Annulus (mathematics)
In mathematics, an annulus (the Latin word for "little ring", with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. It is also used to describe the area between two concentric circles. The adjectival form is annular (for example, an annular eclipse).
The open annulus is topologically equivalent to both the open cylinder S1 × (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 2πρ dρ ( ) and then integrating from ρ = r to ρ = R:
The area of an annulus segment of angle θ, with θ measured in radians, is given by:
Read more about Annulus (mathematics): Complex Structure