Centroid Formula

What is Centroid

Centroid is the geometrical concept which refers to its geometric center of the object.

Centroid formula is used to determine the coordinates of a triangle’s centroid. The centroid of a triangle is the center of the triangle, which can be determined as the point of intersection of all the three medians of a triangle. The median is a line drawn from the midpoint of any one side to the opposite vertex. It should be noted that the centroid divides all the medians of the triangle in 2:1 ratio.

Centroid of the Triangle

Centroid Formula

The Centroid Formula is given by

Where,

• C denotes centroid of the triangle.
• x₁, x₂, x₃ are the x-coordinates of the vertices of a triangle.
• y₁, y₂, y₃ are the y-coordinates of the vertices of a triangle.

Solved Examples

Example 1:

Determine the centroid of a triangle whose vertices are (5,3), (6,1) and (7,8).

Solution

Given parameters are,

(x₁, y₁) = (5,3)

(x₂, y₂) = (6,1)

(x₃, y₃) = (7,8)

The centroid formula is given by

C = [(x₁ + x₂ + x₃)/ 3, (y₁ + y₂ + y₃)/ 3)

C = [(5 + 6 + 7) / 3, (3 + 1 + 8) / 3]

C = (18 / 3, 12 / 3)

C = (6, 4)

Example 2:

Calculate the centroid of a triangle whose vertices are (9,0), (2,8) and (1,4).

Solution

Given parameters are

(x₁, y₁) = (9, 0)

(x₂, y₂) = (2, 8)

(x₃, y₃) = (1, 4)

The centroid formula is given by,

C = [(x₁ + x₂ + x₃)/ 3, (y₁ + y₂ + y₃)/ 3]

C = [(9 + 2 + 1) / 3, (0 + 8 + 4) / 3]

C = (12 / 3, 12 / 3)

C = (4, 4)

Example 3: If the (2, 2) are coordinates of the centroid of a triangle whose vertices are (0, 4), (-2, 0) and (p, 2), then find the value of p.

Solution:
Given vertices of a triangle are:
(x₁, y₁) = (0, 4)
(x₂, y₂) = (-2, 0)
(x₃, y₃) = (p, 2)
Centroid = (x, y) = (2, 2)

Using the centroid formula,

(x, y) = [(x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3]

(2, 2) = [(0 – 2 + p)/3, (4 + 0 + 2)/3]

(2, 2) = [(p-2)/3, 6/3)]

Equating the x-coordinates,
(p – 2)/3 = 2
p – 2 = 2 × 3
p = 6 + 2
p = 8
Hence, the value of p = 8

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