How this calculation works
Sweep a circular cross-section around an external coplanar axis and restrict the inputs to a ring torus. Derive its volume and surface from the tube-circle measures and centerline circumference.
Volume = 2π²Rr²; surface area = 4π²Rr; outside diameter = 2(R+r); hole diameter = 2(R−r)A worked example
With major radius 5 m and tube radius 2 m, outside diameter is 14 m and hole diameter 6 m. Volume is 40π² ≈ 394.784176 m³ and surface is also numerically 40π², but in m² rather than m³.
Good to know
Ordinary ring torus only, with R>r>0. Horn and self-intersecting spindle tori, hollow tube walls, oval cross-sections and deformation are excluded. No gasket, pressure or material-rating determination.
Reference: MathWorld: geometry definitions and formulas (opens in a new tab) ↗
Major radius stops at the tube centerline
The default major radius 5 m runs from the rotation axis to the center of the circular tube cross-section. The minor radius is the tube radius, 2 m. Outside radius is therefore 7 m and hole radius 3 m, giving diameters 14 m and 6 m.
Entering the outside radius as the major radius would make a larger torus than intended. The diagram marks the tube centerline separately from its outside edge. The supported ring domain requires major radius greater than tube radius, rather than including horn or self-intersecting spindle cases.
The circular and annular sector calculator describes a planar region; a ring torus sweeps a circular tube cross-section through three-dimensional space.
Sweep the tube circle around its centerline
The cross-section area is 4π m² and its center travels a 10π m circle. Their product gives volume 40π², about 394.784176 m³. Multiplying tube circumference 4π m by the same centerline circumference gives surface 40π² m².
The equal numerical coefficients in this particular example do not make area and volume the same quantity. They use square and cubic units. Changing tube radius changes volume quadratically but surface linearly when major radius is held fixed.
Ring geometry is not an annular disk or a hollow pipe wall
The circular and annular sector calculator describes a planar region with boundary length, not this three-dimensional swept tube. A face-on torus silhouette can resemble an annulus even though its volume depends on a circular cross-section.
This torus model has a solid circular tube cross-section. It does not subtract an inner tube wall, model an oval section or infer gasket deformation and pressure capacity. Trial rows that would leave the ring-torus domain are omitted rather than evaluated with the same formulas as if nothing changed.
A few questions, answered
Is major radius the outside radius?
No. It runs from the rotation axis to the tube-circle center. Outside radius is major plus minor radius.
Why require R greater than r?
That is the ordinary ring-torus domain supported here. Horn and self-intersecting spindle cases are excluded.
Does volume scale linearly with tube radius?
No. At fixed major radius, volume is proportional to tube radius squared, while surface area is proportional to tube radius.
Does this calculate a hollow pipe’s material volume?
No. The supported tube cross-section is a solid circle; no wall thickness is inferred.