Your result
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Enter your values, then click Calculate result.How this calculator helps
This center of mass calculator combines measured or assumed positive point masses at explicit Cartesian coordinates. It answers a different question from a geometric midpoint because each coordinate is weighted by mass. The three-dimensional result can also represent a planar system when every z coordinate is zero. It does not infer the mass distribution of a real object from its outline; the point model and coordinate convention are supplied by the user.
How to use it
- 1
Choose a common Cartesian coordinate frame and compatible mass and length units.
- 2
Paste each point as positive mass, x, y and z; use zero z for a planar model.
- 3
Click Calculate result and inspect center coordinates, total mass and first moments.
- 4
Check weighted sums and coordinate bounds before using the point model in a larger mechanics problem.
Formula and methodology
Total mass M = Σmᵢ. Center coordinates are x̄ = Σmᵢxᵢ/M, ȳ = Σmᵢyᵢ/M and z̄ = Σmᵢzᵢ/M. First moments are the corresponding mass-coordinate sums.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
Place a 2-unit point mass at (0,0,0) and a 1-unit point mass at (6,3,0). Total mass is 3. First moments are (6,3,0), so the center of mass is (2,1,0). The center lies nearer the heavier point. An unweighted geometric midpoint would be (3,1.5,0), which is different. Doubling both masses keeps the center at (2,1,0) while doubling total mass and each first moment.
How to interpret your result
Read the center, total mass and first moments together. The coordinate must lie within the convex span of the supplied positive-mass positions, so an output beyond every point along one axis signals an input or arithmetic problem. The result depends on the point model and coordinate frame. It is a weighted position, not a force location certified for every loading condition.
For different inputs or formulas, use Midpoint Calculator; Momentum Impulse Calculator; Vector Dot Cross Calculator.
Related questions this calculator covers
- center of mass calculator
- point mass center of mass calculator
- 3d mass center calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| Unequal pair | masses two and one at the example points give (2,1,0). |
| Equal masses | the weighted mean becomes the ordinary coordinate mean. |
| Common translation | shifting every x by ten shifts center x by ten. |
Common mistakes to avoid
- Mixing coordinate frames or length units.
- Averaging positions without their masses.
- Confusing first moments with moments of inertia or torque.
Independently sum masses and each mass-coordinate product for the worked example. Multiply the reported center by total mass and recover the three first moments. Double every mass and verify the center is fixed. Add a common coordinate offset and verify the center receives the same offset. For each axis, check the center falls between the minimum and maximum input coordinates.
Authoritative reference. Reviewed on 6 October 2026 for the specific disclosed method or measurement context. The arithmetic is independently implemented. A linked reference does not certify an individual result or extend the worksheet beyond its stated scope.What can affect the result?
Define one coordinate frame
All positions must use the same origin, axis directions and length unit. Coordinates from different local drawings cannot be combined without first transforming them into a common frame. Negative coordinates are valid and mean positions on the opposite side of an axis relative to the chosen origin. The result uses the same frame and units. A coordinate is not a distance from the origin unless the relevant geometry and axis have been specified.
Represent each point with four entries
Enter one row per point: positive mass, x, y and z. Separate numbers by spaces or commas and use a new line for the next point. One through one hundred rows are supported, with finite values inside the stated numerical bound. Supply z as zero for a planar example rather than omitting the fourth entry. Each point is an independent contribution; identical coordinates may represent separate masses and are not silently deduplicated.
Use compatible positive mass units
All mass entries must use one common unit, such as kilograms or grams. Scaling every mass by the same factor leaves the center unchanged, but mixing units between rows does not. This worksheet uses positive masses and rejects zero or negative mass entries. A negative signed coordinate is different from negative mass. If a modeling technique uses removal or signed density, it needs an explicitly designed method outside this positive point-mass interface.
Mass weighting differs from averaging points
The center is a weighted mean of positions, not an ordinary average of point coordinates unless all masses are equal. A heavier mass influences the result more strongly. Adding a new point shifts the center toward that point in proportion to its share of total mass. Counting one object twice doubles its modeled contribution, so verify whether repeated rows are intentional. The point count is displayed separately from total mass to expose that distinction.
First moments make the result auditable
The worksheet displays the mass-coordinate sums before division. Their units are mass multiplied by length, such as kg·m, rather than force or rotational torque. Dividing each by total mass returns a coordinate in the original length unit. These are first moments associated with the chosen origin and axes. They must not be mistaken for moments of inertia, which involve squared distances and a different physical calculation.
Point models simplify extended objects
An extended component can sometimes be represented by its known mass and its own independently determined center of mass. That representation requires a valid component model. A geometric center is not always its mass center when density is uneven. This page does not find a continuous density integral or calculate the centroid of an arbitrary shape. Verify how each row represents the actual component before interpreting the aggregate center.
Mass center does not establish stability
The computed point tells where the mass-weighted position lies under the supplied model. It does not by itself determine whether a structure tips, a suspension is safe or a vehicle meets handling requirements. Support geometry, forces, acceleration, constraints and uncertainties can matter. The worksheet is appropriate for a transparent mechanics calculation, while safety-critical decisions need the relevant complete assessment. Display digits reflect arithmetic rather than position-measurement certainty.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Mixing coordinate frames or length units. Read the center, total mass and first moments together. The coordinate must lie within the convex span of the supplied positive-mass positions, so an output beyond every point along one axis signals an input or arithmetic problem. The result depends on the point model and coordinate frame. It is a weighted position, not a force location certified for every loading condition. Entries remain on this device for the calculation and are not submitted to a calculation server. Browser floating-point arithmetic and displayed rounding do not establish real-world measurement certainty.
Sources and review information
Frequently asked questions
Can I use a two-dimensional example?＋
Yes. Enter zero for every z coordinate.
Is this the same as a midpoint?＋
Only for two equal masses in a common coordinate frame.
Can masses use different units?＋
No. Convert all masses to one common unit first.
Are negative coordinates allowed?＋
Yes. Positions can be signed; masses must remain positive.
Does this assess tipping safety?＋
No. Center-of-mass arithmetic alone is not a stability assessment.