Calculations at a torus. R ist the distance from the center of the tube to the center of the torus, r is the radius of the tube. With R>r it is a ring torus. A ring torus is a toroid with a circle as base. With R=r this is a horn torus, where the inner side of the tube closes the center of the torus. This calculation is for ring and horn torus. With R<r, it is a spindle torus. Enter both radiuses and choose the number of decimal places. Then click Calculate.

Formulas:
a = R - r
b = 2 * ( R + r )
A = 4 * π² * R * r
V = 2 * π² * R * r²

pi:
π = 3.141592653589793...

The radiuses have the same unit (e.g. meter), the surface area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^{-1}.

The torus is a special case of the toroid in which the rotating base surface is a circle. The slice planes of the ring torus are therefore the circle (or two circles) for a vertical cut and the annulus for a horizontal cut. Two vertical cuts at two different points create a torus sector.
The torus is axially symmetrical to every vertical plane and to the horizontal plane, each through the center. It is also point-symmetrical to this center and rotationally symmetrical for any angle.
The torus as a surface with content is also called a solid torus, if the term torus only refers to the surface. Both terms play a role in topology. A torus is also mentioned in knot theory; a torus knot can be traced back to a torus that is unknotted.
Hypothetical designs for space colonies are often designed in torus form; the best known design of this kind is NASA's Stanford torus. The rotation around the vertical axis would simulate gravity, so that at the outermost edge of the inner side of the torus, usable and living space would be created under Earth-like conditions. The effort required to build such a colony would be very great.