The torus volume formula from Pappus's theorem
How 2 pi squared R r squared is derived, how to measure the major and minor radius, and how volume and surface area differ.
Deriving the volume with Pappus's theorem
A torus is what you get when a circle is swept around an axis that does not touch it. Pappus's centroid theorem says the volume of such a solid equals the area of the swept region times the distance its centroid travels. The tube's cross section is a circle of area pi r squared, and its center sits a distance R from the axis, so it travels a circular path of length 2 pi R. Multiplying gives pi r squared times 2 pi R, which simplifies to 2 pi squared R r squared. That is exactly the formula this calculator evaluates.
Measuring the major and minor radius
To measure a real donut-shaped object, first find the outer radius from the center of the hole to the outer edge, then the inner radius from the center to the hole's edge. The minor radius r is half the difference between them, and the major radius R is their average. For example, an outer radius of 13 and an inner radius of 7 give r equal to 3 and R equal to 10, the tool's default. Getting these two numbers right matters more than any other step, because volume grows with the square of r.
Volume versus surface area
Volume answers how much material fills the ring, while surface area answers how much skin wraps around it. They scale differently: volume uses r squared and surface area uses r to the first power, so doubling the tube radius roughly quadruples the volume but only doubles the surface area. This is why a slightly fatter tube adds material fast. The tool shows both figures side by side so you can compare how a change to R or r pushes each one.
When the formula stops describing a clean donut
The tidy picture assumes R is at least r, so the hole stays open and the tube never overlaps itself. If you set r larger than R, the swept circle passes through the axis and the surfaces intersect, producing a spindle or horn torus rather than a ring. The calculator still returns 2 pi squared R r squared, which remains the correct volume of the swept solid counted once, but it no longer matches the everyday idea of a donut with a hole.