Centroid Calculator

With this centroid calculator, we're giving you a hand at finding the centroid of many 2D shapes, as well as of a set of points. In just a few clicks and several numbers inputted, you can find the centroid of a rectangle, triangle, trapezoid, kite, or any other shape imaginable — the only restrictions are that the polygon should be closed, non-self-intersecting, and consist of a maximum of ten vertices. Also, if you're searching for a simple centroid definition, or formulas explaining how to find the centroid, you won't be disappointed — we have it all.

A centroid, also called a geometric center, is the center of mass of an object of uniform density. To make it easier to understand, you can imagine it as the point on which you should position the tip of a pin to have your geometric figure balanced on it.

We now know the centroid definition, so let's discuss how to localize it. In the following section, we show you the centroid formula.

In general, a centroid is the arithmetic mean of all the points in the shape. The x- and y-coordinate of the centroid read

Gₓ = (x₁ + x₂ + x₃ + ... + x_k) / k

Gᵧ = (y₁ + y₂ + y₃ +... + y_k) / k

where (x₁,y₁), ..., (x_k,y_k) are the vertices of our shape.

For convex shapes, the centroid lays inside the object; for concave ones, the centroid can lay outside (e.g., in a ring-shaped object).

Finding the centroid of a triangle or a set of points is an easy task – the formula is really intuitive. However, if you're searching for the centroid of a polygon – like a rectangle, a trapezoid, a rhombus, a parallelogram, an irregular quadrilateral shape, or another polygon- it is, unfortunately, a bit more complicated.

To find the centroid of a triangle ABC, you need to find the average of vertex coordinates. So if A = (X₁,Y₁), B = (X₂,Y₂), C = (X₃,Y₃), the centroid formula is:

G = [ (X₁ + X₂ + X₃)/3 , (Y₁ + Y₂ + Y₃)/3 ]

If you don't want to do it by hand, just use our centroid calculator! 🙂

For special triangles, you can find the centroid quite easily:

1. Centroid of an equilateral triangle

   If you know the side length, a, you can find the centroid of an equilateral triangle:

   G = (a/2, a√3/6)

   (you can determine the value of a with our equilateral triangle calculator)

2. Centroid of an isosceles triangle

   If your isosceles triangle has legs of length l and height h, then the centroid is described as:

   G = (l/2, h/3)

   (if you don't know the leg length l or the height h, you can find them with our isosceles triangle calculator)

3. Centroid of a right triangle

   For a right triangle, if you're given the two legs, b and h, you can find the right centroid formula straight away:

   G = (b/3, h/3)

   (the right triangle calculator can help you to find the legs of this type of triangle)

Sometimes people wonder what the midpoint of a triangle is — but hey, there's no such thing! The midpoint is a term tied to a line segment. It's the middle point of a line segment and therefore does not apply to 2D shapes.

However, you can say that the midpoint of a segment is both the centroid of the segment and the centroid of the segment's endpoints.

You can check it in this centroid calculator: choose the N-points option from the drop-down list, enter 2 points, and input some random coordinates. The result should be equal to the outcome from the midpoint calculator.

(Keep in mind that calculations won't work if you use the second option, the N-sided polygon. Why? That's because that formula uses the shape area, and a line segment doesn't have one).

To find the centroid of a set of k points, you need to calculate the average of their coordinates:

Gₓ = (x₁ + x₂ + x₃ +... + x_k) / k

Gᵧ = (y₁ + y₂ + y₃ +... + y_k) / k

And that's it! Did you notice that it's the general formula we presented before?

Calculating the centroid of a set of points is used in many different real-life applications, e.g., in data analysis. The most popular method is K-means clustering, where an algorithm tries to minimize the squared distance between the data points and the cluster's centroids.

To calculate a polygon's centroid, G(C_x, C_y), which is defined by its n vertices (x₀,_y), (x₁,y₁), ..., (x_n-1,y_n-1), all you need to do is to use these following three formulas:

where A is a polygon's signed area:

Remember that the vertices should be inputted in order, and the polygon should be closed — meaning that the vertex (x₀, y₀) is the same as the vertex (x_n, y_n).

If that centroid formula scares you a bit, wait no further — use this centroid calculator, as we've implemented that equation for you.

Even though you can find many different formulas for a centroid of a trapezoid on the Internet, the equations presented above are universal — you don't need to have the origin coinciding with one vertex, nor the trapezoid base in line with the x-axis. Here, you can find the centroid position by knowing just the vertices. The same applies to the centroid of a rectangle, rhombus, parallelogram, pentagon, or any other closed, non-self-intersecting polygon.

To use this centroid calculator, simply input the vertices of your shape as Cartesian coordinates. Let's check how to find the centroid of a trapezoid:

1. Choose the type of shape for which you want to calculate the centroid. In our case, we will choose an N-sided polygon.

2. Enter the parameter for N (if required). For our example, we need to input the number of sides of our polygon. As the trapezoid is, of course, the quadrilateral, we type 4 into the N box.

3. The fields for inputting coordinates will then appear. Enter the coordinates of the vertices of your shape. Let's assume our trapezoid vertices are:
   • A = (1,1)
   • B = (2,4)
   • C = (5,4)
   • D = (11,1)
4. Our centroid calculator will then display the answer! The centroid of the trapezoid of our choice is (4.974, 2.231).