Calculate the surface area and volume of a torus (donut shape) from its major radius R and minor radius r. Includes outer/inner diameters, circumferences, cross-section area, SA/V ratio, proportion...
A torus — the mathematical term for a donut shape — is generated by revolving a small circle of radius <em>r</em> (the tube) around a larger circle of radius <em>R</em> (the central ring). Tori appear everywhere: in O-rings and gaskets, bicycle and car tires, life preservers, bagels and donuts, particle accelerators, and even in the topology of the universe (some cosmological models). Two elegant formulas capture its geometry: the <strong>surface area</strong> is SA = 4π²Rr and the <strong>volume</strong> is V = 2π²Rr². Both can be derived using Pappus' centroid theorem — the surface area of a solid of revolution equals the path length of the generating curve times the distance traveled by its centroid. This calculator accepts the major radius R (center of the torus to center of the tube) and minor radius r (radius of the tube itself). It instantly outputs the surface area, volume, outer and inner diameters, tube diameter, mean circumference, tube circumference, cross-section area, and the SA-to-volume ratio. An optional wall-thickness field lets you model a hollow torus (like a tire or inflatable ring) and compute the material volume. Proportion bars visualize the r/R ratio, hole fraction, and SA efficiency. Eight presets load common objects from a small O-ring to a car tire, and the reference table lists pre-computed values for quick comparison.
This calculator is useful when you need to understand how much outer material a donut-shaped object actually has, not just memorize 4π²Rr. Surface area matters for coating, heat transfer, material usage, and cleaning calculations, while the companion volume and cross-section outputs help you see how the same torus dimensions affect capacity and bulk.
It is also practical for real objects such as tires, O-rings, inflatable rings, and molded rubber parts. The optional wall-thickness field lets you estimate shell material instead of treating every torus as solid, which is often the difference between a classroom example and an engineering estimate that matches the physical object.
Surface Area = 4π²Rr Volume = 2π²Rr² Outer Diameter = 2(R + r) Inner Diameter = 2(R − r) Tube Circumference = 2πr Mean Circumference = 2πR Cross-Section Area = πr²
Result: Surface area ≈ 1184.3525 cm², volume ≈ 1776.5288 cm³, outer diameter = 26 cm, inner diameter = 14 cm
For a torus with R = 10 cm and r = 3 cm: • SA = 4π²(10)(3) = 1,184.35 cm² • Volume = 2π²(10)(9) = 1,776.53 cm³ • Outer diameter = 2(10 + 3) = 26 cm • Inner diameter = 2(10 − 3) = 14 cm • Mean circumference = 2π(10) = 62.83 cm
The major radius R is measured from the center of the torus to the center of the tube, not all the way to the outer edge. The minor radius r is the radius of the circular tube itself. Many setup errors come from mixing those with outer diameter or tube diameter. If you keep the centerline picture in mind, the rest of the formulas become much easier to interpret.
The torus surface area depends linearly on both R and r, but the volume depends on r². That means increasing the tube thickness has a much larger effect on material and capacity than increasing the ring radius alone. A thin, large-diameter torus can have a surprisingly high surface area with modest volume, while a thicker torus gains volume rapidly even if the central hole changes very little.
Real parts such as tires, inflatable rings, and seals are often hollow rather than solid. In those cases, the outer torus dimensions tell only part of the story; the wall thickness determines how much material is actually present. Using the shell calculation helps estimate rubber volume, plastic usage, or air space more realistically than a single solid-torus formula would.
A torus is a surface of revolution generated by rotating a circle in 3D space around an axis that does not intersect the circle. The common visualization is a donut or inner tube.
SA = 4π²Rr, where R is the major radius (center to tube center) and r is the minor radius (tube radius). Use this as a practical reminder before finalizing the result.
V = 2π²Rr². It can also be written as V = (πr²)(2πR), i.e., cross-section area times mean circumference.
The major radius R is the distance from the center of the torus hole to the center of the tube. The minor radius r is the radius of the tube cross-section.
If r > R the torus has no hole (a "spindle torus") and the standard formulas still apply, though the shape self-intersects. This calculator requires r ≤ R (ring torus).
Subtract the inner torus volume from the outer: V_shell = 2π²R(r_outer² − r_inner²). Use the wall-thickness field for this.