Torus Calculator
Calculate the volume and surface area of a torus (donut shape) from its major radius and minor (tube) radius.
🍩 What is a Torus?
A torus is a three-dimensional ring-shaped surface produced by revolving a circle around an axis that lies in the same plane as the circle but does not pass through it, like a donut, bagel, or inner tube. Every torus is defined by two measurements: the major radius (R), the distance from the centre of the whole shape to the centre of the circular tube, and the minor radius (r), the radius of the tube itself. Together, R and r fully determine the torus's volume, surface area, and overall proportions.
Tori appear constantly in everyday objects and engineering. A bicycle or car inner tube, a rubber O-ring seal used to prevent leaks in pipes and engines, an inflatable pool ring, and of course food like donuts and bagels are all approximately torus-shaped. In engineering, O-ring seal design depends directly on the tube cross-section area (πr²) to determine how much material compresses against a mating surface. In physics, the magnetic confinement chambers of tokamak fusion reactors are torus-shaped so that plasma can circulate endlessly without hitting a wall. Some proposed space stations and rotating habitats also use a torus shape so that rotation can simulate gravity around the ring.
A common misconception is that any donut-like ring shape is automatically a valid torus. Mathematically, a standard "ring torus" only exists when the major radius R is strictly greater than the minor radius r. If r equals R, the tube's inner edge passes exactly through the centre point, producing a "horn torus" with no hole. If r is greater than R, the tube overlaps itself near the centre, producing a self-intersecting "spindle torus" that the simple volume and surface area formulas below do not describe. This calculator checks that R is greater than r before computing a result, so it always represents a real, non-self-intersecting ring shape.
This calculator instantly computes a torus's volume, surface area, outer diameter, inner diameter, and tube cross-section area from just the major and minor radius, with the underlying geometry shown in a labelled diagram. It is useful for students studying solids of revolution, for anyone estimating the material or capacity of a ring-shaped tank or seal, and for quick sanity checks when comparing donut-shaped designs of different sizes.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Bagel-Sized Torus
Major radius R = 6 units, minor radius r = 3 units
Example 2 — Vehicle Tire Inner Tube
Major radius R = 10 units, minor radius r = 4 units
Example 3 — Large Ring-Shaped Structure
Major radius R = 50 units, minor radius r = 10 units (e.g. a rotating space habitat concept)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for the volume of a torus?
Volume = 2 × π² × R × r², where R is the major radius (centre of torus to centre of tube) and r is the minor radius (tube radius). For R = 6 and r = 3, Volume = 2 × π² × 6 × 9 ≈ 1,065.92 cubic units.
What is the formula for the surface area of a torus?
Surface Area = 4 × π² × R × r. For R = 6 and r = 3, Surface Area = 4 × π² × 6 × 3 ≈ 710.61 square units. This is the total curved surface, similar to unrolling the tube around the full ring.
What is the difference between the major radius and minor radius of a torus?
The major radius (R) is the distance from the centre of the whole torus to the centre of the circular tube. The minor radius (r) is the radius of the tube itself, i.e. how thick the ring is. A bicycle inner tube has a large major radius and a small minor radius, while a thick rubber O-ring has the two much closer in size.
Why does this calculator require R to be greater than r?
A standard ring torus only exists when the major radius R is strictly greater than the minor radius r. If r equals R, the tube passes exactly through the centre point (a horn torus). If r is greater than R, the tube overlaps itself and self-intersects (a spindle torus), and the simple volume and surface area formulas used here no longer describe a single, non-overlapping surface.
What are some real-world examples of a torus shape?
Donuts, bagels, bicycle and car inner tubes, rubber O-ring seals, some space station and habitat ring designs, certain magnetic confinement fusion reactors (tokamaks), and swimming pool inflatable rings are all approximately torus-shaped.
How do I find the outer and inner diameter of a torus?
Outer diameter = 2 × (R + r), the distance straight across the widest point of the whole shape. Inner diameter = 2 × (R - r), the distance across the hole in the middle. For R = 6 and r = 3, outer diameter = 18 units and inner diameter = 6 units.
How is a torus different from a cylinder or a sphere?
A cylinder is a straight tube with flat circular ends. A sphere is a solid ball with a single radius. A torus is what you get by taking a circular tube and bending it around into a closed ring, so it has a hole through the middle that neither a cylinder nor a sphere has. Mathematically a torus also has a different topology (genus 1) than a sphere (genus 0).
Does doubling the minor radius double the volume of a torus?
No, it roughly quadruples it. Volume = 2π²Rr² depends on r squared, so doubling r multiplies the volume term from r² by 4, while R stays fixed. Doubling R instead, with r fixed, only doubles the volume, because volume is linear in R.
What is a horn torus and a spindle torus?
A horn torus is the special case where the minor radius equals the major radius (r = R), so the tube's inner edge meets exactly at the centre point with no hole. A spindle torus occurs when r is greater than R, causing the tube to overlap itself near the centre. This calculator only supports the standard ring torus case where R is strictly greater than r.
How do I measure R and r on a real donut-shaped object?
Measure the total outer diameter and the diameter of the central hole. Major radius R = (outer diameter + inner diameter) / 4, and minor radius r = (outer diameter - inner diameter) / 4. For example, a bagel with a 12 cm outer diameter and a 3 cm hole gives R = 3.75 cm and r = 2.25 cm.
Is the volume formula for a torus related to Pappus's centroid theorem?
Yes. Pappus's theorem states that the volume of a solid of revolution equals the area of the rotated shape times the distance travelled by its centroid. For a torus, the tube's cross-section is a circle of area πr², and its centroid travels a distance of 2πR when revolved around the main axis, giving Volume = πr² × 2πR = 2π²Rr², exactly the formula used here.