# Annulus (mathematics)

__: Region between two concentric circles__

**Short description**In mathematics, an **annulus** (plural **annuli** or **annuluses**) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word *anulus* or *annulus* meaning 'little ring'. 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.

## Area

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*:

- [math]\displaystyle{ A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right). }[/math]

The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, 2*d* in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so *d* and *r* are sides of a right-angled triangle with hypotenuse *R*, and the area of the annulus is given by

- [math]\displaystyle{ A = \pi\left(R^2 - r^2\right) = \pi d^2. }[/math]

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 2π*ρ dρ* and then integrating from *ρ* = *r* to *ρ* = *R*:

- [math]\displaystyle{ A = \int_r^R\!\! 2\pi\rho\, d\rho = \pi\left(R^2 - r^2\right). }[/math]

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

- [math]\displaystyle{ A = \frac{\theta}{2} \left(R^2 - r^2\right). }[/math]

## Complex structure

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

- [math]\displaystyle{ r \lt |z - a| \lt R. }[/math]

If *r* is 0, the region is known as the **punctured disk** (a disk with a point hole in the center) 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

- [math]\displaystyle{ z \mapsto \frac{z - a}{R}. }[/math]

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

- Engineering:Annular cutter
- Annulus theorem/conjecture – In mathematics, on the region between two well-behaved spheres
- Spherical shell
- Torus – Doughnut-shaped surface of revolution

## References

- ↑ Haunsperger, Deanna; Kennedy, Stephen (2006).
*The Edge of the Universe: Celebrating Ten Years of Math Horizons*. ISBN 9780883855553. https://books.google.com/books?id=I9oVP8TlyqIC&pg=PA70. Retrieved 9 May 2017.

## External links

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

Original source: https://en.wikipedia.org/wiki/Annulus (mathematics).
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