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How to use
A torus is formed by revolving a circle of radius r around an axis at major radius R. For R≥r, the standard formulas are V=2π²Rr² and A=4π²Rr. The calculator also reports the hole and outside dimensions.
- Enter major radius R from the torus axis to the tube center.
- Enter tube radius r using the same length unit.
- Confirm R≥r, then calculate volume, area and dimensions.
How it is calculated
Example
Example: R=5 and r=2 give V=40π²≈394.784 cubic units and A=40π²≈394.784 square units. Inner radius is 3 and outer radius is 7.
Important notes
These formulas are applied here to ring tori (R>r) and the horn limit (R=r). R<r is rejected because a self-intersecting spindle torus requires a different geometric interpretation of area and volume.
Worked examples and interpreting results
For a circular torus with major radius 3 cm and tube radius 1 cm, volume is 6π² cm³ and surface area 12π² cm². The major radius runs from the axis to the center of the tube cross-section.
| Case | Calculation | Result |
|---|---|---|
| Volume | 2 × π² × 3 × 1² | 59.217626 cm³ |
| Surface area | 4 × π² × 3 × 1 | 118.435253 cm² |
How to check the result
A usual ring torus with a hole needs major radius greater than tube radius. Here the outer radius is 4 cm and inner radius 2 cm; substituting outer radius for major radius changes the result.
Frequently asked questions
What is the difference between R and r?
R is the distance from the rotation axis to the tube center; r is the radius of the circular tube cross-section.
Do units matter?
Use the same length unit for both radii; volume uses its cube and area its square.
What happens at R=r?
The inner hole radius becomes zero, the horn-torus limit.
What condition defines the usual ring torus?
For a non-self-intersecting ring torus, major radius R must exceed tube radius r. Equal radii form a horn torus; when R is smaller, the surface self-intersects and the geometric interpretation changes.