Center of Mass Calculator

List what each piece weighs and where it sits, and see the balance point with the weighted sum behind every axis.

System

Dimensions
Weight each entry by

Use one unit for every entry and one unit for every coordinate. The answer comes back in the coordinate unit you typed; nothing is converted for you.

Entries

One row per point mass or per part. Paste a block from a spreadsheet into any cell to fill the rows from there. Leave a coordinate blank to read it as 0, and use a negative entry for a hole or a cut-out.

#NameMassxyShare
1—
2—
3—

Balance point

Center of mass

—

AxisWeighted sumCoordinate

System map

Formula and substitution

M = Σ wᵢ

x_cm = Σ(wᵢ·xᵢ) ÷ M

y_cm = Σ(wᵢ·yᵢ) ÷ M

    Fill in a row to find the balance point.

    Your entries and every coordinate stay in this browser and are never uploaded.

    FAQ

    Is the center of mass the same as the center of gravity?

    They coincide whenever gravity pulls equally on every part, which covers almost anything you can hold or build. The center of mass depends only on how matter is spread out; the center of gravity is where the weight appears to act. The two separate on objects tall enough for gravity to weaken measurably from bottom to top, such as a space tether or a mountain-scale structure.

    Can the balance point land somewhere with nothing in it?

    Yes, and that is normal rather than an error. A ring, a boomerang, a horseshoe and a pair of separated weights all balance around empty space. It is also why the Earth–Moon system turns about a point buried inside the Earth instead of midway between the two.

    How do I enter a shape rather than a point?

    Give the shape one row, using its own balance point as the coordinates: a rectangle balances at the middle of its diagonals, a triangle a third of the way up from the base, a half-disc about 0.4244 of its radius from the flat edge. Build a complicated outline from several such rows, and subtract a hole by entering it as a negative row at the hole’s own balance point.

    Does it matter where I put the origin?

    No — pick whichever corner or edge makes the coordinates easy to read off, as long as every row uses the same one. The answer arrives in that same frame, so moving the origin moves the reported coordinates by exactly the same amount while the physical point stays put.

    Written and reviewed by the BroBroGo editorial team.Last updated:

    Understanding the Center of Mass and Centroid

    The balance point of a physical system is the specific location where its entire distribution of matter balances in all directions. Depending on how you choose to weight your system, this point is mathematically defined in one of two ways:

    • Center of Mass: When you weight your entries by Mass, the calculator determines the physical center of mass. This is the unique point where the weighted relative position of the distributed mass sums to zero.
    • Centroid: When you weight your entries by Area or Volume, the calculator determines the geometric centroid. The centroid represents the purely geometric center of a shape.

    It is important to note that the geometric centroid and the physical center of mass are only identical if the system features a completely uniform density throughout. If any of your parts have a variable internal density, they must be split into smaller, uniform parts and entered as separate rows to maintain physical accuracy.

    How to Use the Center of Mass Calculator

    The calculator allows you to find the balance point of point masses and composite parts in one, two, or three dimensions.

    1. Configure the System Settings

    In the System section, establish the parameters of your calculation:

    • Dimensions: Select whether your system is configured in 1D, 2D, or 3D.
    • Weight each entry by: Choose the physical basis for your calculation. You can select Mass, Area, or Volume.
    • Displayed decimals: Adjust this setting to control the decimal precision of your displayed results.

    2. Manage Your Entries

    In the Entries table, you can add up to 500 rows to represent the individual pieces of your system:

    • Name: You can optionally label each entry (such as "Motor", "Battery", or "Camera"). If left blank, a numbered placeholder like "Item ‹n›" is shown.
    • Weight: Enter the weight value of the entry based on your chosen weighting basis. You can enter a negative value to represent a physical hole or a cut-out.
    • Coordinates: Enter the position of the entry along the active coordinate axes. Leaving a coordinate blank causes the tool to read it as 0.

    To speed up data entry, you can click Load example to populate the table with a pre-filled system, or click Clear entries to empty the table. If you have data prepared in an external spreadsheet, you can copy it and use the Paste block feature to paste it directly into any cell, which automatically fills multiple rows and columns. To expand your system, click the Add entry button.

    Calculation Rules and Physical Assumptions

    To ensure accurate results, the calculator operates under a specific set of mathematical rules and physical assumptions:

    • Unit Consistency: The tool does not perform unit conversions. You must use a single consistent unit for all weights and a single consistent unit for all coordinates. The final balance point coordinates are returned in the exact coordinate unit you typed.
    • Point Mass Simplification: Every row in the table is treated mathematically as a single point. This can represent a simple particle, or a composite part whose individual balance point is already known and entered as the coordinates.
    • Origin Independence: The choice of coordinate origin does not affect the physical location of the balance point. Shifting your coordinate origin shifts the output coordinates by the exact same amount.
    • Modeling Negative Space: Entering a negative weight value subtracts that entry from the total. This is the standard mathematical method for modeling a physical hole, cut-out, or cavity in a solid object.

