Torus Volume Calculator
Donut ring volume and surface from two radii.
What it does
A torus is the math name for a donut: a tube of radius r swept around a ring of radius R. Bagels, lifebuoys, and toroidal tanks all answer to the same two radius line.
The tool above takes major radius R with tube radius r. It returns volume with surface area beside it, in cubic and square inches or centimeters, so material and coating estimates come from one stop.
The defaults set R 8 with r 3: volume 1,421.22 cu in with surface 947.48 sq in. The metric mirror (20.32 cm, 7.62 cm) gives 23,289.67 cu cm with 6,112.78 sq cm.
The line follows Pappus centroid theorem as taught in OpenStax Calculus moments and centers of mass, where the worked torus example matches this formula exactly. MIT single variable calculus covers the same disk and shell ground the theorem stands on.
How to use
- Pick imperial (in) or metric (cm).
- Enter major radius R to the tube center.
- Enter tube radius r, smaller than R.
- Read volume first.
- Read surface area beside it.
- Multiply by density or coat rate for material.
Formula
- R
- Major radius: hole center to tube center.
- r
- Tube radius, must stay under R.
- V
- Tube area times centroid travel distance.
- S
- Tube circumference times centroid travel distance.
Volume multiplies tube area pi r squared by travel 2 pi R; surface multiplies tube circumference 2 pi r by the same travel. Self crossing spindles with r above R break the theorem, so the tool stops those.
Worked example
Example: R 8, r 3
Defaults: imperial, major radius 8 in, tube radius 3 in.
- Tube area: pi x 9 = 28.27 sq in.
- Travel: 2 x pi x 8 = 50.27 in.
- Volume: 28.27 x 50.27 = 1,421.22 cu in.
- Surface: 4 x pi squared x 8 x 3 = 947.48 sq in.
- Tool shows 1,421.22 cu in, 947.48 sq in.
- Metric mirror gives 23,289.67 cu cm, 6,112.78 sq cm.
Reference table
| Ring | Volume | Surface |
|---|---|---|
| R 8, r 3 | 1,421.22 cu in | 947.48 sq in |
| R 4, r 2 | 315.83 cu in | 315.83 sq in |
| R 20.32 cm, r 7.62 cm | 23,289.67 cu cm | 6,112.78 sq cm |
Limits
- Ring torus only; r must stay under R.
- Hollow wall thickness never enters.
- Horn and spindle shapes need other math.
- Inputs are ideal circles, not lumpy dough.
Common mistakes
- Swapping R and r in the boxes.
- Entering diameters instead of radii.
- Mixing inches and centimeters.
- Reading surface where volume belongs.
- Forcing r past R and expecting sense.
Try it yourself
Quick checks. Answers sit tucked under each question.
R 4, r 2 gives what?
315.83 cu in with 315.83 sq in.
R 6, r 1 gives what?
118.44 cu in with 236.87 sq in.
Tube bigger than ring?
Invalid ring torus; the tool asks for r under R.
Questions
What is a torus?
A donut solid: a disk swept in a circle around an outside axis.
What is the torus volume formula?
2 pi squared R r squared, from tube area times centroid travel.
What is R vs r?
R runs hole center to tube center; r is the tube radius itself.
Why must r stay under R?
Past R the tube crosses the axis and the ring math breaks.
Hollow torus?
Subtract the inner ring volume from the outer one with this page twice.
Donut vs bagel?
Same solid; only the snack size and the hole change.