# Center of Mass Calculator

URL: https://speedycalculator.com/calculators/math/center-of-mass/
Last updated: 2026-10-30

> The center of mass calculator finds the balance point of up to six point masses on a line, in a plane or in space, as the mass-weighted average of their coordinates.

## Inputs

- Dimension
- Number of point masses
- Mass m₁
- x₁
- y₁
- z₁
- Mass m₂
- x₂
- y₂
- z₂
- Mass m₃
- x₃
- y₃
- z₃
- Mass m₄
- x₄
- y₄
- z₄
- Mass m₅
- x₅
- y₅
- z₅
- Mass m₆
- x₆
- y₆
- z₆

## Results

- x of the center of mass
- y of the center of mass
- z of the center of mass
- Total mass

## Formula

```
x_cm = Σ mᵢxᵢ / Σ mᵢ,  y_cm = Σ mᵢyᵢ / Σ mᵢ,  z_cm = Σ mᵢzᵢ / Σ mᵢ
```

## Explanation

The center of mass is where an object or system balances. Engineers need it for vehicle stability, aircraft loading and cranes; physicists use it to describe the motion of a system as if all the mass were in one point.

### How to use the Center of Mass Calculator

- **Dimension**
- **Number of point masses**
- **Mass m₁**
- **x₁**
- **y₁**
- **z₁**
- **Mass m₂**
- **x₂**
- **y₂**
- **z₂**
- **Mass m₃**
- **x₃**
- **y₃**
- **z₃**
- **Mass m₄**
- **x₄**
- **y₄**
- **z₄**
- **Mass m₅**
- **x₅**
- **y₅**
- **z₅**
- **Mass m₆**
- **x₆**
- **y₆**
- **z₆**

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

### How it works

Choose the dimension and number of masses, then enter each mass and its coordinates (any consistent units). Heavier masses pull the centre toward themselves.

### Inputs explained

FieldWhat to enterDimensionChoose one: 1D (on a line), 2D (in a plane), 3D (in space)Number of point massesChoose one: 2, 3, 4, 5 …Mass m₁A numberx₁A numbery₁A numberz₁A numberMass m₂A numberx₂A numbery₂A numberz₂A numberMass m₃A numberx₃A numbery₃A numberz₃A numberMass m₄A numberx₄A numbery₄A numberz₄A numberMass m₅A numberx₅A numbery₅A numberz₅A numberMass m₆A numberx₆A numbery₆A numberz₆A number

### Example values

How the main result changes when **Mass m₁** changes and everything else stays at the example values above:

Mass m₁x of the center of mass10.33331.50.368420.42.50.428630.454540.5

## Worked example

Inputs — Dimension: 2D (in a plane); Number of point masses: 3; Mass m₁: 2; x₁: 1; y₁: 1; Mass m₂: 3; x₂: 4; y₂: 2; Mass m₃: 5; x₃: -2; y₃: 3.

Results — **x of the center of mass = 0.4**; y of the center of mass = 2.3; Total mass = 10.

## Key terms

- **Center of mass**: The mass-weighted average position.

## FAQ

**Is the center of mass the same as the centre of gravity?**
In a uniform gravitational field, yes.

**Can the center of mass be outside the object?**
Yes – a ring or a boomerang balances at an empty point.

**What units should I use?**
Any, as long as all masses share one unit and all coordinates another.

**What if one mass is zero?**
It simply does not affect the result.


## Sources

- [Wolfram MathWorld – Center of mass](https://mathworld.wolfram.com/CenterofMass.html)
- [OpenStax – Calculus Volume 2](https://openstax.org/details/books/calculus-volume-2)