What is the mathematical name for a doughnut shape?

The mathematical name for the shape of a doughnut’s surface is a torus (plural tori). It’s formed by rotating a circle around an axis that lies in the same plane but doesn’t cross the circle. In topology, a torus is a surface with one hole, which is why mathematicians joke that a coffee mug and a doughnut are the same shape.

Doughnut Shape Mathematical Name: What Is a Torus?

How a torus is made

Take a circle of radius r. Spin it around a line that sits a distance R from the circle’s center, where R > r. The surface it sweeps out is a torus. The word comes from Latin torus, meaning a cushion or swelling.

Measure	Formula
Surface area	4π²Rr
Volume	2π²Rr²

Example: A ring doughnut with R = 4 cm and r = 1.5 cm has a volume of about 178 cm³.

A coffee mug is a doughnut

Topology studies properties that don’t change when a shape is stretched or bent, as long as you don’t tear it or glue parts together. To a topologist:

Same as a sphere (no holes)	Same as a torus (one hole)	Two holes
Ball, bowl, plate, spoon	Doughnut, coffee mug, ring, bagel	Pretzel-like shapes, a figure-eight

A clay mug can be smoothly reshaped into a doughnut: the handle’s hole becomes the doughnut’s hole. The number of holes is called the genus.

Torus shapes in the real world

Tokamak fusion reactors confine super-hot plasma in a torus-shaped chamber.
Inflatable swim rings, bagels and life buoys.
Some space station designs use a spinning ring to create artificial gravity.
Smoke rings are moving tori of air.

The video game torus

In some classic games such as Asteroids and Pac-Man, leaving one edge of the screen brings you back on the opposite edge. Mathematically, a screen that wraps both left-right and top-bottom is a flat torus.

A horn torus occurs when R = r, so the hole shrinks to a point. A spindle torus forms when R < r and the surface intersects itself.

For another branch of everyday science, see Newton’s three laws of motion.