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A ring torus

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.

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, write. The distance from the center of the tube to the center of the torus is called the major radius, write.

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.