Find the mass-weighted position of a system
Calculate the center of mass of point masses in one, two or three dimensions. Add up to 20 points with mixed mass and length units, or solve one unknown mass or position from a target center.
Last updated: July 30, 2026- 1D, 2D and 3D
- Mixed units
- Exact decimal arithmetic
- Inverse solving
- No sign-up
Center of Mass Calculator
Use one common origin and axis direction for every coordinate. Mass must be positive. Coordinates may be positive, negative or zero.
Result
Ready for your values
Add point masses, choose the active dimensions, then select Calculate.
Center of mass
| Quantity | Displayed value | SI equivalent |
|---|
What Is the Center of Mass?
The center of mass is the mass-weighted average position of a system. It is a single representative point for the system’s translational motion, not a complete description of its mass distribution. For separate particles, components or objects, each point contributes in proportion to its mass and coordinate.
A heavier point pulls the center closer to its position. Equal masses place the center at the ordinary coordinate average. Signed coordinates matter because the answer depends on the chosen origin and positive axis direction. A negative result does not mean negative mass. It means the center lies on the negative side of the selected origin.
This calculator treats every row as a point mass. If a row represents an extended part, place the point at that part’s own center of mass. The method then works for composite objects when each part’s mass and center location are known.
How to Use This Center of Mass Calculator
Keep one coordinate frame for the full calculation. Do not change the origin or reverse an axis between rows.
Choose the calculation
Select center of mass, one unknown mass, or one unknown position.
Select the dimensions
Use 1D for x only, 2D for x and y, or 3D for x, y and z.
Enter the known points
Add each positive mass and its active signed coordinates. Choose units for every row.
Complete inverse inputs
For an inverse mode, enter the target center and the unknown point’s known mass or position.
Choose answer units
Select the length and mass units used for the main result and total mass.
Set display precision
Choose up to 4, 6, 8, 10 or 12 significant digits. Internal arithmetic stays exact.
Review the calculation
Check the weighted sums, mass shares, SI equivalents and model assumptions.
Center of Mass Formula
For point masses, first add every mass to obtain total mass M. Then divide each mass-weighted coordinate sum by M.
In Cartesian coordinates, calculate every active component separately:
The mass units disappear from each coordinate ratio, but mixed mass units must still be converted before the terms are added. Length units also need conversion when points use different units.
Unknown mass formula
For known-system mass M, known moment S, target coordinate R and unknown position ru:
In 2D or 3D, every informative axis must produce the same positive mass. Otherwise the requested target and point position are inconsistent.
Unknown position formula
For entered positive mass mu and target coordinate R:
Apply the same equation to x, y and z. The result uses the same origin and axis directions as the known points.
Worked Center of Mass Examples
Two-dimensional point masses
Take masses 1 kg, 2 kg and 3 kg at (0, 0), (3, 0) and (0, 4) metres. Total mass is 6 kg. The x-moment is 1(0) + 2(3) + 3(0) = 6 kg·m, so xCM = 6/6 = 1 m. The y-moment is 1(0) + 2(0) + 3(4) = 12 kg·m, so yCM = 12/6 = 2 m. The center of mass is (1, 2) m.
Finding an unknown mass
Suppose known masses 2 kg at 0 m and 3 kg at 10 m have total mass 5 kg and moment 30 kg·m. Add an unknown mass at 20 m so the target center is 10 m. The unknown mass is [10(5) − 30] / (20 − 10) = 2 kg. Checking the full system gives [30 + 2(20)] / 7 = 10 m.
How to Interpret the Result
| Result feature | Meaning | Check |
|---|---|---|
| Signed coordinate | Location relative to your origin and positive axis | Confirm every row uses the same coordinate frame |
| Mass share | Each point’s fraction of total mass | All shares should add to 100% |
| Coordinate contribution | (mi/M) times the point coordinate | Contributions add to the center coordinate |
| Bounds | Positive masses create a weighted average | Each component lies between the smallest and largest entered component |
| Inverse solution | A mass or point position that produces the target | Recalculate the full system and recover the target |
A center of mass may lie in empty space. A ring, hollow object or separated group of masses can have a center where no material exists. For positive point masses, however, the vector result stays inside their convex hull.
