BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Shape
- What it is
- The shape whose centre of gravity you want. Each option names what A, B, C and the angle mean, and the point or edge the result distance is measured from.
- Units and format
- Choose one of 15 options from the list: Triangle (C = height): from base; Semicircle area (A = radius): from diameter; Circle sector (A = radius, angle): from centre; Circular segment (A = radius, angle): from centre; Circular arc (A = radius, angle): from centre; Trapezoid (A = base, B = top, C = height): from base; Parabolic segment (C = height): from base chord; Solid cone or pyramid (C = height): from base; Frustum of cone (A = base dia, B = top dia, C = height): from base; Solid hemisphere (A = radius): from flat face; Hollow hemisphere (A = outer radius, B = inner radius): from flat face; Solid paraboloid (C = height): from base; Spherical segment (A = sphere radius, C = height): from sphere centre; Spherical sector (A = sphere radius, C = cap height): from sphere centre; Two bodies (A = mass 1, B = mass 2, C = distance apart): from body 1.
- Example
- The form opens with “Frustum of cone (A = base dia, B = top dia, C = height): from base” selected as a worked example. Choose the option that matches your case.
- Get it right
- Note the reference in the option text (base, centre, flat face or body 1); the result is a distance from that point.
What this means: Dimension A
- What it is
- The first size of the chosen shape: a radius (semicircle, sector, segment, arc, hemisphere, sphere), the base length of a trapezoid or the base diameter of a frustum. For two bodies it is the mass of body 1.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For two bodies A and B are masses in any one consistent unit, despite the mm label.
What this means: Dimension B
- What it is
- The second size where the shape needs one: the top length of a trapezoid, the top diameter of a frustum or the inner radius of a hollow hemisphere. For two bodies it is the mass of body 2.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a hollow hemisphere it must be smaller than A. For two bodies use the same mass unit as A.
What this means: Dimension C
- What it is
- The height of a triangle, trapezoid, parabolic segment, cone, frustum, paraboloid or spherical segment or sector, measured square to the base. For two bodies it is the distance between their centres.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- For the triangle, cone, paraboloid and parabolic segment the result is a fixed fraction of this height.
- Get it right
- For a spherical segment the height cannot exceed the sphere diameter.
What this means: Included angle
- What it is
- The full included angle of a circle sector, circular segment or circular arc, measured at the circle’s centre. Other shapes ignore it.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.001° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Give the full angle, not the half angle; the method halves it internally.
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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