Learn everything about the centroid of a triangle, including its formula, properties, differences, solved examples, and other frequently asked questions.
Centroid of a Triangle Formula Properties and Example Questions.pdf
1. Centroid of a Triangle – Formula,
Properties and Example Questions
In mathematics, physics, and engineering, the centroid is one of the most important concepts or
topics to understand. It is helpful to study the center of a plane figure, gravity, and moments of
inertia. The centroid or geometric center always seems within the triangle. Professors and math
tutors teach the uses of centroid with real-life examples. To find the center of a specific area in any
field, the concept of the centroid is used. In this article, we have covered the most asked questions
about the centroid of a triangle, such as its definition, properties, formulas, theorems, example
questions, and other important points.
Centroid of the Triangle
Centroid is one of the important concepts and properties of triangles. Triangle is a 2D geometric
shape with three sides and three interior angles. they are classified into different types based on
their angles and sides, which are:
2. • Acute Angled triangle
• Obtuse Angled triangle
• Right Angled triangle
• Scalene triangle
• Isosceles triangle
• Equilateral triangle
Before understanding the concept, definition, and other properties of the centroid of a triangle, we
must go through the concept of medians. First, the question that begs to be asked is, what are the
medians of a triangle?
Medians of a triangle have equal areas, and each median divides the triangle into two parts or
smaller triangles. Medians are the line segments drawn from the vertex of a triangle to the midpoint
of the opposite side of the vertex. Here, a centroid is the point of intersection of the medians of a
triangle. It is always inside the triangle.
What is the centroid of a triangle?
By definition, the centroid is the center point of any object. In geometry, the point where all the
three medians of a triangle intersect is known as the centroid of a triangle. In other words, you can
say the centroid is the point of concurrency of all the three medians of a triangle. observe the
following figure to understand the concept of centroid more clearly.
Take an average of x coordinate and y coordinate points of triangles’ all vertices to find out the
centroid of the respective triangle.
3. Centroid of Right-Angle Triangle
A point where three medians of a triangle intersect, drawn to the midpoints of the opposite sites
from the vertices of the triangle, is known as the centroid of a right-angle triangle.
Centroid of a Square
The centroid of the square is a point where the square’s diagonals intersect each other. Finding the
centroid of a square is not a big deal compared to the triangle because squares have equal sides.
4. Formula of Centroid
Centroid formula is required to determine the coordinates of the centroid of a triangle. To calculate
centroid, you must find out the vertices of the given triangle. It could be any type of triangle. As
triangles always have three vertices, taking the average of coordinate points is required.
Following is the formula of the centroid of the triangle:
C (x, y) = ((x1+x2+x3)/3, (y1+y2+y3)/3)
5. Where,
⇒ C = centroid of a triangle
⇒ x1+x2+x3 = x coordinates of triangle
⇒ y1+y2+y3 = y coordinates of triangles
Centroid Theorem
The triangle’s centroid theorem states that,
The centroid is at 2/3 of the distance from each vertex of a triangle to the midpoint of the opposite
side.
6. According to the image given above, V is the centroid of the PQR triangle. Where PQ, QR, and PR
have the midpoints denoted as S, T, and U, respectively.
Thus,
QV = 2/3 QU,
PV = 2/3 PT and
RV = 2/3 RS
Properties of Centroid of Triangle
Among different points of concurrency, discovering the triangles’ centroid is a bit tricky. However,
the following properties are quite helpful to distinguish it. Give them a read:
• The geometric center of the object is called the centroid
• The medians’ intersection forms centroid
• Like the incenter of the triangle, the centroid is also located inside the triangle.
• Medians of the triangle are divided into the ratio of 2:1 by the centroid.
• Centroid is also known as the center of gravity.
• Centroid is one of the triangle’s points of concurrency.
Centroid formula for other shapes
Different geometrical shapes have their centroid formula. Below are the shapes with their figure, x̄, ȳ
and area
7. Triangular Area
x̄ = null
ȳ = h/3
Area = bh /2
Parabolic Area
x̄ = 0
ȳ = 3h/5
Area = 4ah/3
13. Solved Examples of Centroid
Question 1: If vertices of the triangle are (0,4), (4,0), and (0,0), find its
centroid.
Solution:
Given that
(x1, y1) = (0,0)
(x2, y2) = (4,0)
(x3, y3) = (0,4)
As formula of centroid is
C (x, y) = ((x1+x2+x3)/3, (y1+y2+y3)/3)
By putting values, we get
C = (0+4+0)/3, (0+0+4)/3
C = (4/3, 4/3)
Hence, the centroid of the given triangle is (4/3, 4/3)
Question 2: Suppose a triangle has TUM vertices and their x, y coordinates
points are (1, 5), (2, 6) and (4, 10) respectively. Find out its centroid by using
formula.
14. Solution:
Given vertices are
⇒ T (1, 5),
⇒ U (2, 6)
⇒ M (4, 10)
As formula of centroid is
Centroid = ((x1+x2+x3)/3, (y1+y2+y3)/3)
By putting values, we get
C = (1+2+4)/3, (5+6+10)/3
C = (7/3, 21/3)
C = (7/3,7)
Hence, the triangle with vertices T (1, 5), U (2, 6), and M (4, 10) has centroid C = (7/3,7)
Centroid of Triangle: Frequently Asked Questions
How would you define the centroid of a triangle?
The centroid is the point that lies inside a triangle where its medians intersect with each other.
How to find the centroid of a triangle? Explain step by step.
The following steps are the easiest way to find a triangle’s centroid:
• Identify the coordinates of each vertex of a triangle. Both x and y.
• List all the three coordinates.
• Add the x coordinates of all 3 vertices and divide the sum by 3.
• Add the y coordinates of all 3 vertices and divide the sum by 3.
• Calculated data will be the centroid of the given triangle.
What is the formula of centroid?
Centroid = C (x, y) = ((x1+x2+x3)/3, (y1+y2+y3)/3)
In the given formula, x1+x2+x3 and y1+y2+y3 are the x and y coordinates of triangles, respectively.
What is the main difference between the centroid and incenter of the triangle?
One of the main differences between the centroid and the incenter of the triangle are:
15. • Centroid: it is the intersection point of the medians of the triangle.
• Incenter: it is the intersection point of the angle bisectors of the triangle.
Where does the centroid of a triangle located or lie?
It lies inside the triangle.
What is a common point between the incenter and centroid of a triangle?
Both incenter and centroid lie inside the triangle.
What is the difference between the centroid and orthocenter of a triangle?
• The centroid lies inside the triangle, whereas the orthocenter lies outside of the
triangle.
• The orthocenter is the intersection point of the altitudes, whereas the centroid is the
intersection point of the medians of a triangle.
Is there any centroid calculator?
Yes, there are plenty of online centroid calculator tools. All you have to do is insert the given
coordinate points of the triangle, and the tool will display the answer in a fraction of seconds.
What is the median of a triangle?
By definition, the line segment drawn from a vertex of the triangle to the midpoint of the opposite
side of the vertex is known as the median of a triangle. Medians are concurrent at a triangle point,
and the concurrency point is called the triangle’s centroid.
Write the ratio of centroid?
The ratio of the centroid is formed by the intersection of the triangle’s medians. The ratio will always
be 2:1 or 2/3 along any median.
