What Is a Torus?
A torus is the three-dimensional shape formed by revolving a circle around an external axis in the same plane as the circle. It looks like a donut, an inner tube, or a ring. It is defined by two radii: the major radius R (from the centre of the torus to the centre of the tube) and the minor radius r (the radius of the tube itself).
In practice, it is easier to think of a torus in terms of its outer diameter and inner diameter — the outer diameter of the ring and the diameter of the hole through the middle. This calculator accepts both input methods and derives all torus properties automatically.
Torus Formulas
- Volume = 2π² × R × r²
- Surface Area = 4π² × R × r
- From outer/inner diameter: R = (outer + inner) / 4
- From outer/inner diameter: r = (outer − inner) / 4
- Condition: minor radius r must be less than major radius R (otherwise the torus self-intersects).
Real-World Applications of Torus Geometry
Torus geometry appears in engineering, architecture, and industrial design. Understanding volume and surface area is essential for material estimation, weight calculation, and heat transfer calculations.
- O-rings and gaskets: torus cross-sections define the sealing geometry of pipe fittings and hydraulic systems.
- Donuts and ring-shaped baked goods: volume calculations for portion control and recipe scaling.
- Pool rings and flotation devices: volume determines buoyancy.
- Architectural features: toroidal forms appear in modern building design and parametric architecture.
- Tokamak fusion reactors: plasma is confined in a toroidal magnetic chamber.
Frequently Asked Questions
What is the difference between major and minor radius?
The major radius R is the distance from the central axis of the torus (the centre of the hole) to the centre of the tube. The minor radius r is the radius of the circular cross-section of the tube. For a standard donut, R is roughly half the distance between the outer and inner edges of the donut, and r is the thickness of the dough.
Can the torus formula be used for a ring or O-ring?
Yes. For an O-ring with a known outer diameter, inner diameter, and cross-section diameter, set the outer diameter to the full outer diameter of the ring and the inner diameter to the full inner diameter (the hole). The calculator gives the volume of rubber material in the ring and its surface area.
What happens if the inner diameter equals zero?
When the inner diameter approaches zero, the major radius R equals the minor radius r. This is a horn torus — the inner edge collapses to a point. The formulas still apply mathematically, but the shape is no longer a standard donut ring. This calculator requires a positive inner diameter smaller than the outer diameter.