Centroid Calculator

Find the centroid of a polygon by entering vertex coordinates. Supports triangles, quadrilaterals, and general polygons.

Enter Vertex Coordinates

Vertex 1 (A)

Vertex 2 (B)

Vertex 3 (C)

Result

Centroid Coordinates
(3, 3)
coordinate units
Centroid x --
Centroid y --
Polygon Area --
Number of Vertices --

Step-by-Step Solution

C = ((x1+x2+x3)/3, (y1+y2+y3)/3)

What Is a Centroid?

The centroid of a polygon is the geometric center (center of mass) of the shape, assuming a uniform density. For a triangle, the centroid is the point where all three medians intersect. For more complex polygons, the centroid is calculated using the Shoelace formula combined with weighted coordinate averages.

The centroid divides each median of a triangle in a 2:1 ratio, with the longer segment being closer to the vertex. This property makes the centroid the balance point of the triangle: if you cut out a triangular shape from uniform material, it would balance perfectly on the centroid.

Centroid Formulas

Triangle Centroid

The simplest centroid calculation: average the coordinates of the three vertices.

C = ((x1+x2+x3)/3, (y1+y2+y3)/3)

General Polygon Centroid

Uses the Shoelace formula to weight vertex contributions by area.

Cx = (1/6A) * Sum[(xi+xi+1)(xi*yi+1 - xi+1*yi)]

Shoelace Area Formula

Compute the signed area of a polygon from vertex coordinates.

A = (1/2)|Sum(xi*yi+1 - xi+1*yi)|

Weighted Average

For a set of discrete points, the centroid is the arithmetic mean of coordinates.

C = (Sum(xi)/n, Sum(yi)/n)

How to Find the Centroid

  1. For a triangle: Simply average all x-coordinates and all y-coordinates of the three vertices. This gives the centroid directly.
  2. For a general polygon: Use the Shoelace formula to first compute the signed area A. Then compute the centroid x and y coordinates using the weighted formula involving cross products of consecutive vertices.
  3. Verify: The centroid should lie inside the polygon for convex shapes. For concave polygons, the centroid may lie outside the shape.

Properties of the Centroid

  • The centroid always lies inside a convex polygon.
  • For a triangle, the centroid divides each median in a 2:1 ratio from the vertex.
  • The centroid minimizes the sum of squared distances to all points within the shape.
  • For symmetric shapes, the centroid lies on every axis of symmetry.
  • The centroid of a composite shape can be found by combining the centroids and areas of its component parts.

Applications of Centroid Calculations

Centroid calculations have many practical uses:

  • Structural Engineering: Determining the center of mass of beams and cross-sections for stress analysis.
  • Computer Graphics: Placing labels, computing mesh properties, and determining object centers.
  • Geography/GIS: Finding the geographic center of regions, countries, or land parcels.
  • Robotics: Computing the balance point for robotic components and assemblies.
  • Physics: Locating the center of gravity for stability analysis.