Mass and Center of Gravity of a 3D Part

This interactive calculator works with 3D geometry directly in the browser. You can either create standard primitives – box, sphere, cylinder, cone or upload an STL file. Once the geometry and material density are provided the tool computes: volume, mass, center of mass (CoM), surface area, inertia tensor and principal moments. The calculations run online desktop and mobile. Use the sample primitives to preview results or upload your CAD/STL model for a full analysis.

• Sample: simple part

How it works

• Scene: an interactive 3D view shows the current model; rotate, pan and zoom using mouse or touch.
• Geometry source: choose a primitive and enter dimensions, or upload an STL and select the units it was exported in (mm, cm, m, in).
• Material: set density (kg/m³) or pick a preset (PLA, ABS, aluminum, steel, copper, etc.).
• Samples / accuracy: the CoM and inertia can be refined by increasing the Monte-Carlo / sampling count used for volume sampling — more samples → more stable inertia estimates for complex shapes.
• Output: table with volume, mass, CoM (X,Y,Z), surface area, full inertia tensor and principal moments; export as JSON for further use.

Important formulas

Volume & mass:

$$\text{Volume } V = \int_{\Omega} dV \qquad (\text{in } m^3)$$
$$\text{Mass } m = \rho \cdot V
$$
$$
\rho \ \text{is density in } kg/m^3$$

Center of mass (centroid) coordinates:

$$x_{c} = \frac{1}{V}\int_{\Omega} x \, dV,
$$
$$
y_{c} = \frac{1}{V}\int_{\Omega} y \, dV,
$$
$$
z_{c} = \frac{1}{V}\int_{\Omega} z \, dV$$

Inertia tensor about the centroid (3×3 matrix):

I = [[ Ixx, Ixy, Ixz ],
[ Ixy, Iyy, Iyz ],
[ Ixz, Iyz, Izz ]]

Principal moments are eigenvalues of the inertia tensor and describe resistance to rotation about principal axes.

Interface & controls

1. Geometry Source — Primitive or STL upload.
2. Primitive parameters — enter W / H / L or radius/height depending on the chosen shape; set position offsets (X, Y, Z).
3. STL upload — choose file and the units the model was exported in; correct unit selection is essential for proper scaling.
4. Units — choose display/working units (mm, cm, m, inch); internal computations use SI (meters).
5. Density — enter kg/m³ or pick a preset material.
6. Samples — number of random points / sampling resolution used for inertia estimation (increase for better accuracy).
7. Compute — runs the calculations and fills the results table; Export JSON downloads the numeric results and meta information.

Notes about STL files

• Select correct units for the STL (mm/cm/m/in). If you choose the wrong unit, computed volumes, masses and inertia will be off by the cube of the scale factor.
• Meshes must be manifold / watertight for exact volume from a triangle-based algorithm. If the mesh has holes the tool attempts a signed-volume approach and will warn you if the computed volume is near zero.
• If the mesh orientation produces negative volume the tool will flip triangle orientation and recompute automatically.

Material density reference

Material	Density, kg/m³	Notes
PLA	1240	Common FDM filament
ABS	1040	Sturdy thermoplastic
PETG	1270	Durable, moisture resistant
Aluminum	2700	Light metal
Steel	7850	Structural steel, approximate
Copper	8960	High conductivity
Brass	8500	Copper-zinc alloy
Titanium	4500	High strength-to-weight

Worked examples

Example A — rectangular box, material: PLA

Dimensions: Width W = 150 mm, Height H = 80 mm, Depth L = 60 mm. Density = 1240 kg/m³ (PLA).

Convert to meters: W = 0.150 m, H = 0.080 m, L = 0.060 m.

Volume:

\(V = 0.150 \times 0.080 \times 0.060\)
\(0.150 \times 0.080 = 0.012\)
\(0.012 \times 0.060 = 0.00072\ m^3\)

Mass:

\(m = \rho \cdot V = 1240 \times 0.00072 = 0.8928\ kg\)

Surface area, all faces:

\(A = 2(WH + WL + HL)\)
\(WH = 0.150 \times 0.080 = 0.012\)
\(WL = 0.150 \times 0.060 = 0.009\)
\(HL = 0.080 \times 0.060 = 0.0048\)
Sum = 0.0258 → \(A = 2 \times 0.0258 = 0.0516\ m^2\)

Approximate inertia – about centroid, standard box formulas:

For a solid rectangular block (mass m, sides W,H,L) the moments are:

\(I_x = \frac{1}{12} m (H^2 + L^2)\)   around axis along W
\(I_y = \frac{1}{12} m (W^2 + L^2)\)   around axis along H
\(I_z = \frac{1}{12} m (W^2 + H^2)\)   around axis along L

Plugging the numbers gives, rounded:

• \(I_x \approx 0.000744\ kg\cdot m^2\)
• \(I_y \approx 0.001942\ kg\cdot m^2\)
• \(I_z \approx 0.002150\ kg\cdot m^2\)

Example B — solid cylinder

Dimensions: Radius R = 25 mm, Height H = 100 mm. Density = 2700 kg/m³ (Al).

Convert to meters: R = 0.025 m, H = 0.100 m.

Volume:

\(V = \pi \times (0.025)^2 \times 0.100 \approx 0.000196\ m^3\)

Mass:

\(m = \rho \cdot V \approx 2700 \times 0.000196 \approx 0.5301\ kg\)

Surface area (total, including caps):

\(A = 2\pi R(R + H) \approx 0.01963\ m^2\)

Inertia, solid cylinder:

Principal moments, about centroid:

\(I_z = \frac{1}{2} m R^2\) (axis along cylinder axis)

\(I_x = I_y = \frac{1}{12} m (3R^2 + H^2)\)

Numerical results:

• \(I_z \approx 0.000166\ kg\cdot m^2\)
• \(I_x = I_y \approx 0.000525\ kg\cdot m^2\)

Practical guidance & tips

• Units: Always confirm the STL units — a model exported in millimetres but imported as metres will be off by 1e9 in volume.
• Mesh quality: For exact analytical volume/inertia the mesh must be closed (watertight). If not, the tool attempts signed-volume methods and will issue a warning when results are unreliable.
• Sampling & accuracy: Monte-Carlo sampling is used to estimate inertia for complex, non-convex shapes; increase samples (e.g. 5k–50k) for more accurate principal moments at cost of runtime.
• Hollow models: If your model is a thin shell (e.g. STL exported as surface only), the tool treats the mesh as a surface and attempts a signed-volume approach — results may not reflect internal voids correctly.
• Export: Use the JSON export to save: geometry metadata, units, density, computed V, m, CoM, surface area, inertia tensor and principal axes.

Use cases

• Quick mass and CoM estimation for 3D printed parts (select appropriate filament density).
• Preliminary inertia data for simple mechanical assembly calculations.
• Educational demonstrations of how geometry and density affect mass distribution.
• Quality check before printing or manufacturing: check mass and surface area for cost/print time estimation.

Disclaimer: calculated values are engineering estimates. For final design and safety-critical work use detailed CAD analysis, FE solvers or certified measurement methods.

References

1. Engineering Mechanics: Dynamics — J.L. Meriam, L.G. Kraige, Wiley.
2. Mechanics of Materials — Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, McGraw-Hill.
3. Introduction to Solid Modeling Using SolidWorks — Joseph Musto, McGraw-Hill.
4. Principles of Computer-Aided Design and Manufacturing — Farid Amirouche, Pearson.