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Torus
doughnut-shaped surface of revolution, based on a revolved figure of a circle / From Wikipedia, the free encyclopedia
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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/17/Torus.png/640px-Torus.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Torus_revolution.gif/320px-Torus_revolution.gif)
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 . The distance from the center of the tube to the center of the torus is called the major radius, written as
.
The surface area of a torus is given by
.
This area is the same as the area of a straight tube that has a radius and length
.
The volume of a solid torus is given by
.
This volume is the same as the volume of a straight rod that has a radius and length
.