Mathwords logoMathwords

Centroid of a Triangle — Definition, Formula & Examples

The centroid of a triangle is the point where the three medians of the triangle meet. It acts as the triangle's balance point — the spot where the triangle would balance perfectly on the tip of a pencil.

The centroid is the point of concurrency of the three medians of a triangle, where each median connects a vertex to the midpoint of the opposite side. For a triangle with vertices A(x1,y1)A(x_1, y_1), B(x2,y2)B(x_2, y_2), and C(x3,y3)C(x_3, y_3), the centroid GG has coordinates (x1+x2+x33, y1+y2+y33)\left(\frac{x_1+x_2+x_3}{3},\, \frac{y_1+y_2+y_3}{3}\right). The centroid divides each median in a 2:12:1 ratio measured from vertex to midpoint.

Key Formula

G=(x1+x2+x33,  y1+y2+y33)G = \left(\frac{x_1 + x_2 + x_3}{3},\; \frac{y_1 + y_2 + y_3}{3}\right)
Where:
  • GG = The centroid of the triangle
  • (x1,y1)(x_1, y_1) = Coordinates of the first vertex
  • (x2,y2)(x_2, y_2) = Coordinates of the second vertex
  • (x3,y3)(x_3, y_3) = Coordinates of the third vertex

How It Works

To find the centroid, you simply average the xx-coordinates and yy-coordinates of the three vertices. No matter the shape of the triangle — acute, right, or obtuse — the centroid always lies inside the triangle. A key property is the 2:1 ratio: the centroid sits two-thirds of the way along each median from the vertex toward the opposite side's midpoint. This means the distance from a vertex to the centroid is exactly twice the distance from the centroid to the midpoint of the opposite side.

Worked Example

Problem: Find the centroid of the triangle with vertices A(0, 0), B(6, 0), and C(3, 9).
Step 1: Average the x-coordinates of the three vertices.
xG=0+6+33=93=3x_G = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3
Step 2: Average the y-coordinates of the three vertices.
yG=0+0+93=93=3y_G = \frac{0 + 0 + 9}{3} = \frac{9}{3} = 3
Step 3: Write the centroid as a coordinate point.
G=(3,3)G = (3, 3)
Answer: The centroid is at the point (3, 3).

Another Example

Problem: Verify that the centroid G(3, 3) divides the median from vertex C(3, 9) to the midpoint of AB in a 2:1 ratio.
Step 1: Find the midpoint M of side AB, where A(0, 0) and B(6, 0).
M=(0+62,  0+02)=(3,0)M = \left(\frac{0+6}{2},\; \frac{0+0}{2}\right) = (3, 0)
Step 2: Compute the distance from C(3, 9) to G(3, 3).
CG=(3−3)2+(3−9)2=36=6CG = \sqrt{(3-3)^2 + (3-9)^2} = \sqrt{36} = 6
Step 3: Compute the distance from G(3, 3) to M(3, 0).
GM=(3−3)2+(0−3)2=9=3GM = \sqrt{(3-3)^2 + (0-3)^2} = \sqrt{9} = 3
Step 4: Check the ratio CG : GM.
CGGM=63=21\frac{CG}{GM} = \frac{6}{3} = \frac{2}{1}
Answer: The centroid divides the median from C in a 2:1 ratio, confirming the property.

Why It Matters

The centroid appears throughout high school geometry courses and on standardized tests like the SAT and ACT, where coordinate geometry problems frequently ask for triangle centers. In physics and engineering, the centroid represents the center of mass of a uniform triangular plate, which is essential for structural design and balance calculations.

Common Mistakes

Mistake: Reversing the 2:1 ratio and thinking the centroid is one-third of the way from the vertex instead of two-thirds.
Correction: The centroid is two-thirds of the way from each vertex to the midpoint of the opposite side. The shorter segment (one-third) is from the centroid to the midpoint.
Mistake: Confusing the centroid with the circumcenter or incenter and using perpendicular bisectors or angle bisectors instead of medians.
Correction: The centroid is defined only by medians — segments from each vertex to the midpoint of the opposite side. Perpendicular bisectors locate the circumcenter, and angle bisectors locate the incenter.

Related Terms