doughnutshaped

Doughnut-shaped describes a three-dimensional form that resembles a torus: a ring with a central hole. The term is used in everyday language for objects such as doughnuts and bagels and, more broadly, for any object whose cross-section is a circle rotated around an external axis to create a hollow center.

Geometrically, a standard torus is generated by revolving a circle of radius r around an axis in the same plane at a distance R from the circle’s center, with R > r. The major radius is R (distance from the hole’s center to the tube’s center) and the minor radius is r (the tube’s radius). The volume of a torus is V = 2 π^2 R r^2, and its surface area is S = 4 π^2 R r.

A common parametric description of the torus uses two angles: x = (R + r cos v) cos u, y = (R + r cos v) sin u, z = r sin v, where u and v each range from 0 to 2π. This expresses the torus as a smooth surface of revolution with a circular cross-section.

Variants and terminology include ring torus (R > r), horn torus (R = r, inner hole shrinks to a point), and spindle torus (R < r, a self-intersecting form). In topology, a torus is a surface of genus 1.

Applications span engineering, physics, and design. Toroidal shapes are used in components such as toroidal inductors and transformers, magnetic confinement devices in plasma physics (tokamaks), and various decorative and functional parts in architecture and art.