Center of Mass, Center of Gravity and Centroid
| Term | Depends on | When it matches center of mass |
|---|---|---|
| Center of mass | Mass distribution | It is the reference quantity |
| Center of gravity | Mass distribution and gravitational field | When the gravitational field is effectively uniform across the object |
| Geometric centroid | Shape or geometric measure | When density is uniform under the chosen geometric model |
Weights measured in one uniform gravitational field are proportional to mass, so the common gravitational factor cancels in the weighted average. Do not combine weights measured under different gravitational fields as though they were masses.
Point Masses, Composite Bodies and Continuous Objects
The discrete formula applies directly to separate particles and to composite systems represented by components. For a uniform rectangular plate, rod or cylinder, first use that component’s known geometric center as its point location. Then combine the components using their masses.
A continuous object with density that varies through space requires integration:
This page does not integrate a density function or reconstruct a solid from geometry. You may still use it after dividing the object into parts whose masses and individual centers are known. Finer divisions usually improve an approximation, but measurement and modeling uncertainty still limit the result.
Accuracy, Units and Numerical Precision
- Use mass, not force. Enter avoirdupois pound mass in lb, never pound-force or newtons.
- Use one physical origin and one direction for each axis, even when rows use different length units.
- Keep more working precision than the measured data supports, then round the final result appropriately.
- Do not infer measurement accuracy from extra displayed digits.
- For an inverse problem, verify that the solved mass is physically available or that the solved position fits the actual assembly.
The calculator parses entered decimals and exact conversion factors as rational numbers. It does not use ordinary floating-point multiplication for the weighted sums. This prevents common failures such as overflow from a large mass-coordinate product, false cancellation between large signed coordinates and loss of a one-unit difference near 1016.
Related Calculators
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Method Sources and Use Notice
This calculator provides educational and preliminary engineering results for an ideal point-mass model. It does not replace validated mass-property analysis, calibrated measurements, tolerance studies, nonuniform-gravity modeling or qualified engineering review. Verify consequential lifting, stability, vehicle, aerospace, structural and safety decisions independently.
Center of Mass Calculator FAQs
What is the formula for center of mass?
For point masses, total mass is M = Σmi and the center position is r⃗CM = Σ(mir⃗i)/M. Calculate x, y and z components separately.
Can center of mass coordinates be negative?
Yes. A negative coordinate means the center lies on the negative side of your chosen origin. It does not mean mass is negative.
Do all masses and coordinates need the same unit?
No. This calculator converts each row to kilograms and metres before adding terms. Every coordinate must still use the same physical origin and axis direction.
Is the center of mass always inside an object?
No. It can lie in empty space for a hollow, curved or separated system. With positive point masses, the result lies inside the points’ convex hull.
What is the difference between center of mass and center of gravity?
Center of mass depends only on mass distribution. Center of gravity also depends on the gravitational field. They coincide when that field is effectively uniform across the object.
How do I calculate center of mass in 2D or 3D?
Use the weighted-average formula independently for x, y and z. The total mass denominator is the same for every component.
Can I enter zero or negative mass?
No. Each active row must have positive mass. Remove a point that has no mass. The physical point-mass model used here does not support negative mass.
What does the point-mass assumption mean?
Each row places all of one component’s mass at one coordinate. For an extended component, use its own center of mass as that coordinate.
How does the calculator solve an unknown mass?
It rearranges the weighted-average equation using the target center and unknown point position. In 2D or 3D, every informative axis must give exactly the same positive mass.
Is the centroid the same as the center of mass?
Only under the applicable uniform-density model. A geometric centroid depends on shape, while center of mass depends on how mass is distributed.