حلقة (هندسة رياضية)

Illustration of Mamikon's visual calculus method showing that the areas of two annuli with the same chord length are the same regardless of inner and outer radii.^[1]

في الرياضيات، وتحديداً في الهندسة الرياضية يعطى شكل الحلقة في المستوي على شكل قرص مفرغ بدائرة مشتركة المركز مع مركز القرص. تعادل الحلقة طوبولوجياً إسطوانة مفرغة $S^{1}\times (0,1)$ والمستوي المثقوب.

تعطى مساحة الحلقة بأخذ فرق مساحة الدائرتين ذاتي أنصاف الأقطار R و r على الشكل التالي:

A=\pi (R^{2}-r^{2})\,.

The open annulus is topologically equivalent to both the open cylinder $S 1 \times (0,1)$ and the punctured plane.

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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 $r$ :

A=\pi R^{2}-\pi r^{2}=\pi \left(R^{2}-r^{2}\right).

As a corollary of the chord formula, the area bounded by the circumcircle and incircle of every unit convex regular polygon is

π

/4

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

A=\pi \left(R^{2}-r^{2}\right)=\pi d^{2}.

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 $ρ = R$ :

A=\int _{r}^{R}\!\!2\pi \rho \,d\rho =\pi \left(R^{2}-r^{2}\right).

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

A={\frac {\theta }{2}}\left(R^{2}-r^{2}\right).

Complex structure

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

r<|z-a|<R.

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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}r/R$ . 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