Centroid Formula — Definition, Examples & How to Find

Centroid Formula

The coordinates of the centroid of a triangle are found by averaging the x- and y-coordinates of the vertices. This method will also find the centroid (center of mass) of any set of points on the x-y plane.

Centroid formula: for points (x₁,y₁)…(xₙ,yₙ), centroid = ((x₁+x₂+…+xₙ)/n, (y₁+y₂+…+yₙ)/n)

See also

Formula

Key Formula

\left(\frac{x_1 + x_2 + x_3}{3},\;\frac{y_1 + y_2 + y_3}{3}\right)

Where:

$(x_1, y_1)$ = Coordinates of the first vertex of the triangle
$(x_2, y_2)$ = Coordinates of the second vertex of the triangle
$(x_3, y_3)$ = Coordinates of the third vertex of the triangle

Worked Example

Problem: Find the centroid of the triangle with vertices A(2, 4), B(8, 6), and C(5, 2).

Step 1: Write down the coordinates of each vertex.

(x_1, y_1) = (2, 4),\quad (x_2, y_2) = (8, 6),\quad (x_3, y_3) = (5, 2)

Step 2: Average the x-coordinates by adding them and dividing by 3.

\bar{x} = \frac{2 + 8 + 5}{3} = \frac{15}{3} = 5

Step 3: Average the y-coordinates the same way.

\bar{y} = \frac{4 + 6 + 2}{3} = \frac{12}{3} = 4

Step 4: Combine the results to state the centroid.

\text{Centroid} = (5,\; 4)

Answer: The centroid of the triangle is at (5, 4).

Another Example

This example extends the formula beyond a triangle to a set of four points, showing how the same averaging method works for any number of points.

Problem: Find the centroid of a set of four points: P(0, 0), Q(6, 0), R(6, 8), and S(0, 8).

Step 1: For a set of n points, the centroid formula generalizes to the average of all x-coordinates and the average of all y-coordinates.

\left(\frac{x_1 + x_2 + \cdots + x_n}{n},\;\frac{y_1 + y_2 + \cdots + y_n}{n}\right)

Step 2: Average the four x-coordinates.

\bar{x} = \frac{0 + 6 + 6 + 0}{4} = \frac{12}{4} = 3

Step 3: Average the four y-coordinates.

\bar{y} = \frac{0 + 0 + 8 + 8}{4} = \frac{16}{4} = 4

Step 4: The centroid of the four-point set is the center of this rectangle, as expected.

\text{Centroid} = (3,\; 4)

Answer: The centroid of the four points is (3, 4).

Frequently Asked Questions

What is the difference between the centroid and the center of mass?

For a uniform shape or a set of equal-mass points, the centroid and the center of mass are the same point. They differ only when objects have unequal masses or non-uniform density. In that case, the center of mass formula uses a weighted average, while the centroid formula uses a simple (unweighted) average.

Is the centroid always inside the triangle?

Yes. Unlike the circumcenter or orthocenter, the centroid always lies inside the triangle regardless of the triangle's shape. It sits exactly one-third of the way from each side toward the opposite vertex, measured along each median.

How does the centroid relate to the medians of a triangle?

The centroid is the intersection point of a triangle's three medians. A median connects a vertex to the midpoint of the opposite side. The centroid divides each median in a 2:1 ratio, with the longer segment on the vertex side.

Centroid Formula vs. Center of Mass Formula

	Centroid Formula	Center of Mass Formula
Definition	Average of the coordinates of the vertices (equal weight)	Weighted average of coordinates using each point's mass
Formula (x-coordinate)	(x₁ + x₂ + x₃) / 3	(m₁x₁ + m₂x₂ + m₃x₃) / (m₁ + m₂ + m₃)
When to use	All points have equal weight or you need the geometric center	Points have different masses or importance
Result when weights are equal	Same as center of mass	Same as centroid

Why It Matters

The centroid formula appears frequently in geometry courses when you work with triangles on the coordinate plane, and it is a standard topic on standardized tests. In physics, the same idea underpins finding the center of mass, and in engineering it helps locate the balance point of structures and materials. Understanding this formula also builds your skill with coordinate geometry and averaging, which are foundational for more advanced topics like vectors and integration.

Common Mistakes

Mistake: Dividing the sum of coordinates by 2 instead of 3 for a triangle.

Correction: You divide by the number of vertices. A triangle has 3 vertices, so always divide by 3. Only use 2 if you are averaging two values, such as finding a midpoint.

Mistake: Confusing the centroid with the midpoint of one side.

Correction: The midpoint formula applies to two endpoints of a single segment. The centroid uses all three vertices. Mixing these up gives a point on one side rather than the interior balance point.

Related Terms

Centroid — The point this formula calculates
Center of Mass Formula — Weighted version of the centroid formula
Coordinates — The ordered pairs used in the formula
Triangle — Most common shape the formula is applied to
Average — Core operation in computing the centroid
Vertex — The triangle's corners whose coordinates are averaged
Point — Formula extends to any set of points
x-y Plane — The coordinate plane where the formula is applied