Centroid is a geometric point that represents the center of mass or the average position of all points in a shape or object, often used in mathematics, physics, and engineering for various analytical purposes. Centroid always lies within the figure and is not only related to triangles; it can be determined for every geometric figure as well.
In this article, we will explore the concept of the centroid in detail, including the centroid of various geometric shapes such as triangles, quadrilaterals, polygons, as well as circles. Additionally, we will learn about the formula to calculate the centroid of a triangle using the coordinates of its vertices.
What is Centroid?
In geometry, a centroid is a point defined as the arithmetic mean position of all the points in a shape. It is often referred to as the “center of mass” or the “center of gravity” of a geometric object. The centroid depends on the distribution of mass or points within the object and is a useful concept in various branches of geometry, physics, and engineering.
Definition of Centroid
For a two-dimensional object, such as a triangle or a polygon, the centroid is the point where the medians intersect.
The median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side. The centroid divides each median into two segments, with one segment twice as long as the other.
For a three-dimensional object, like a solid with uniform density, the centroid is the point where the medians (or analogously, the lines connecting opposite faces’ centroids) intersect.
Properties of Centroid
- Centroid is the center point of object.
- Centroid always lies inside the object.
- Centroid is center point of gravity.
- Centroid never lies outside the object.
- For different objects with different shape , centroid are calculated separately.
Centroid in Triangle
Similarly, we can derive centroid of more geometric figure by calculating mid point that lies inside the figure. In mathematics centroid is mainly concerned with triangles. Centroid is point inside triangle , where all three medians of triangle intersect. In further section we will derive the formula of centroid of triangle and discuss some problems based on it.
Note: Median of triangle is defined as the line joining the vertex of triangle with the opposite side and bisecting the opposite side.
Centroid Definition in Triangle
Centroid is the point of triangle , where all medians of triangle meet.
In other words , the point of intersection medians of triangle is known as centroid of triangle. The centroid of a triangle always lies inside the triangle. Centroid is also called as geometric center of triangle.
In the given triangle ABC, the coordinates of triangle are (x1 , y1), (x2 , y2), (x3 , y3) and the medians from all three vertex A, B and C meet at point G which is centroid of the triangle.
Properties of Centroid in Triangles
Some of the properties of the point named as centroid of triangle
- Centroid is the meeting point of all the three medians of triangle.
- Centroid lies inside the triangle.
- Centroid divides the median in ratio of 2:1.
- Median bisects the opposite side of vertex .
- Centroid never lies outside the triangle.
Centroid Formula for Triangle
The centroid of a triangle can be calculated by using centroid formula. If the vertices of triangle are in the form of (x1 , y1) , (x2 , y2) and (x3 , y3) then centroid of triangle can be define as:
Centroid of Triangle; (x , y) = (x1+x2+x3/3, y1+y2+y3/3)
Where,
- x1 , x2 and x3 are x coordinates of vertices of triangle and
- y1 , y2 and y3 are y coordinates of vertices of triangle.
Centroid for some common plane figures are discussed as follows:
Centroid of Quadrilateral
Centroid of a quadrilateral can be easily calculated by moving half the distance in length direction and half distance in breadth direction. As quadrilateral is a two dimensional figure it has two dimensions of length and breadth only. The mid point inside quadrilateral is centroid point.
Centroid of Rectangle
Rectangle is a solid closed structure bounded by four lines. The opposites sides of rectangle are equal and parallel. Centroid of rectangle lies in the mid of triangle at half distance of length and half distance of breadth. Suppose if length of rectangle is L unit and breadth of rectangle is B unit then centroid of rectangle will lie on point (L/2 , B/2).
Centroid of Rectangle
Centroid of Square
Square is closed geometrical figure bounded by four lines. Unlike rectangles, all the sides of square are equal and all interior angles of square is equal to 90 deg. Centroid of square lies at the mid point of square. All sides of square are equal so suppose if side of square is S unit then centroid of square lie on point ( S/2 , S/2).
Centroid of Square
Centroid of Polygon
Centroid of a polygon is calculated by taking integration of all points on polygon and dividing it by total number of sides present in it. Centroid is calculated by using formula,
Centroid of Polygon = 1/n(integration of all points in polygon)
Centroid of Circle
A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed shape, two-dimensional shape, curved shape. For circle centroid overlap on center of circle. If diameter of circle is D then centroid lies on point D/2.
Centroid of Circle
| Triangle |
Intersection of medians |
| Square |
Intersection of diagonals |
| Rectangle |
Intersection of diagonals |
| Circle |
Center of the circle |
| Regular Polygon |
Center of the polygon |
Centroid in a solid figure is point of center of gravity. As centroid is center point of object.
