toroidshaped

Toroid-shaped describes an object that resembles a torus, a doughnut-like ring created by revolving a circle around an axis in the same plane as the circle. The term is used across mathematics, physics, and engineering to indicate a hollow ring form with a central hole.

In geometric terms, a toroid-shaped object has a cross-sectional circle of radius r and a central (major) radius R, with R greater than r. The cross section remains circular, and the hole’s radius is R − r. The surface of a torus can be parametrized by two angles and is a classic example of a surface of revolution. In practical contexts, a torus is often treated as a solid of revolution with a circular cross-section, sometimes simply called a toroid.

Two common formulas describe its size: the surface area is 4π^2Rr, and the volume is 2π^2Rr^2, valid for R > r. These relationships show how changing the major or minor radius alters both the outer area and the contained volume.

Term usage varies: mathematicians typically distinguish between a torus (the hollow surface) and a solid torus or toroid (the thicker, donut-shaped solid). In technology, toroid-shaped components include doughnut-shaped transformers and inductors, while in physics the toroidal geometry appears in devices such as tokamaks for magnetic confinement.

Examples of toroid-shaped objects include everyday items like doughnuts and lifebuoys, as well as engineered components and scientific instruments that exploit the ring geometry for magnetic, fluid, or structural reasons.