Torus Calculator

Volume and surface area of a torus (ring or doughnut shape) from its major and minor radii, plus its inner and outer diameters.

From the center of the hole to the center of the tube.
Radius of the tube itself (half its thickness).
Volume
394.78
Surface area
394.78
Outer diameter
14
Inner (hole) diameter
6

Estimate generated with SortedSums: https://sortedsums.com/torus-calculator/. For informational purposes only; verify before relying on it.

How it’s calculated

By Pappus’s theorem, a solid of revolution has volume equal to the area of the revolved shape times the distance its centroid travels. For a torus the circle πr² travels 2πR, and the circumference 2πr sweeps the surface the same way.

V = (π r²)(2π R) = 2 π² R r² A = (2π r)(2π R) = 4 π² R r Outer diameter = 2(R + r), inner diameter = 2(R − r)

Example: R = 5 in, r = 2 in. V = 2π² × 5 × 4 = 40π² ≈ 394.78 in³; A = 4π² × 5 × 2 = 40π² ≈ 394.78 in²; outer diameter 14 in, hole 6 in.

Frequently asked questions

How do I measure R and r on a real ring?

Measure the outer diameter D and the hole diameter d. Then r = (D − d)/4 and R = (D + d)/4.

Why must r be no larger than R?

If the tube radius exceeds the major radius the tube passes through the axis and overlaps itself (a spindle torus), and these formulas no longer give the enclosed volume.

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Sources

Formulas are taken from the free public references above. Results are provided “as is” for informational and educational purposes only. See our disclaimer.

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