This article deals with centroid formula and its derivation. Centroid is certainly a very simple concept. Centroid of an object refers to its geometric center. This is a very useful concept in engineering to find the center of an object. Furthermore, if it contains one axis of symmetry then the controls would be in that particular axis. Moreover, if it has two axes of symmetry, then the intersecting point of the two axes would be the centroid.

**What is Centroid?**

Centroid is an interesting concept in mathematics and physics. Furthermore, one can say that centroid refers to the geometric center of a particular plane figure. Moreover, it is the arithmetic mean position of all the points which exist in the figure. Centroid is the point at which a cutout of the shape could be balanced in a perfect manner on the tip of a pin. Also, the concept of centroid can apply to any object in n-dimensional space.

One, two, or three coordinates may be required depending on the shape of the object, in order to define the exact position of centroid in space. Furthermore, if the shape contains an axis of symmetry, then the location of the centroid would on that axis. In case there are multiple axes of symmetry, then the centroid would coincide with the intersection of the axis.

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**Centroid Formula and Derivation**

The centroid of a triangle refers to the intersection of the three medians. Furthermore, it refers to the average of the three vertices. One can find the centroid by using coordinates. Therefore, a triangle which has vertices at L = \(\left ( x_{L, y_{L}} \right )\) , M = \(\left ( x_{M,y_{M}} \right )\), N = \(\left ( x_{N},y_{N}\right )\) would have the centroid at

C = \(\frac{1}{3}\left ( L+M+N \right )\)

C = \(\left ( \frac{1}{3}\left ( x_{L}+x_{M}+x_{N} \right ), \frac{1}{3}\left ( y_{L}+y_{M}+y_{N} \right ) \right )\)

Most noteworthy, the centroid is at \(\frac{1}{3}:\frac{1}{3}:\frac{1}{3}\) when it comes to barycentric coordinates.

Furthermore, the expression of centroids in trilinear coordinates can take place in any of these equivalent ways in terms of the side lengths and vertex angles. Moreover, the side lengths are a,b,c and the vertex angles are L,M,N.

C = \(\frac{1}{a}:\frac{1}{b}:\frac{1}{c}\) = bc : ca : ab = csc L : csc M : csc N

= cos L + cos M.cos N : cos M + cos N.cos L : cos N + cos L.cos M

= sec L + sec M.sec N : sec M + sec N.sec L : sec N + sec L.sec M.

**Solved Examples on Centroid Formula **

Q1 There is a triangle which has three vertices: A = (3,4), B = ( 5,2), and C = (8,15). Find out the coordinates of the centroid of triangle ABC?

A1 In order to find out the coordinates of the centroid of this particular triangle, the formula of centroid must be applied which is below:

\(\left ( \frac{1}{3}\left ( x_{L}+x_{M}+x_{N} \right ), \frac{1}{3}\left ( y_{L}+y_{M}+y_{N} \right ) \right )\)

Therefore, the centroid lies at

\(\left ( \frac{3+5+8}{3},\frac{4+12+15}{3} \right )\)

= \(\left ( \frac{16}{3},\frac{31}{3} \right )\).

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