# Torus

doughnut-shaped surface of revolution

A torus (plural: tori or toruses) is a tube shape that is similar to the surface of a doughnut or an inner tube. In geometry, a torus is obtained by rotating a circle in three dimensional space. To make a torus, the circle is rotated around a line (called the axis of rotation) that is in the same plane as the circle. Usually the line does not touch the circle, so the torus has a hole through the center, and the torus is called a ring torus. For a ring torus, the axis of rotation passes through the center of the hole.

A ring torus

In topology, sizes don't matter, and a torus is any shape that has one hole through it.

The ring torus is the most well-known type of torus, but other types exist. If the line the circle rotates around is tangent to the circle, then it becomes a horn torus, and if it passes through the circle then it is a spindle torus. A toroid is a suface made by rotating any shape around a line, so a torus is one kind of toroid.

If the torus is filled to make a solid shape, it is called a solid torus. A solid torus is often simply called a torus. A solid torus is made by rotating a disk (a filled-in circle) around a line. Common objects that have the shape of a solid torus are a doughnut, a bagel and an O-ring.

A torus is like a tube that is bent into a circle so it connects to itself. The radius of the tube or circle is called the minor radius, written ${\displaystyle r}$. The distance from the center of the tube to the center of the torus is called the major radius, written ${\displaystyle R}$.

The surface area of a torus is given by

{\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr\end{aligned}}}.

This area is the same as the area of a straight tube that has radius ${\displaystyle r}$ and length ${\displaystyle 2\pi R}$.

The volume of a solid torus is given by

{\displaystyle {\begin{aligned}V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}\end{aligned}}}.

This volume is the same as the volume of a straight rod that has radius ${\displaystyle r}$ and length ${\displaystyle 2\pi R}$.