Editing Annulus (mathematics) (section)

== Area ==
The area of an annulus is the difference in the areas of the larger [[circle]] of radius {{math|''R''}} and the smaller one of radius {{math|''r''}}:
:<math>A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right) = \pi (R+r)(R-r) .</math>
[[File:annuli_with_same_area_around_unit_regular_polygons.svg|thumb|upright=0.8|As a corollary of the chord formula, the area bounded by the [[circumcircle]] and [[incircle]] of every unit convex regular polygon is {{pi}}/4]]
The area of an annulus is determined by the length of the longest [[line segment]] within the annulus, which is the [[chord (geometry)|chord]] tangent to the inner circle, {{math|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 {{math|''d''}} and {{math|''r''}} are sides of a right-angled triangle with hypotenuse {{math|''R''}}, and the area of the annulus is given by
:<math>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 {{math|''dρ''}} and area {{math|2π''ρ dρ''}} and then [[Integral|integrating]] from {{math|1=''ρ'' = ''r''}} to {{math|1=''ρ'' = ''R''}}:
:<math>A = \int_r^R\!\! 2\pi\rho\, d\rho = \pi\left(R^2 - r^2\right).</math>

{{anchor|Sector}}The area of an ''annulus sector'' (the region between two [[circular sector]]s with overlapping radii) of angle {{math|''θ''}}, with {{math|''θ''}} measured in radians, is given by
:<math> A = \frac{\theta}{2} \left(R^2 - r^2\right). </math>