Centroid of Triangle (Centroid)
The centroid of a triangle is a point of concurrency of the medians of a triangle. Before understanding the point of concurrency, let us discuss the medians of a triangle. Medians are the line segments that are drawn from the vertex to the midpoint of the opposite side of the vertex. Each median of a triangle divides the triangle into two smaller triangles that have equal areas. The point of intersection of the medians of a triangle is known as centroid. The centroid always lies inside a triangle, unlike other points of concurrencies of a triangle.
In this article, we will explore the concept of the centroid of a triangle, also commonly called centroid, along with its formula, and its properties. Let us learn more about the centroid of a triangle along with a few solved examples and practice questions.
What is Centroid of a Triangle?
The centroid of a triangle is formed when three medians of a triangle intersect. It is one of the four points of concurrencies of a triangle. The medians of a triangle are constructed when the vertices of a triangle are joined with the midpoint of the opposite sides of the triangle. Observe the following figure that shows the centroid of a triangle.
Properties of the Centroid of Triangle
The following points show the properties of the centroid of a triangle which are very helpful to distinguish the centroid from all the other points of concurrencies.
 The centroid is also known as the geometric center of the object.
 The centroid of a triangle is the point of intersection of all the three medians of a triangle.
 The medians are divided into a 2:1 ratio by the centroid.
 The centroid of a triangle is always within a triangle.
Centroid of Triangle Formula
The centroid of a triangle formula is used to find the centroid of a triangle uses the coordinates of the vertices of a triangle. The coordinates of the centroid of a triangle can only be calculated if we know the coordinates of the vertices of the triangle. The formula for the centroid of the triangle is:
C(x,y) = (x_{1} + x_{2} + x_{3})/3, (y_{1} + y_{2} + y_{3})/3
where, x_{1}, x_{2}, and x_{3} are the 'xcoordinates' of the vertices of the triangle; and y_{1}, y_{2}, and y_{3} are the 'ycoordinates of the vertices of the triangle.
Observe the following figure which shows the vertices of the triangle in the form of coordinates.
Difference Between Orthocentre and Centroid of Triangle
There are various types of differences between the orthocenter and the centroid of the triangle. The main three differences between the orthocenter and the centroid of a triangle are explained in the table given below:
Orthocenter  Centroid 

The orthocenter is the intersection point of the altitudes.  The centroid is the intersection point of the medians. 
It may lie outside of the triangle  It always lies inside the triangle. 
There is not a particular ratio into which it divides the altitudes.  The medians are divided into a 2:1 ratio by the centroid. 
Difference Between Incentre and Centroid of Triangle
The centroid and the incenter have various types of differences between them depending upon the type of triangle it lies in. The important differences between the orthocenter and the centroid of a triangle are explained in the table given below:
Incenter  Centroid 

The incenter is the intersection point of the angle bisectors.  The centroid is the intersection point of the medians. 
It always lies inside the triangle.  It always lies inside the triangle. 
There is not a particular ratio into which it divides the angle bisectors.  The medians are divided into a 2:1 ratio by the centroid. 
Important Notes on Centroid of Triangle
 The centroid of a triangle is the point of intersection of the medians of a triangle.
 It always lies inside the triangle.
 Centroid divides the medians in the ratio 2:1.
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Centroid of a Triangle Examples

Example 1: If the coordinates of the vertices of a triangle are given as (4,3), (6,5), and (5,4), find the position of the centroid of the triangle.
Solution:
To find the centroid of a triangle, the given parameters are:
(x_{1}, y_{1}) = (4, 3)
(x_{2}, y_{2})= (6, 5)
(x_{3}, y_{3}) = (5, 4)
Using the centroid formula:
The centroid of a triangle = C(x,y) = ((x_{1} + x_{2} + x_{3})/3, (y_{1} + y_{2} + y_{3})/3)
= ((4 + 6 + 5)/3, (3 + 5 + 4)/3)
= (15/3 , 12/3)
= (5 , 4)
Answer: The centroid of the triangle is (5 , 4).

