Torus

A torus (plural: tori or toruses) is a tube shape that looks like a doughnut or an inner tube. In geometry, a torus is made 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

A circle spinning around a line to make a 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 surface 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 as $r$. The distance from the center of the tube to the center of the torus is called the major radius, written as $R$.

The surface area of a torus is given by

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

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

The volume of a solid torus is given by

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

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