Centroid of a Triangle

Summary

This math recipe will help you find the centroid of a triangle, coordinates of whose vertices are known.

Centroid of a Triangle

Skill Level

Easy

Time

Approx. 2 min

Ingredients

Coordinates of the three vertices: \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\)

Method

To find the centroid \(G\) of \(\Delta ABC\), we’ll simply plug their coordinates into the formula below:

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

This formula is sometimes called the Centroid Formula.

Examples

Example 1 Find the centroid of the triangle, whose vertices are \(P(4, 6)\), \(Q(-2, 2)\), and \(R(1, 1)\).

Solution The figure shows \(\Delta PQR\) on the plane.

Centroid of a Triangle

To find the centroid \(G\), we’ll use the centroid formula directly.

\(G \equiv \left ( \frac{4+(-2)+1}{3},\frac{6+2+1}{3} \right ) \)

\( \equiv \left ( \frac{3}{3},\frac{9}{3} \right ) \)

\( \equiv (1,3) \)

Example 2 Find the centroid of the triangle, whose vertices are \(A(-1, 5)\), \(B(7, -3)\), and \(C(0, 2)\).

Solution The figure shows \(\Delta ABC\) on the plane.

Centroid of a Triangle

To find the centroid \(G\), we’ll again use the centroid formula.

\(G \equiv \left ( \frac{-1+7+0}{3},\frac{5+(-3)+(-2)}{3} \right ) \)

\( \equiv \left ( \frac{6}{3},\frac{0}{3} \right ) \)

\( \equiv  (2,0) \)

That’s it for this recipe. Hope you found it helpful.

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