the centroid is that it divides every median in a 2:1 ratio measured from the vertex. If the full median from vertex A has length m_a, then the distance from A to G is (2/3)*m_a, and the distance from G to the midpoint of BC is (1/3)*m_a. You can calculate the length of each median using the formula: m_a = (1/2) * sqrt(2*b^2 + 2*c^2 - a^2), where a, b, c are the side lengths opposite to vertices A, B, C respectively. This calculator computes all three median lengths so you can verify the 2:1 ratio and cross-check your geometry.
Area, perimeter, and triangle type
Beyond the centroid coordinates, this calculator also outputs the triangle area (using the Shoelace formula: Area = |x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2)| / 2), the perimeter (sum of the three side lengths), and the triangle type: equilateral if all sides are equal, isosceles if exactly two sides are equal, or scalene if all three sides differ. The centroid always exists and lies strictly inside the triangle regardless of the triangle type.
Key centroid properties at a glance
| Property | Value / Rule |
|---|---|
| Centroid divides each median | 2:1 ratio from vertex to opposite side midpoint |
| Distance from vertex to G | 2/3 of the full median length |
| Distance from midpoint to G | 1/3 of the full median length |
| Sub-triangles formed by G | Three triangles of equal area |
| Equilateral triangle centroid | Coincides with circumcenter, incenter, and orthocenter |
| Median length formula | m_a = (1/2) * sqrt(2b^2 + 2c^2 - a^2) |
| Centroid x-coordinate | x_G = (x1 + x2 + x3) / 3 |
| Centroid y-coordinate | y_G = (y1 + y2 + y3) / 3 |
These properties hold for any triangle and are used to verify centroid calculations.
Frequently asked questions
What is the formula for the centroid of a triangle?
The centroid G of a triangle with vertices A(x1, y1), B(x2, y2), C(x3, y3) is given by x_G = (x1 + x2 + x3) / 3 and y_G = (y1 + y2 + y3) / 3. It is simply the arithmetic mean (average) of the x-coordinates and the y-coordinates separately.
Does the centroid always lie inside the triangle?
Yes. Unlike the orthocenter (which can lie outside for obtuse triangles) or the circumcenter, the centroid always lies strictly inside the triangle regardless of whether it is acute, right, or obtuse. This is because it is the average of the three vertices, which is always a convex combination of points inside the convex hull.
What is the 2:1 ratio property of the centroid?
The centroid divides each median in the ratio 2:1, measured from the vertex to the opposite midpoint. If the median from vertex A has total length m_a, then the centroid lies (2/3)*m_a from A and (1/3)*m_a from the midpoint of BC. This holds for all three medians simultaneously.
How does the centroid relate to the center of mass?
For a thin flat triangular plate of uniform density, the centroid coincides with the center of mass (or center of gravity). This means the plate balances perfectly on a knife edge or pin placed at the centroid. The property stems from the fact that the centroid is the weighted average of all points in the region, with equal weight everywhere for uniform density.
Is the centroid the same as the circumcenter or incenter?
Only for an equilateral triangle. In an equilateral triangle, the centroid, circumcenter (center of the circumscribed circle), incenter (center of the inscribed circle), and orthocenter (intersection of altitudes) all coincide at the same point. For all other triangle types, these four centers are distinct points.
How do I find the centroid from side lengths only (no coordinates)?
If you only have side lengths rather than vertex coordinates, you cannot find the centroid as an (x, y) coordinate without placing the triangle on a coordinate system. However, you can still calculate all three median lengths using the formula m_a = (1/2)*sqrt(2*b^2 + 2*c^2 - a^2), which this calculator also outputs. Place one vertex at the origin and another on the x-axis to convert your side lengths into coordinates.