# Definition:Centroid

## Definition

### Centroid of Set of Points

Let $S = \set {A_1, A_2, \ldots, A_n}$ be a set of $n$ points in Euclidean space.

### Definition 1

Let the position vectors of the elements of $S$ be given by $\mathbf a_1, \mathbf a_2, \dotsc, \mathbf a_n$ respectively.

Let $G$ be the point whose position vector is given by:

$\vec {OG} = \dfrac 1 n \paren {\mathbf a_1 + \mathbf a_2 + \dotsb + \mathbf a_n}$

Then $G$ is known as the centroid of $S$.

### Definition 2

Let the Cartesian coordinates of the elements of $S$ be $\tuple {x_j, y_j, z_j}$ for each $j \in \set {1, 2, \ldots, n}$.

Let $G$ be the point whose Cartesian coordinates are given by:

$G = \tuple {\dfrac 1 n \ds \sum_{j \mathop = 1}^n x_j, \dfrac 1 n \ds \sum_{j \mathop = 1}^n y_j, \dfrac 1 n \ds \sum_{j \mathop = 1}^n z_j}$

That is, the arithmetic mean of the Cartesian coordinates of the elements of $S$

Then $G$ is known as the centroid of $S$.

### Centroid of Weighted Set of Points

Let $S = \set {A_1, A_2, \ldots, A_n}$ be a set of $n$ points in Euclidean space whose position vectors are given by $\mathbf a_1, \mathbf a_2, \dotsc, \mathbf a_n$ repectively.

Let $W: S \to \R$ be a weight function on $S$.

Let $G$ be the point whose position vector is given by:

$\vec {OG} = \dfrac {w_1 \mathbf a_1 + w_2 \mathbf a_2 + \dotsb + w_n \mathbf a_n} {w_1 + w_2 + \dotsb + w_n}$

where $w_i = \map W {A_i}$ for each $i$.

Then $G$ is known as the centroid of $S$ with weights $w_i, w_2, \dotsc, w_n$.

### Centroid of Surface

Let $S$ be a surface.

Let $S$ be divided into a large number $n$ of small elements.

Consider one point of each of these elements.

Let a weight function be associated with this set of points.

Let $G$ be the centroid of each of these weighted points.

Let $n$ increase indefinitely, such that each element of $S$ converges to a point.

Then the limiting position of $G$ is the centroid of $S$.

### Centroid of Solid Figure

Let $F$ be a solid figure.

Let $F$ be divided into a large number $n$ of small elements.

Consider one point of each of these elements.

Let a weight function be associated with this set of points.

Let $G$ be the centroid of each of these weighted points.

Let $n$ increase indefinitely, such that each element of $F$ converges to a point.

Then the limiting position of $G$ is the centroid of $F$.

### Centroid of Triangle

Let $\triangle ABC$ be a triangle.

The centroid of $\triangle ABC$ is the point $G$ where its three medians $AL$, $MB$ and $CN$ meet.

## Also known as

A centroid is also referred to as a center of mean position.

Some sources refer to it as a mean point.

Approaches to this subject from the direction of physics and mechanics can be seen referring to it as a center of gravity.

However, it needs to be noted that the latter is merely a special case of a centroid.

## Also see

• Results about centroids can be found here.