Centroid Formula

The geometric center of the object is known as the centroid. For determining the coordinates of the triangle’s centroid we use the centroid formula. The centroid of a triangle can be determined as the point of intersection of all the three medians of a triangle. The centroid of a triangle divides all the medians in a 2:1 ratio. Let us learn about the centroid formula with few solved examples at the end.

What Is a Centroid Formula?

The centroid of a triangle is the center of the triangle. It is referred to as the point of concurrency of medians of a triangle.

Centroid Formula

The centroid formula of a given triangle can be expressed as,

C = \( \left(\dfrac{x_1+ x_2+ x_3}{3} , \dfrac{y_1+ y_2+ y_3}{3}\right)\)

where,

• C denotes the centroid of a triangle
• \(x_1, x_2, x_3\) are the x-coordinates of the 3 vertices.
• \(y_1, y_2, y_3\) are the y-coordinates of the 3 vertices.

Derivation of Centroid Formula

We apply the section formula to derive the centroid of a triangle formula. Let PQR be any triangle with the coordinates of vertices as P(\(x_1\), \(y_1\)), Q(\(x_2\),\(y_2\)), and R(\(x_3\),\(y_3\)), such that D, E, and F are midpoints of the side PQ, QR, and PR respectively. We represent the centroid of a triangle as G. Since, D is the midpoint of side PQ, applying the midpoint formula, we get its coordinates as,
D((\(x_1\) + \(x_2\))/2)

The centroid of a triangle divides the medians in the ratio 2:1. Therefore, from the coordinates of D, we can find the coordinates of G as,

X-coordinate of G: [(2(\(x_1\) + \(x_2\))/2) + 1(\(x_3\))]/(2+1) =  (\(x_1\) + \(x_2\) + \(x_3\))/3

Y-coordinate of G: [(2(\(y_1\) + \(y_2\))/2) + 1(\(y_3\))]/(2+1) =  (\(y_1\) + \(y_2\) + \(y_3\))/3

Therefore, the coordinates of G are given as, ((\(x_1\) + \(x_2\) + \(x_3\))/3, (\(y_1\) + \(y_2\) + \(y_3\))/3)

Let us have a look at a few solved examples to understand the centroid formula better.

Examples Using Centroid Formula

Example 1: Vertices of the triangle are (4,3), (6,5), and (5,4). Determine the centroid of a triangle using the centroid formula.

Solution: 

To find: Centroid of a triangle.

Given parameters are,

\((x_1, y_1) = (4,3)\)

\((x_2, y_2) = (6,5)\)

\((x_3, y_3) = (5,4)\)

Using centroid formula,

The centroid of a triangle = \(\left(\dfrac{x_1 + x_2 + x_3}{3} ,  \dfrac{y_1 + y_2 + y_3}{3}\right)\)

=  \( \left(\dfrac{4 + 6 + 5}{3} , \dfrac{3 + 5 + 4}{3} \right)\)

= \(\dfrac{15}{3} , \dfrac{12}{3}\)

= (5, 4)

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

Example 2: If the coordinates of the centroid of a triangle are (3, 3) and the vertices of the triangle are (1, 5), (-1, 1), and (k, 3), then find the value of k.

Solution: 

To find: The value of k

Given parameters are,

The centroid of a triangle is (3, 3)

\((x_1, y_1) = (1, 5)\)

\((x_2, y_2) = (-1, 1)\)

\((x_3, y_3) = (k, 3)\)

Using the centroid formula,

The centroid of a triangle = \(\dfrac{x_1+ x_2+ x_3}{3} , \dfrac{y_1+ y_2+ y_3}{3}\)

(3, 3)  = \(\dfrac{1+(-1)+ k}{3} , \dfrac{5+1+3}{3}\)

(3, 3)  = \(\dfrac{k}{3} , \dfrac{9}{3}\)

Equating the x-coordinates,

\(\dfrac{k}{3} = 3\)

k = 9

Answer: The value of k is 9.

Example 3: Calculate the centroid of a triangle with vertices (1,3), (2,1), and (3,2).

Solution: 

To find: Centroid of a triangle

Using the centroid formula, we know, Centroid, G = \(\dfrac{x_1+ x_2+ x_3}{3} , \dfrac{y_1+ y_2+ y_3}{3}\)

⇒ G = ((1+2+3)/3, (3+1+2)/3) = (2,2)

Answer: Centroid of given triangle = G(2, 2)