# Definition:Centroid of Set of Points/Definition 1

## Definition

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

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 $S$.

## 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**.

Beware that some sources use the term **center of gravity** even when approaching the topic from a pure mathematical perspective, which can cause confusion.

## Also see

- Results about
**centroids**can be found**here**.

## Sources

- 1921: C.E. Weatherburn:
*Elementary Vector Analysis*... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Centroids: $9$. Centroid, or centre of mean position