Torus Calculator

The major radius R runs from the center of the hole to the center of the tube, while the minor radius r is the radius of the tube itself.

For a standard ring torus the major radius must be at least as large as the minor radius so the tube does not intersect itself. If R = r the inner hole closes (horn torus); if R < r the surface self-intersects.

The volume is V = 2 x pi^2 x R x r^2, using the major radius R and the minor tube radius r. This comes from Pappus's theorem: the area of the tube's cross section (pi*r^2) times the distance its centroid travels around the axis (2*pi*R). The surface area is A = 4 x pi^2 x R x r.

It depends on how the tube radius r compares to the center-to-tube distance R. A ring torus has R greater than r (a normal donut with a hole), a horn torus has R equal to r (the hole closes to a point), and a spindle torus has R less than r (the tube self-intersects).

The surface area is A = 4π²Rr, where R is the distance from the center to the middle of the tube and r is the tube radius. It follows from Pappus's theorem, as does the volume V = 2π²Rr².