Torus Calculator
Compute the volume, surface area, and outer and inner diameters of a torus from its major radius and tube radius.
units
units
Enclosed volume of the doughnut
- 1
Tube radius squared
2² = 4 - 2
π²
9.869604Used because the torus is a body of revolution swept through a full circle. - 3
Volume = 2 × π² × R × r²
2 × 9.869604 × 5 × 4 = 394.78
How does this calculator work?
A torus has volume V = 2π²·R·r² and surface area A = 4π²·R·r, where R is the major radius (centre to tube axis) and r is the tube radius. Its outer diameter is 2(R + r) and inner diameter is 2(R − r). The shape is a valid ring only when R ≥ r > 0.
Formula
How this is calculated
A torus is the ring (doughnut) shape swept by revolving a circle of radius r around an axis lying in the same plane, where the centre of that circle stays a distance R from the axis. R is the major radius (from the torus centre to the tube axis) and r is the tube (minor) radius. For a genuine ring torus the major radius must satisfy R ≥ r > 0; if r equals R the hole closes to a point (horn torus) and if r exceeds R the surface self-intersects, so those cases are rejected.
Using Pappus's theorems, the volume equals the tube's cross-sectional area (πr²) times the distance travelled by its centroid (2πR), giving V = 2π²·R·r². The surface area equals the tube's circumference (2πr) times the same centroid path (2πR), giving A = 4π²·R·r. The outer diameter is measured across the widest part, 2(R + r), and the inner diameter across the hole, 2(R − r).
All lengths share the same unit you enter, so volume is in cubic units and surface area in square units. The calculator also reports the tube circumference (2πr). Because the formulas scale exactly with R and r, you can work in centimetres, inches, or any consistent unit and simply read the output in the matching power of that unit.
Frequently asked questions
The major radius R is the distance from the centre of the torus to the centre of the circular tube, while the tube radius r is the radius of the tube itself. The outer diameter is 2(R + r) and the inner (hole) diameter is 2(R − r).
When R ≥ r the torus is a proper ring with an open hole. If r equals R the hole shrinks to a single point (a horn torus), and if r is larger than R the surface overlaps itself (a spindle torus), so the standard volume and area formulas no longer describe a simple ring.
By Pappus's centroid theorem the volume is the area of the rotating circle (πr²) multiplied by the distance its centroid travels (2πR), which gives V = 2π²·R·r². The same theorem applied to the tube's circumference yields the surface area A = 4π²·R·r.
Also known as
TG we-Calculate Editorial Team. (2026). Torus Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/torus-calculator
TG we-Calculate Editorial Team. "Torus Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/torus-calculator.
TG we-Calculate Editorial Team, "Torus Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/torus-calculator
@misc{wecalculate_torus_calculator, title = {Torus Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/torus-calculator}}, year = {2026}, note = {TG we-Calculate} }
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