Torus Surface Area

Pappus's other centroid theorem covers surfaces: the area swept by a curve equals the curve's length times the distance its centroid travels. Sweeping a circle of circumference $2\pi r$ around a path of length $2\pi R$ gives $A = 4\pi^2 R r$, a result so tidy it looks like a typo next to the volume's $2\pi^2 R r^2$.

Surface area is what you need when something has to coat, wet or transfer across the doughnut: the paint on a torus-shaped handrail, the plating on a ring terminal, the heat exchanged by a coiled tube, the amount of glaze per doughnut in an industrial fryer. It also sets how much lubricant an O-ring carries and how much of it is exposed to attack by a chemical.

One elegant consequence is that a torus with $R = r$, the so-called horn torus whose hole has shrunk to a point, has surface area $4\pi^2 r^2$, which is exactly $\pi$ times the surface area of a sphere of the same tube radius. The doughnut, in that sense, has more than three times the skin of the ball it was rolled from.