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Ring and Horn Torus Surface from Inner and Outer Radii

Ring and Horn Torus Surface from Inner and Outer Radii. Convert compatible units, recover unknown dimensions and distinguish curved surfaces, bases and total measures.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Example in initial displayed units: Inner radius (a) 2, Outer radius (b) 4 gives Tube radius (r) 1, Radius of revolution (R) 3, Surface area 118.43525.

r = (b−a)/2; R = (a+b)/2; A = 4π²Rr

Worked example

Enter these known values and leave the other values blank.

Inner radius (a)
2 m
Outer radius (b)
4 m
Tube radius (r)
1 m
Radius of revolution (R)
3 m
Surface area
118.44 m²

Assumptions and limitations

  • The inner and outer radii are measured from the axis, while tube radius measures the circular tube and revolution radius reaches the tube center. Ring tori have revolution radius greater than tube radius. Equality is the retained horn-torus limit; self-intersecting spindle tori are excluded.
  • Select units before entering dimensions. Clear an unknown result before solving backward, and supply enough independent measurements. Display rounding does not represent measurement uncertainty.
  • Calculations use 64 significant digits and full-precision π. Original reference drawings explain dimensions and are not drawn to the entered scale.

Common questions

What is the difference between tube radius and revolution radius?

Tube radius measures the circular cross section. Revolution radius measures the distance from the rotation axis to that circle center. Inner and outer radii equal R−r and R+r.

References

Require positive physical dimensions and real nondegenerate inverse roots while preserving documented limiting shapes.

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