A ring doughnut approximates a torus, the doughnut-shaped mathematical surface formed by rotating a circle around an external coplanar axis.
Strictly speaking, an ordinary doughnut is closer to a solid torus: it includes the volume inside the curved surface. The torus itself is only the two-dimensional boundary, while the solid torus is topologically equivalent to a circle multiplied by a disk.
The torus has one handle, or genus one. That is why a coffee mug with one handle is topologically equivalent to a doughnut, even though their shapes differ dramatically. A sphere, by contrast, has no hole and genus zero.
The word torus comes from Latin for a swelling or rounded molding. In geometry, related forms include horn tori and spindle tori, but the familiar ring torus is the one that most closely matches a ring doughnut.