The shape of a donut
A torus is the surface you get by sweeping a circle around an axis that lies in the same plane but does not cross the circle. Two numbers describe it:
- the major radius R, from the center of the hole to the center of the tube;
- the minor radius r, the radius of the tube itself.
Donuts, bagels, inner tubes, lifebuoys, O-rings and the magnetic chambers of fusion reactors called tokamaks are all close to a torus.
The formulas
Volume: V = 2π² × R × r²
Surface area: A = 4π² × R × r
Both follow from the theorems of Pappus, proved by the Greek mathematician Pappus of Alexandria around 300 AD: the volume of a solid of revolution equals the area of the shape times the distance traveled by its center. Here the area is πr² and the distance is 2πR, so V = πr² × 2πR. In the same way, the surface area is the circumference 2πr times 2πR.
From diameters
You can often measure the outer and inner diameters more easily:
- R = (outer diameter + inner diameter) ÷ 4
- r = (outer diameter − inner diameter) ÷ 4
Example
A torus with R = 10 and r = 3, so outer diameter 26 and inner diameter 14:
- Volume: 2 × π² × 10 × 9 = 1,776.5 cubic units
- Surface area: 4 × π² × 10 × 3 = 1,184.4 square units
- Cross-section of the tube: π × 3² = 28.3 square units
If the lengths are in inches, the volume is in cubic inches; divide by 231 to get US gallons. In centimeters, divide by 1,000 to get liters.
Practical uses
- O-rings: the volume tells you how much rubber a seal contains, and the groove must hold that volume with room to spare, usually 10 to 25%.
- Inner tubes and pool floats: the volume gives the amount of air inside.
- Baking: the volume of a ring cake or donut helps scale recipes between molds of different sizes.
- Engineering: toroidal tanks and transformer cores use the same formulas.
Special cases
When r equals R, the hole closes up to a single point and the shape is called a horn torus. When r is larger than R, the tube crosses the axis and the surface intersects itself; the formulas then no longer give the true volume of the solid. A torus with r much smaller than R behaves almost like a long, thin cylinder of length 2πR bent into a ring.