Geometric Centroid — Definition, Formula & Examples

The geometric centroid is the point where a shape would perfectly balance if cut from a uniform material. For a triangle, it is the intersection of the three medians, located one-third of the way from each side toward the opposite vertex.

The centroid of a plane figure is the arithmetic mean position of all points in the region. For a set of vertices $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ of a polygon treated as point masses, the centroid is $\left(\frac{1}{n}\sum x_i,\, \frac{1}{n}\sum y_i\right)$ . For continuous regions, it is computed via integration over the area.

Key Formula

G = \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
$(x_3, y_3)$ = Coordinates of the third vertex
$G$ = The centroid (center of area) of the triangle

How It Works

For a triangle with known vertices, average the

x

-coordinates and the

y

-coordinates separately. This gives you the centroid directly — no need to find the medians and intersect them. For irregular shapes in calculus, you integrate to find the first moments of area and divide by the total area. The centroid always lies inside a convex shape, but it can fall outside a concave one.

Worked Example

Problem: Find the centroid of the triangle with vertices A(0, 0), B(6, 0), and C(3, 9).

Average the x-coordinates: Add the three x-values and divide by 3.

\bar{x} = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3

Average the y-coordinates: Add the three y-values and divide by 3.

\bar{y} = \frac{0 + 0 + 9}{3} = \frac{9}{3} = 3

State the centroid: Combine the results into a coordinate pair.

G = (3,\, 3)

Answer: The centroid is at (3, 3).

Why It Matters

Engineers and architects use centroids to determine where loads concentrate on beams and structural panels. In AP Calculus, you compute centroids of irregular regions using integration — a standard free-response topic. Physics courses rely on the centroid concept when analyzing torque and center of mass.

Common Mistakes

Mistake: Confusing the centroid with the circumcenter or incenter of a triangle.

Correction: The centroid is the intersection of medians (and always lies inside the triangle). The circumcenter is equidistant from all vertices and can lie outside; the incenter is equidistant from all sides. These are three distinct points.