Torus Calculator
Volume and surface area of a ring, by Pappus.
Work out Torus. Volume and surface area of a ring, by Pappus. Handles the case the formula sheet skips.
Centre of the hole to the centre of the tube.
The thickness of the ring itself. Must be smaller than the ring radius.
Volume
394.7842
surface area 394.784 · outer radius 7
V = 2π²Rr² and A = 4π²Rr, and both fall straight out of Pappus's centroid theorem: revolve a shape around an axis and the volume is the shape's area times the distance its centroid travels. Here that is πr² × 2πR, which multiplies out to exactly the volume above. The same theorem gives the surface area as the circle's circumference times that path. The tube radius has to be smaller than the ring radius. Equal, and the hole closes to a point; larger, and the tube passes through itself, giving a shape these formulas no longer describe.
How the Torus Calculator works
Enter the ring radius and the tube radius for the volume, surface area and both outer and inner radii. Both formulas fall straight out of Pappus's centroid theorem, which is a more memorable route than the π² expressions suggest.
Also known as: torus volume calculator · donut volume calculator · torus surface area calculator · ring volume formula
Frequently asked questions
What is the volume of a torus?
V = 2π²Rr², with R the ring radius and r the tube radius. It is the tube's cross-sectional area πr² multiplied by the distance 2πR that its centre travels around the ring.
What is Pappus's centroid theorem?
Revolve a shape around an axis and the resulting volume is the shape's area times the distance its centroid travels. It gives both the torus volume and its surface area in one step, without integration.
Why must the tube radius be smaller than the ring radius?
Otherwise there is no hole. Equal radii close it to a point, and a larger tube passes through itself — a shape these formulas no longer describe.
What are the outer and inner radii?
R + r and R − r. Those are what you would measure with callipers across a real ring, whereas R and r are the values the formulas need — so converting between them is usually the first step.
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