BIZOMAT 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). |
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: Profile
- What it is
- The profile that is swept around the axis: a circle, which makes a torus such as an O-ring, or any flat profile whose length, area and centroid positions you know.
- Units and format
- Choose one of 2 options from the list: Circle (torus or O-ring); Any profile with known centroid.
- Example
- The form opens with “Circle (torus or O-ring)” selected as a worked example. Choose the option that matches your case.
- Get it right
- With the custom option you supply the centroid radii yourself; the torus option finds them from the circle.
What this means: Circle diameter (torus)
- What it is
- The diameter of the round cross-section that is swept around the axis, such as the cord section of an O-ring. Used only for the torus profile.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Surface area rises in proportion to it and volume with its square.
- Get it right
- This is the cord diameter, not the ring’s inside or outside diameter; it must not exceed the centroid path diameter.
What this means: Diameter of the centroid path (torus)
- What it is
- The diameter of the circle traced by the centre of the round cross-section as it goes round the axis. For an O-ring it is the inside diameter plus one cord diameter.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Both surface area and volume are proportional to it.
- Get it right
- Do not use the ring’s inside or outside diameter; use the mean diameter through the cord centre.
What this means: Profile line length (custom)
- What it is
- The length of the profile line (straight or curved) that sweeps out the surface, for the custom profile. Measure along the line itself, as you would unroll it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Surface area is directly proportional to the line length.
- Get it right
- Include only the line that forms the surface you want; leave out edges that lie on the axis.
What this means: Profile area (custom)
- What it is
- The area of the flat profile that sweeps out the solid, for the custom profile, such as the cross-section of a groove or bead seen in a half-section drawing.
- Units and format
- Enter a number in mm²: square millimetres. The form accepts 0 mm² or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Volume is directly proportional to the profile area.
- Get it right
- The profile must lie entirely on one side of the axis; CAD section properties give the area directly.
What this means: Radius of the line’s centroid from the axis (cu…
- What it is
- The distance from the axis of revolution to the centroid of the profile line, for the custom profile. The centroid of a line is its balance point along its length.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Surface area is directly proportional to this radius.
- Get it right
- The centroid of the line and of the area are generally at different radii; do not use one for the other.
What this means: Radius of the area’s centroid from the axis (cu…
- What it is
- The distance from the axis of revolution to the centroid of the profile area, for the custom profile. CAD section properties or a centroid table give it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Volume is directly proportional to this radius.
- Get it right
- Measure from the axis of revolution, not from an edge of the profile.
What this means: Angle of revolution
- What it is
- How far the profile is swept around the axis: a full turn for a complete ring, or less for a part ring or segment.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 360° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Surface area and volume are both proportional to the angle.
- Get it right
- Use 360 for a full revolution; a half ring is 180.
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.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Sizing the blank for a dome
In Hand Forming Sheet Metal by the Numbers
What it covers: Size a dome's blank by sinking, raising or both, predict the mean thickness and the rim's shrink for each, make a checking template, and catch a blank cut with no trim allowance.
Method explained here · BIZOMAT original book
Cones and reducers by radial lines
In Sheet Metal Patterns by the Numbers
What it covers: Develop a concentric reducer as part of a cone's sector: slant height, apex distance, sector angle and chords, what to do when the apex is out of reach, and an apex found from diameters.
Method explained here · BIZOMAT original book
Extruded, revolved and swept surfaces
What it covers: Build surfaces by moving one profile: extrude it, revolve it or sweep it along a path, and predict when a sweep will fold back on itself before it fails.
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