KEVOS® MAKEOS · Geometry
Surface and Volume of Revolution (Pappus–Guldinus)
What surface area and volume does a profile sweep out when it is revolved about an axis?
Area of a surface of revolution (profile length × path of its centroid) and volume of a solid of revolution (profile area × path of its centroid) by the Pappus or Guldinus rules, for a torus or any profile with known centroid, over a full or partial revolution.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Profile | choice | A circle swept into a torus, or any profile whose centroid you know. |
| Circle diameter (torus) | mm | Diameter of the circular cross-section. |
| Diameter of the centroid path (torus) | mm | Diameter of the circle traced by the cross-section’s centre. |
| Profile line length (custom) | mm | Length of the profile line generating the surface. |
| Profile area (custom) | mm² | Area of the profile generating the solid. |
| Radius of the line’s centroid from the axis (custom) | mm | Distance from the axis to the centroid of the line. |
| Radius of the area’s centroid from the axis (custom) | mm | Distance from the axis to the centroid of the area. |
| Angle of revolution | ° | Angle swept (360° for a full revolution). |
Method and formulas
- Surface area = length of the line × length of the path of its centre of gravity.
- Volume = area of the profile × length of the path of its centre of gravity.
- For a partial revolution the path is 2π r × angle ÷ 360°.
- Example: a 3 mm circle swept on a 10 mm path gives 296.09 mm² and 222.07 mm³ (the handbook’s torus example in inches gives 296.09 in² and 222.07 in³).
Limitations
- The profile must lie wholly on one side of the axis and in a plane containing it.
- Centroid positions must be exact for exact results.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the volume of an O-ring, ring groove or turned bead
- Find the surface area of a turned part for coating or plating
- Estimate material in a rolled or spun ring
Who it is for
- Design engineers
- Estimators
- Toolmakers
What you get
- Surface area of revolution
- Volume of revolution
- Centroid path length
Interpreting the result
- For an O-ring groove, compare the ring’s volume with the groove’s to check fill.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 66–67. Topic reference; no source tables or recommended settings are reproduced.
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