Math

Centroid Calculator

Calculate the centroid (center of mass) of a triangle given the coordinates of its three vertices.

Point A (x₁, y₁)

Point B (x₂, y₂)

Point C (x₃, y₃)

Centroid
(3, 2)
Point A
(0, 0)
Point B
(6, 0)
Point C
(3, 6)
Centroid (G)
(3, 2)

The centroid is the point where all three medians of a triangle intersect. It is also the center of mass of the triangle.

What is a centroid?

The centroid of a triangle is the point where all three medians intersect. A median is a line segment connecting a vertex to the midpoint of the opposite side. The centroid is also known as the center of mass or center of gravity of the triangle.

Formula

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

Centroid=(x1+x2+x33,y1+y2+y33)\text{Centroid} = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)

Properties of the centroid

  • The centroid always lies inside the triangle
  • It divides each median in a 2:1 ratio from the vertex
  • The centroid is the balance point of the triangle
  • All three medians pass through the centroid

How to find the centroid

  1. Identify the coordinates of all three vertices
  2. Add the x-coordinates of all three vertices and divide by 3
  3. Add the y-coordinates of all three vertices and divide by 3
  4. The result is the centroid (x, y)

Example

For a triangle with vertices at A(0, 0), B(6, 0), and C(3, 6):

xG=0+6+33=93=3yG=0+0+63=63=2\begin{aligned} x_G &= \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3 \\[0.5em] y_G &= \frac{0 + 0 + 6}{3} = \frac{6}{3} = 2 \end{aligned}

The centroid is at (3, 2).

Applications

  • Physics: Finding the center of mass of triangular objects
  • Engineering: Structural analysis and load distribution
  • Computer graphics: Mesh processing and 3D modeling
  • Navigation: Geographic calculations and mapping