KEVOS® MAKEOS · Geometry
Centre of Gravity of Common Shapes
Where is the centre of gravity of this shape or solid?
Position of the centroid (centre of gravity) of plane areas, arcs and uniform solids: triangle, sector, segment, arc, semicircle, trapezoid, parabolic area, cone, pyramid, frustum, hemisphere, hollow hemisphere, paraboloid, spherical segment and sector, and two bodies.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Shape | choice | Shape; the option text names each dimension and where the result is measured from. |
| Dimension A | mm | First dimension (or mass 1 for two bodies, in any consistent unit). |
| Dimension B | mm | Second dimension (or mass 2). |
| Dimension C | mm | Height, or distance between two bodies. |
| Included angle | ° | Full included angle of a sector, segment or arc. |
Method and formulas
- Triangle h/3; semicircle 4r/3π; sector 2r sin(θ/2) ÷ (3 × θ/2); arc r sin(θ/2) ÷ (θ/2); segment c³ ÷ 12A from the centre.
- Trapezoid h(a + 2b) ÷ 3(a + b); parabolic segment 2h/5; cone and pyramid h/4; paraboloid h/3; hemisphere 3r/8; hollow hemisphere 3(R⁴ − r⁴) ÷ 8(R³ − r³).
- Frustum h(R² + 2Rr + 3r²) ÷ 4(R² + Rr + r²); spherical segment 3(2r − h)² ÷ 4(3r − h); spherical sector 3(2r − h)/8; two bodies m₂ d ÷ (m₁ + m₂).
- Example: a cone frustum 100/50 mm over 80 mm has its centre of gravity 31.429 mm above the base.
Limitations
- Uniform density and ideal shapes.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Locate lifting points and balance features
- Find centroids for section and moment calculations
- Check CAD mass properties
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- Distance of the centre of gravity from the stated reference
- Area or volume of the shape
Interpreting the result
- Combine several shapes with the combined centre of gravity tool.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 221–227. Topic reference; no source tables or recommended settings are reproduced.
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