# Centroid Calculator

Calculate the centroid of a triangle from three vertex coordinates. Find the center of mass with the area included. Free online centroid tool.

## What this calculates

Find the centroid of a triangle by entering the coordinates of its three vertices. The centroid is the point where all three medians intersect, and this calculator also computes the triangle's area.

## Inputs

- **Vertex A: X**
- **Vertex A: Y**
- **Vertex B: X**
- **Vertex B: Y**
- **Vertex C: X**
- **Vertex C: Y**

## Outputs

- **Centroid X** — The x-coordinate of the centroid.
- **Centroid Y** — The y-coordinate of the centroid.
- **Centroid (x, y)** — formatted as text — The centroid as a coordinate pair.
- **Triangle Area** — The area of the triangle formed by the three vertices.

## Details

The centroid of a triangle is its geometric center, the point where the three medians meet. A median is a line segment from a vertex to the midpoint of the opposite side.

**The Formula:**

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

It is simply the average of the three x-coordinates and the average of the three y-coordinates.

**Example:**

For a triangle with vertices at A(0, 0), B(6, 0), and C(3, 9):

- Centroid X = (0 + 6 + 3) / 3 = 3
- Centroid Y = (0 + 0 + 9) / 3 = 3
- Centroid = (3, 3)

**Properties of the Centroid:**

- The centroid always lies inside the triangle, regardless of the triangle's shape.
- It divides each median into a 2:1 ratio. The centroid is two-thirds of the way from each vertex to the opposite midpoint.
- For a triangle made of uniform material, the centroid is the center of mass (the balance point).
- The centroid is one of the four classical triangle centers, alongside the incenter, circumcenter, and orthocenter.

**Area Calculation:**

This calculator also computes the triangle's area using the shoelace formula:

Area = |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| / 2

## Frequently Asked Questions

**Q: What is the centroid of a triangle?**

A: The centroid is the point where all three medians of a triangle intersect. A median connects a vertex to the midpoint of the opposite side. You can think of it as the balance point: if you cut the triangle out of cardboard, the centroid is where it would balance on a pin.

**Q: Is the centroid always inside the triangle?**

A: Yes, the centroid always lies inside the triangle, no matter what shape it is. This is different from the circumcenter and orthocenter, which can be outside the triangle for obtuse triangles. The centroid divides each median in a 2:1 ratio from vertex to midpoint.

**Q: What is the difference between centroid, incenter, and circumcenter?**

A: The centroid is the intersection of the medians (center of mass). The incenter is the intersection of the angle bisectors (center of the inscribed circle). The circumcenter is the intersection of the perpendicular bisectors (center of the circumscribed circle). All three are the same point only for equilateral triangles.

**Q: How is the centroid used in real life?**

A: Engineers and architects use centroids to find the center of mass of structural components. In computer graphics, centroids help position labels on polygon regions. In physics, the centroid determines where to apply a force for uniform loading. Robotics uses centroids for balance calculations.

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Source: https://vastcalc.com/calculators/math/centroid
Category: Math
Last updated: 2026-04-08