Torus Calculator

Torus (Donut Shape)

A torus is a 3D surface generated by rotating a circle around an external axis in the same plane. A donut, a life preserver, an O-ring and an inflatable swimming ring are all tori.

The two radii:

• R = major radius: distance from the center of the tube to the center of the torus
• r = minor radius: radius of the circular tube itself
• The minor radius must be smaller than the major radius (r < R) for a proper ring torus

Formulas (from Pappus centroid theorem):

Pappus theorem (why these formulas work): A torus is formed by rotating a circle of radius r through a full 360 degrees around an axis at distance R.

• Volume = (area of circle) × (distance traveled by centroid) = pi * r^2 × 2piR
• Surface Area = (circumference of circle) × (distance traveled) = 2pir × 2piR

Worked example, a standard donut shape (R = 4 cm, r = 1.5 cm):

• Volume: 2 × pi^2 × 4 × 1.5^2 = 2 × 9.87 × 4 × 2.25 ≈ 177.7 cm^3
• Surface Area: 4 × pi^2 × 4 × 1.5 = 4 × 9.87 × 6 ≈ 236.9 cm^2
• Outer diameter: 2(R + r) = 11 cm, and the hole in the middle is 2(R − r) = 5 cm across

That last line is the one to measure against, because R and r are both invisible on a real donut. Nobody can point at the centre of the tube. What you CAN measure with a ruler is the outside, 11 cm, and the hole, 5 cm, and those give you back R = (11 + 5)/4 = 4 and r = (11 − 5)/4 = 1.5.

Special torus types:

• R = r: the hole shuts to a single point, and this is a horn torus
• R < r: the surface passes through itself, and this is a spindle torus
• This calculator requires R > r, the standard ring torus, and says so rather than returning a number for a shape that does not exist

Real-world tori: Tyre inner tubes, life preservers, O-rings, magnetic confinement fusion reactors (the tokamak is a torus by design, because a plasma bottle with ends leaks at the ends), and toroidal transformer cores. Topology cares about the torus for a different reason: it is the surface with exactly one hole, and no amount of stretching turns it into a sphere. That is the joke about a topologist and a coffee cup, whose single handle-hole makes it a torus.