Formula for calculating the centroid in solid figures :
Cx = ∫xdV/V , Cy = ∫ydV/V and Cz = ∫zdV/V
The perpendicular distance in the x direction from the yz-plane to the centroid is Cx
The perpendicular distance in the y direction from the zx-plane to the centroid is Cy
The perpendicular distance in the z direction from the xy-plane to the centroid is Cz
The coordinates of the centroid are (Cx , Cy , Cz).
Centroid Vs Circumcenter Vs Incenter
The key difference between centroid, circumcenter and incenter are listed in the following table:
| Definition |
The point where the medians of a triangle intersect. Each median divides the side it joins into a 2:1 ratio. |
The center of the circle that passes through all three vertices of the triangle. |
The point where the angle bisectors of a triangle intersect. |
| Location |
Two-thirds of the way from each vertex to the midpoint of the opposite side. |
Equidistant from all three vertices. |
Inside the triangle, closer to the vertex with the larger angle. |
| Properties |
Divides the triangle into three smaller triangles with equal areas. |
The circumradius (radius of the circle) is equal to half the longest side of the triangle. |
The incircle (circle tangent to all three sides) is always inside the triangle, except for equilateral triangles where it coincides with the circumcircle. |
Sample Problems on Centroid
Problem 1: Find the centroid of triangle if the coordinates of triangle are (9,8) , (6,7) and (2,3).
Solution:
As we know, Centroid of Triangle = (x1+x2+x3/3 , y1+y2+y3/3)
⇒ x-coordinate of centroid = (9+6+2)/3 , y-coordinate of centroid = (8+7+3)/3
⇒ Centroid of Triangle = (5.66 , 6)
Problem 2: If the centroid of triangle is (5,4) , and two coordinates of the triangle is (2,3) and (4,5), then find its third coordinate.
Solution:
To find the third coordinate of triangle , let us suppose the third coordinate of triangle is (x3 , y3)
the first two coordinate of triangle are (2,3) and (4,5)
⇒ 5 = (2+3+x3)/3, and 4 = (3+5+y3)/3
⇒ x3 = 15-3-2 = 10, and y3 = 12-5-3 = 4
⇒ (x3, y3) = (10 , 4)
Problem 3: If the three coordinates of triangle are (2 , 3) , (x , 5) and (3 , y) and centroid of triangle is (4, 3) then find the value of x and y.
Solution:
Using above centroid formula i.e., (x, y) = [(x1+x2+x3)/3 , (y1+y2+y3)/3]
⇒ 4 = 2+x+3/3 , and 3 = (3+5+y)/3
⇒ x = 12-5 = 7, and y = 9-8 = 1
So , the value of x is 7 and value of y is 1.
Problem 4: What is coordinate of centroid of a rectangle of length 10cm and breadth 6cm ?
Solution:
For rectangle coordinate of centroid is (L/2 , B/2)
Given: L = 10 cm and B = 6cm
Centroid of Given Rectangle = (10/2 , 6/2)
Thus, (5 , 3) is coordinate of centroid of given rectangle.
Practice Problems on Centroid
Here are some practice problems on centroids:
Problem 1: Consider a triangle with vertices at (0, 0), (4, 0), and (2, 3). Find the coordinates of the centroid.
Problem 2: For a quadrilateral with vertices at (-1, 2), (3, 5), (6, 1), and (2, -3), determine the coordinates of the centroid.
Problem 3: A regular hexagon is inscribed in a circle of radius 5 units. Find the coordinates of the centroid of this hexagon.
Problem 4: Given a composite figure composed of a rectangle with vertices at (0, 0), (0, 4), (3, 4), and (3, 0), and a triangle with vertices at (3, 0), (5, 4), and (6, 0), calculate the coordinates of the centroid of the entire figure.
Problem 5: For a line segment with endpoints at (2, -3) and (6, 4), find the coordinates of the point which divides the segment in the ratio 2:1.
Centroid: FAQs
1. What do you mean by Centroid?
Centroid is the center point of any geometric figure. It is center of gravity.
2. What is Centroid of Triangle?
Centroid is the point of triangle , where all medians of triangle meet . In other words , the point of intersection medians of triangle is known as centroid of triangle.
3. What is Median?
A median of a triangle is a line segment that meet the vertex at the mid-point of the side that is opposite to that vertex.
4. What is the Formula of Centroid of Triangle?
Coordinates of centroid of triangle are given by (x1+x2+x3/3 , y1+y2+y3/3), where (x1 , y1) , (x2 , y2) and (x3 , y3) are the vertices of triangles.
5. Can Centroid lie outside the Geometric Figure?
No, centroid can never lie outside the geometric figure.
6. How do you Calculate the Coordinates of Centroid?
We can calculate the coordinates of centroid using various formulas discussed in this article.
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