Triangle Centroid Calculator

Welcome to the Triangle Centroid Calculator — an interactive tool that finds the centroid (center of mass) of any triangle from its three vertex coordinates, with a live diagram, step-by-step solution, and full triangle analysis. Whether you are a student, engineer, or math enthusiast, this calculator makes centroid computation instant and visual.

What is the Centroid of a Triangle?

The centroid of a triangle is the point where all three medians intersect. A median is a line segment drawn from any vertex to the midpoint of the opposite side. The centroid is also the triangle's center of mass (or center of gravity) — if you cut a triangle from uniform cardboard, it would balance perfectly on the centroid.

Centroid Formula

For a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃):

The centroid coordinates are simply the arithmetic mean of the three vertices' x-coordinates and y-coordinates, respectively.

Key Properties of the Centroid

• Always interior: Unlike the orthocenter or circumcenter, the centroid always lies inside the triangle, for all triangle types.
• 2:1 median ratio: The centroid divides each median in a 2:1 ratio from vertex to midpoint. The distance from a vertex to the centroid is 2/3 of the median length; from the centroid to the midpoint is 1/3.
• Center of mass: A uniform triangular lamina balances at the centroid — it is the average position of all points in the triangle.
• Euler line: For scalene triangles, the centroid lies on the Euler line, which also passes through the circumcenter and orthocenter.

How to Use This Calculator

1. Enter coordinates: Input x and y values for vertices A, B, and C. Negative numbers and decimals are supported.
2. Choose precision: Select your preferred number of decimal places.
3. Click Calculate: The centroid G = (Gx, Gy) is displayed with a full breakdown and interactive diagram.
4. Explore the diagram: See the triangle, its three color-coded medians, midpoints, and the animated centroid marker.

Centroid vs Other Triangle Centers

Frequently Asked Questions

What is the centroid of a triangle?

The centroid is the point where the three medians of a triangle intersect. It is the triangle's center of mass — a uniform triangular lamina balances at this point. The centroid always lies inside the triangle.

How do you find the centroid of a triangle with coordinates?

Average the x-coordinates and y-coordinates of the three vertices: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). This calculator does that instantly with step-by-step working.

What is the 2:1 median ratio property?

The centroid divides each median in a 2:1 ratio from vertex to midpoint. So the distance from any vertex to G is exactly 2/3 of that median's total length, and the distance from G to the opposite side midpoint is 1/3.

Does the centroid always lie inside the triangle?

Yes — unlike the orthocenter or circumcenter (which can be outside for obtuse triangles), the centroid always lies strictly inside the triangle for any non-degenerate triangle.

What is the difference between centroid, circumcenter, and orthocenter?

The centroid (median intersection) is the center of mass. The circumcenter is equidistant from all three vertices. The orthocenter is the altitude intersection. For equilateral triangles, all three coincide. For other triangles, they are collinear on the Euler line in the order O, G, H with G dividing OH in a 1:2 ratio.

Additional Resources

• Median (Geometry) - Wikipedia
• Euler Line - Wikipedia