# Centroid Calculator

URL: https://speedycalculator.com/calculators/math/centroid/
Last updated: 2026-10-30

> The centroid calculator finds the geometric centre of a triangle from its three vertices, of a set of points, or of a polygon (area centroid) from its vertex coordinates.

## Inputs

- Shape
- x₁
- y₁
- x₂
- y₂
- x₃
- y₃
- x-coordinates (in order)
- y-coordinates (same order)

## Results

- Centroid x
- Centroid y
- Area

## Formula

```
Triangle: G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
Polygon: Cx = Σ(xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) / (6A)
```

## Explanation

The centroid is the balance point of a uniform flat plate. In a triangle it is where the three medians meet, two-thirds of the way from each vertex. Structural engineers use centroids to find neutral axes of beams.

### How to use the Centroid Calculator

- **Shape**
- **x₁**
- **y₁**
- **x₂**
- **y₂**
- **x₃**
- **y₃**
- **x-coordinates (in order)**
- **y-coordinates (same order)**

Results update as you type, and everything is calculated in your browser.

### How it works

For a triangle or a set of points the centroid is the average of the coordinates. For a polygon the area-weighted formula is used, which differs from the plain average for most shapes.

### Inputs explained

FieldWhat to enterShapeChoose one: Triangle, Set of points (average), Polygon (area centroid)x₁A numbery₁A numberx₂A numbery₂A numberx₃A numbery₃A numberx-coordinates (in order)A list of numbers separated by commas, spaces or new linesy-coordinates (same order)A list of numbers separated by commas, spaces or new lines

## Worked example

Inputs — Shape: Triangle; x₁: 0; y₁: 0; x₂: 6; y₂: 0; x₃: 3; y₃: 6.

Results — **Centroid x = 3**; Centroid y = 2; Area = 18.

## Key terms

- **Median**: A segment from a vertex to the midpoint of the opposite side.

## FAQ

**Where is the centroid of a triangle?**
At the intersection of the medians, 2/3 of the way from each vertex.

**Is the polygon centroid the average of its vertices?**
Not in general – only for triangles and some symmetric shapes.

**Can the centroid lie outside the shape?**
Yes, for non-convex shapes like an L or a crescent.

**What is the difference from the center of mass?**
The centroid is the center of mass of a uniform shape.


## Sources

- [Wolfram MathWorld – Geometric centroid](https://mathworld.wolfram.com/GeometricCentroid.html)
- [OpenStax – Calculus Volume 2](https://openstax.org/details/books/calculus-volume-2)