# Annulus (mathematics)

In mathematics, an **annulus** (the Latin word for "little ring", with plural *annuli*) is a ring-shaped object, especially a region bounded by two concentric circles. The adjectival form is **annular** (as in annular eclipse).

The open annulus is topologically equivalent to both the open cylinder *S*^{1} × (0,1) and the punctured plane.

The area of an annulus is the difference in the areas of the larger circle of radius *R* and the smaller one of radius *r*:

The area of an annulus can be obtained from the length of the longest interval that can lie completely inside the annulus, 2**d* in the accompanying diagram. This can be proven by the Pythagorean theorem; the length of the longest interval that can lie completely inside the annulus will be tangent to the smaller circle and form a right angle with its radius at that point. Therefore *d* and *r* are the sides of a right angled triangle with hypotenuse *R* and the area is given by:

The area can also be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width *dρ* and area 2*πρ dρ* and then integrating from *ρ* = *r* to *ρ* = *R*:

The area of an annulus sector of angle *θ*, with *θ* measured in radians, is given by:

## Complex structure

In complex analysis an **annulus** ann*(a; r, R)* in the complex plane is an open region defined by:

If *r* is 0, the region is known as the **punctured disk** of radius *R* around the point *a*.

As a subset of the complex plane, an annulus can be considered as a Riemann surface. The complex structure of an annulus depends only on the ratio *r*/*R*. Each annulus ann*(a; r, R)* can be holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map

The inner radius is then *r*/*R* < 1.

The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus.

## See also

- Annulus theorem (or conjecture)
- Spherical shell
- Torus
- List of geometric shapes

## External links

- Annulus definition and properties With interactive animation
- Area of an annulus, formula With interactive animation