    Interpreting the Results

    Once you fill in a row, the calculator instantly processes the data and displays the following outputs in the Balance point section:

    • Hero Display: Displays the calculated coordinates of the balance point, labeled as Center of mass (when weighted by Mass) or Centroid (when weighted by Area or Volume). You can click Copy result to copy these coordinates to your clipboard.
    • Total Weight: The sum of all entries, labeled as Total mass, Total area, or Total volume depending on your chosen weighting basis.
    • Entries Count: Displays the total number of active entries included in the calculation.
    • Calculation Table: A breakdown showing each coordinate Axis, its Weighted sum (the sum of the products of each entry's weight and its coordinate along that axis), and the final calculated Coordinate.
    • System Map: A visual plot of your entries drawn as circles. The circles are sized relative to their share of the total weight, and the balance point is marked by a cross. You can use the Projection selector to change the view, and refer to the legend labeling the "entries" and the "balance point".
    • Formula and Substitution: A step-by-step derivation with your numbers substituted in. The first line states the number of entries and the total weight (Σ w). One line per active axis then states that axis's weighted sum Σ(w · axis), divides it by the total, and gives the resulting coordinate.

    Troubleshooting and Status Messages

    The calculator displays real-time status and error messages to guide your calculations:

    • Fill in a row to find the balance point.: This invitation is shown while the table is empty.
    • Balance point from ‹n› entries.: This confirmation appears once the balance point successfully resolves.
    • With one entry the balance point is simply where that entry sits.: This is displayed when only a single row is filled.
    • A negative entry is subtracted, which is how a hole or a cut-out is normally modelled.: This appears when a negative weight is present in your table.
    • The total came out negative, so more was subtracted than added. The arithmetic still resolves, but check the entries.: This warning appears if the sum of your weights is negative.
    • Row ‹position›: “‹token›” in the ‹column› column is not a number.: This error is reported if a cell contains non-numeric input.
    • The entries add up to zero overall, so there is nothing to divide by and no balance point exists. Look for a negative entry cancelling the rest.: This error occurs if the sum of all weights equals zero, making division mathematically impossible.
    • Keep the system under 500 entries.: This error is triggered if you attempt to exceed the maximum limit of 500 entries.
    • A value or intermediate result exceeds the supported number range.: This error is displayed if any input value or intermediate calculation causes a numerical overflow.

    Data Privacy and Processing

    Your entries and every coordinate stay in this browser and are never uploaded. All calculations and processing happen locally on your device.

    Frequently Asked Questions

    Is the center of mass the same as the center of gravity?

    They coincide whenever gravity pulls equally on every part, which covers almost anything you can hold or build. The center of mass depends only on how matter is spread out; the center of gravity is where the weight appears to act. The two separate on objects tall enough for gravity to weaken measurably from bottom to top, such as a space tether or a mountain-scale structure.

    Can the balance point land somewhere with nothing in it?

    Yes, and that is normal rather than an error. A ring, a boomerang, a horseshoe and a pair of separated weights all balance around empty space. It is also why the Earth–Moon system turns about a point buried inside the Earth instead of midway between the two.

    How do I enter a shape rather than a point?

    Give the shape one row, using its own balance point as the coordinates: a rectangle balances at the middle of its diagonals, a triangle a third of the way up from the base, a half-disc about 0.4244 of its radius from the flat edge. Build a complicated outline from several such rows, and subtract a hole by entering it as a negative row at the hole’s own balance point.

    Does it matter where I put the origin?

    No — pick whichever corner or edge makes the coordinates easy to read off, as long as every row uses the same one. The answer arrives in that same frame, so moving the origin moves the reported coordinates by exactly the same amount while the physical point stays put.

    Ohm's Law CalculatorEnter any two of voltage, current, resistance or power, and get the other two worked out instantly.Survey Weighting CalculatorRake respondent data to known population margins, then inspect every final weight, target residual and effective sample size.Unit ConverterConvert length, weight, temperature, area, volume and speed between any units, instantly.Significant Figures CalculatorCount the significant figures in any value, round to a chosen precision, or evaluate + − × ÷ expressions with the rounding rules shown step by step.Blackbody Radiation CalculatorEnter a temperature — or a measured peak wavelength — and read off where the thermal spectrum peaks, how much power it radiates, and the Planck curve behind it.Doppler Effect CalculatorChoose the medium, set the source and the observer in motion, and get the shifted frequency with the pass-by and the wavefronts drawn in.Collision LabSet two balls in motion in one or two dimensions, choose how bouncy the impact is, and watch velocity, momentum and kinetic energy before and after they collide.Concrete Volume CalculatorBuild a pour schedule from rectangular, solid round and hollow round sections, then see net volume, ordering allowance, unit conversions and every substituted formula.