Centroid Calculator

Calculate the centroid of a triangle, two-dimensional figures, and set of points. The centroid calculator takes a second to compute the coordinates of the point where the medians intersect each other. Get a pictorial view and calculations to understand the solution better.

The calculator supports a maximum of 10 vertices for a polygon that is non-intersecting and closed.

What Is The Centroid of a Triangle?

“The point through which all the three medians of a triangle pass is said to be the centroid of a triangle”

The concept of the centroid relates to that of the midpoint of a line segment.

What Is the General Formula for Centroid?

For the triangle above, we have three vertices:\(A = (x_1, y_1), B = (x_2, y_2), C = (x_3, y_3)\)

Their centroid is the average of x and y coordinates and is calculated with the formula below:

\(Centroid = \dfrac{(x_1 + x_2 + x_3)}{3}, \dfrac{(y_1 + y_2 + y_3)}{3}\)

Centroid Formula for Other Triangles:

Centroid of an Equilateral Triangle:

If the side length is known, you can find the centroid of an equilateral triangle:

\(G = \left(\dfrac{a}{2}, a\dfrac{√3}{6}\right)\)

Centroid of an Isosceles Triangle:

If an isosceles triangle has legs of length ‘l’ and height ‘h’, then the centroid is:

\(G = \left(\dfrac{l}{2}, \dfrac{h}{3}\right)\)

Centroid of a Right Triangle:

If the two legs ‘b’ and ‘h’ are given, then the right centroid formula is:

\(G = \left(\dfrac{b}{3}, \dfrac{h}{3}\right)\)

Find the Centroid of a Polygon (Trapezoid, Rectangle, and others):

For a closed polygon (Vertex \((x_0, y_0)\) = Vertex \((x_n, y_n)\)), the Centroid can be calculated by using the following equations:

\(C_{x}=\frac{1}{6A}\sum_{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i})\)

\(C_{y}=\frac{1}{6A}\sum_{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i})\)

\(A=\frac{1}{2}\sum_{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})\)

How to Find the Centroid of a Set of Points?

If you need to find the centroid of N points, then you can calculate the average of their coordinates, such as:\(G_x = \dfrac{\left(x_1 + x_2 + x_3 +… + x_N\right)}{N}\)

Example Problems:

Centroid of a Triangle:

Suppose that a triangle ABC has the following vertex coordinates:

A = (4,5)
B = (20,25)
C = (30,6)

Centroid Calculations:

\(Centroid = \dfrac{(x_1 + x_2 + x_3)}{3}, \dfrac{(y_1 + y_2 + y_3)}{3}\)

\(Centroid = \dfrac{(4+20+30)}{3}, \dfrac{(5+25+6)}{3}\)

\(Centroid = \dfrac{(54)}{3}, \dfrac{(36)}{3}\)

\(Centroid = (18, 12)\)

Properties of the Centroid:

The intersection of the medians always forms a centroid
It is said to be one of the points of concurrency of a triangle
Centroid of a triangle is always located inside it
Centroid divides each median in a ratio of 2:1. In other words, it will always be 2/3 of the way along any given median

Scholars Also Ask:

Can a Centroid Be Outside of a Shape?

Yes! It can be.

Remember that if a shape possesses an axis of symmetry, then its centroid point will always be located on that axis. Moreover, it is possible for the centroid of an object to be located outside of its geometric boundaries.

How Do You Find the Centroid of a 3d Object?

When it comes to finding the centroid of a 3D shape, you have to look for x, y, and z coordinates (x̄, ȳ, and z̄). This can be referred to as the x, y, and z coordinates of the point, which is the centroid of the shape.