How this calculator works
A ring torus is the shape of a doughnut: a circle of radius r swept around a larger circle of radius R. Its surface is that of a cylinder of length 2πR and circumference 2πr, which gives 4π²Rr. Its volume is the tube cross section πr² multiplied by the same path length 2πR, giving 2π²Rr². The inner diameter is 2(R - r) and the outer diameter is 2(R + r). If the tube radius equals or exceeds the major radius the hole disappears and the surface formula no longer describes a simple ring torus, so that input is rejected.
Formula
Surface area = 4π²Rr; volume = 2π²Rr²; outer diameter = 2(R + r); inner diameter = 2(R - r).
Worked example
A torus with a major radius of 5 m and a tube radius of 2 m has a surface area of 4π² × 10 ≈ 394.784 m², a volume of about 394.784 m³, an outer diameter of 14 m and an inner diameter of 6 m.
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Frequently asked questions
What is the difference between the two radii?
The major radius R measures from the centre of the hole to the centre of the tube. The tube radius r measures the thickness of the ring itself. Their sum and difference give the outer and inner edges.
Why must the tube radius be smaller?
If the tube radius reaches the major radius, the inner hole closes and the surfaces overlap. That is a different shape called a horn torus or spindle torus, which the ring torus formula does not cover.
Is the surface area the same as the volume?
Only by coincidence at some sizes. At R = 5 m and r = 2 m both round to about 394.784, but the formulas differ and they diverge at other dimensions.