Centroid Calculator

Geometry
Hint: Uses the shoelace formula for polygon area and the standard area-weighted centroid formula: Cx = (1/(6A)) Σ (xi + xi+1)·(xi·yi+1 − xi+1·yi), and similarly for Cy.

Understanding the concept of a shape’s centroid is essential in fields such as mathematics, physics, engineering, architecture, and structural design. Whether you are examining a triangle, a complicated polygon, or engaging with CAD-based geometry, the centroid provides you with the precise balance point of the figure. 

Our Centroid Calculator simplifies the entire procedure by instantly calculating the centroid using the coordinates you input.

What Is a Centroid?

The centroid (also called the geometric center or center of mass of a uniform-density shape) is the average position of all the points within a shape. It is located at the average of all x-coordinates and y-coordinates. The centroid is independent of size; only the relative coordinates matter.

Centroid Formula for a Triangle

The centroid of a triangle is located at the intersection of its medians, and it divides each median in a 2:1 ratio.

If the triangle has vertices at:

A (x1,y1), B (x2,y2), C (x3,y3)

Then the centroid (Xc, Yc) is:

Xc = x1 + x2 + x3 3 Yc = y1 + y2 + y3 3

Centroid of an N-Sided Polygon

For a polygon with vertices:

( x1 , y1 ) , ( x2 , y2 ) , … ( xn , yn )

The centroid is calculated using the shoelace formula and area-weighted averages.

Step 1: Compute the polygon area

A = 1 2 ∑ i=1 ( xi yi+1 − xi+1 yi )

Where ( xn+1 , yn+1 ) = ( x1 , y1 ) .


Step 2: Compute centroid coordinates

Xc = 1 6A ∑ i=1 ( xi + xi+1 ) ( xi yi+1 − xi+1 yi ) Yc = 1 6A ∑ i=1 ( yi + yi+1 ) ( xi yi+1 − xi+1 yi )

How to find the Centroid using our Calculator?

  1. Select the “Type of Shape” whose centroid is to be calculated.
  2. Enter the “Co-ordinates of the Vertices” of the above shape.
  3. Click “Calculate” to find the Centroid.

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