Center of Gravity Calculator
Calculate the weighted center of gravity (centroid) of a set of point masses. Enter each point as X, Y, Mass (label optional, negative mass for a cutout) — get the total mass, the CG coordinates, a per-point contribution breakdown, and a plotted diagram.
Input
One point per line as "X, Y, Mass", with an optional leading label. X/Y are position coordinates; Mass is the weight (or area/volume, for composite-shape centroids) at that point — any consistent units work. Enter a negative mass for a cutout or void, subtracted from the total.
Output
Center of Gravity Result
| Metric | Value |
|---|---|
| No data yet | |
Enter at least one point to see the plotted diagram.
Per-Point Breakdown
| Point | X | Y | Mass | % of Total Mass | Distance from CG |
|---|---|---|---|---|---|
| No data yet | |||||
Guides
Find the weighted center of gravity (centroid) of a set of point masses — a bracket assembly, a composite plate with a cutout, or any collection of weighted points on a plane.
How to use it
- Enter one point per line as X, Y, Mass, or Label, X, Y, Mass if you want the breakdown table to read like an actual parts list instead of "Point 1, Point 2, …".
- X and Y are position coordinates; Mass is the weight (or area/volume, if you're finding the centroid of a distributed shape) at that point. Any consistent units work — the calculator doesn't care whether you're entering millimeters and kilograms or inches and pounds.
- For a hole or cutout, enter a negative mass. It subtracts from the total the same way a void subtracts material from a solid shape.
The math
The center of gravity is the mass-weighted average position:
X̄ = Σ(mᵢ·xᵢ) / Σmᵢ · Ȳ = Σ(mᵢ·yᵢ) / Σmᵢ
This is exact whether each line is a literal point mass or the centroid of a larger component — first moments (mass × distance) combine linearly regardless of a component's own shape or extent. That's also why splitting a distributed shape into known sub-component centroids and entering each one's mass or area as the weight (the standard "negative-area method" for composite shapes) gives the right combined centroid.
What you get back
- Total mass, the point count, and the CG coordinates.
- A per-point breakdown: each point's percentage share of the total mass and its distance from the computed CG — not just the final answer.
- A plotted diagram: every point on an SVG scatter plot, sized by its relative mass, with cutouts (negative mass) shown dashed in amber and the CG marked with a crosshair.
Cutouts and zero net mass
A negative-mass point is treated as a void — its position still pulls the CG away from the solid material, just in the opposite direction. If the total mass nets to exactly zero (a cutout that exactly cancels the surrounding mass), the center of gravity is mathematically undefined and the calculator flags it instead of dividing by zero.
Limits
This computes the centroid (first moment) correctly for any combination of point masses or component centroids. It does not compute a moment of inertia — that would need each component's own local rotational inertia added in via the parallel-axis theorem, which a plain point-mass/centroid list doesn't carry. If you need that, treat this as the first step (finding where the combined CG sits) before a separate moment-of-inertia calculation.
Privacy
Everything runs in your browser — no coordinates or masses leave your machine.
Related tools
For the weight of a metal plate itself, see the Plate Weight Calculator. For tolerance analysis across a chain of dimensions, see the Tolerance Stackup Calculator.
Use it from code
From 3 credits per callREST API
curl -X POST https://api.iotools.cloud/v1/tool/center-of-gravity-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"points": "Bracket, 0, 0, 1\nFlange, 1.2, 0.8, 8\nBoss, 2.4, …"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `center-of-gravity-calculator` tool (Center of Gravity Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.