Example 2: Determine the centroid of a rightangled triangle using the centroid formula, if the vertices of the triangle are (0,5), (5,0), and (0,0).
Solution:
To find the centroid of a triangle, the given parameters are:
(x_{1}, y_{1}) = (0,5)
(x_{2}, y_{2}) = (5,0)
(x_{3}, y_{3}) = (0,0)
Using the centroid formula,
C(x,y) = ((x_{1} + x_{2} + x_{3})/3, (y_{1} + y_{2} + y_{3})/3)
= ((0 + 5 + 0)/3, (5 + 0 + 0)/3)
= (5/3 , 5/3)
Answer: The centroid of a triangle is (5/3, 5/3).

Example 3: The vertices of a triangle are (1, 2), (h, −3), and (−4, k) and the coordinates of the centroid of the triangle are (5, –1). Find the value of h and k.
Solution: We have
(x_{1}, y_{1}) = (1, 2)
(x_{2}, y_{2}) = (h, –3)
(x_{3}, y_{3}) = (–4, k)
Using centroid of triangle formula, we have
C(x,y) = ((x_{1} + x_{2} + x_{3})/3, (y_{1} + y_{2} + y_{3})/3)
⇒ (5, 1) = ((1 + h – 4)/3, (2  3 + k)/3)
⇒ (5, 1) = ( (3 + h)/3, (1 + k)/3 )
⇒ (h  3)/3 = 5 and (k  1)/3 = 1
⇒ h  3 = 15 and k  1 = 3
⇒ h = 18 and k = 2
Answer: The values of h and k are h = 18 and k = 2.
FAQs on Centroid of Triangle
What is the Centroid of Triangle?
The centroid of a triangle is formed when three medians of a triangle intersect. It is one of the four points of concurrencies of a triangle.
What is the Formula for the Centroid of Triangle?
The formula used to calculate the centroid of the triangle is: C(x,y) = ((x_{1} + x_{2} + x_{3})/3, (y_{1} + y_{2} + y_{3})/3), where, x_{1}, x_{2}, and x_{3} are the 'xcoordinates' of the vertices of the triangle; and y_{1}, y_{2}, and y_{3} are the ycoordinates of the vertices of the triangle.
What are the Properties of Centroid of Triangle?
The centroid of a triangle is formed when three medians of a triangle intersect. The properties of a centroid are as follows:
 The centroid is also known as the geometric center of the object.
 It is the point of intersection of all the three medians of a triangle.
 The medians are divided into a 2:1 ratio by the centroid.
 The centroid of a triangle is always within a triangle.
What is the Easiest Way to Find the Centroid of a Triangle?
There are three basic steps that can be followed to find the centroid of a triangle:
 Identify and list all the three coordinates of each vertex.
 Add the xcoordinates of all the three vertices and divide the sum by 3.
 Add the ycoordinates of all the three vertices and divide the sum by 3.
This will be the centroid of the given triangle.
Where is the Centroid of Any Given Triangle Located?
Centroid is the intersection of the medians of a triangle. If we construct the medians of a triangle, the point where the medians intersect is the centroid of the triangle. It is located inside the triangle.
How do You Find the Centroid of a Triangle When the Vertices are Given?
If the vertices of the triangle are known to us, we use the following formula to calculate the centroid of a triangle. The formula for the centroid of the triangle is:
C(x,y) = ((x_{1} + x_{2} + x_{3})/3, (y_{1} + y_{2} + y_{3})/3), where, x_{1}, x_{2}, and x_{3} are the 'xcoordinates' of the vertices of the triangle; and y_{1}, y_{2}, and y_{3} are the ycoordinates of the vertices of the triangle.
How do You Find the Centroid of a Triangle?
We can construct the medians of a triangle and mark their intersection point as the centroid or we can use the formula to find the centroid of a triangle.
What is the Relationship Between the Orthocentre, Circumcentre, and Centroid of Triangle?
Orthocenter, circumcenter, and centroid always lie in a straight line, known as the Euler's line. Centroid always lies in between the orthocenter and the circumcenter of the triangle. In an equilateral triangle, the orthocenter, circumcenter, and the centroid, all lie at the same point, inside of the triangle. For the obtuseangled triangle, the orthocenter, circumcenter, both lie outside of the triangle and the centroid lies inside of the triangle.
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