How this calculator works
Pappus's centroid theorem gives the volume of a torus without any calculus: sweep a circle of area πr² around a circular path of length 2πR, and the volume is the cross section multiplied by the path length. That gives V = 2π²Rr². The same theorem gives the surface area as the tube circumference 2πr times the path length, 4π²Rr. The centre line length is the distance a point at the middle of the tube travels on one complete loop.
Formula
Volume = 2π²Rr²; tube cross section = πr²; centre line length = 2πR.
Worked example
A torus with a major radius of 5 m and a tube radius of 2 m has a tube cross section of 4π ≈ 12.566 m², a centre line length of 10π ≈ 31.416 m and a volume of 2π² × 5 × 4 ≈ 394.784 m³.
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Frequently asked questions
Why does the volume use the radius squared?
The tube cross section is a circle, so it contributes r², while the path around the torus contributes one power of R. That is why doubling the tube radius multiplies the volume by four.
Does this work for a square-section torus?
No. This formula assumes a circular tube. The Pappus theorem still applies, but the cross sectional area would be the area of the square instead of πr².
What if the tube is as thick as the major radius?
The hole closes exactly at r = R, producing a horn torus. Beyond that the inner surface intersects itself and the simple ring volume formula no longer applies, so the calculator rejects the input.