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method showing that the areas of two annuli with the same chord length are the same regardless of inner and outer radii.
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is a statement about the maximum value a holomorphic function may take inside an annulus.
66:) is the region between two concentric circles. Informally, it is shaped like a ring or a
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mapped to a standard one centered at the origin and with outer radius 1 by the map
624:
521:{\displaystyle A=\int _{r}^{R}\!\!2\pi \rho \,d\rho =\pi \left(R^{2}-r^{2}\right).}
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to the smaller circle and perpendicular to its radius at that point, so
910: – In mathematics, on the region between two well-behaved spheres
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1148:
121:
The area of an annulus is the difference in the areas of the larger
1136:
404:
383:
246:
227:{\displaystyle A=\pi R^{2}-\pi r^{2}=\pi \left(R^{2}-r^{2}\right).}
100:
257:
The area of an annulus is determined by the length of the longest
889:
277:
948:
The Edge of the
Universe: Celebrating Ten Years of Math Horizons
756:. The complex structure of an annulus depends only on the ratio
386:
by dividing the annulus up into an infinite number of annuli of
1044:
608:{\displaystyle A={\frac {\theta }{2}}\left(R^{2}-r^{2}\right).}
122:
1050:
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241:
As a corollary of the chord formula, the area bounded by the
71:
372:{\displaystyle A=\pi \left(R^{2}-r^{2}\right)=\pi d^{2}.}
272:
in the accompanying diagram. That can be shown using the
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Compact topological surfaces and their immersions in 3D
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are sides of a right-angled triangle with hypotenuse
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459:
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1281:
945:Haunsperger, Deanna; Kennedy, Stephen (2006).
82:meaning 'little ring'. The adjectival form is
1010:
927: – Doughnut-shaped surface of revolution
1017:
1003:
838:{\displaystyle z\mapsto {\frac {z-a}{R}}.}
298:, and the area of the annulus is given by
70:. The word "annulus" is borrowed from the
921: – Three-dimensional geometric shape
469:
249:of every unit convex regular polygon is
236:
33:
531:The area of an annulus sector of angle
1282:
998:
16:Region between two concentric circles
752:, an annulus can be considered as a
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382:The area can also be obtained via
25:
14:
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983:Annulus definition and properties
976:
543:measured in radians, is given by
261:within the annulus, which is the
701:{\displaystyle r<|z-a|<R.}
938:
892:with a slit cut between foci.
811:
733:hole in the center) of radius
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671:
131:and the smaller one of radius
1:
931:
876:Hadamard three-circle theorem
721:, the region is known as the
265:tangent to the inner circle,
7:
989:Area of an annulus, formula
895:
748:As a subset of the complex
10:
1321:
991:With interactive animation
985:With interactive animation
908:Annulus theorem/conjecture
904: – Form of core drill
38:Illustration of Mamikon's
18:
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848:The inner radius is then
914:List of geometric shapes
97:topologically equivalent
1156:Sphere with three holes
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1074:Real projective plane
1059:Pretzel (genus 3) ...
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1229:Euler characteristic
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95:The open annulus is
19:For other uses, see
1295:Elementary geometry
888:an annulus onto an
883:Joukowsky transform
457:
276:since this line is
274:Pythagorean theorem
1056:Number 8 (genus 2)
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739:around the point
619:Complex structure
567:
99:to both the open
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1300:Geometric shapes
1192:Triangulatedness
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1031:Without boundary
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796:holomorphically
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111:punctured plane
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90:annular eclipse
68:hardware washer
40:visual calculus
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1067:Non-orientable
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977:External links
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1112:With boundary
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1100:(genus 3) ...
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1085:Roman surface
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1080:Boy's surface
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1144:Möbius strip
1131:
1092:Klein bottle
962:. Retrieved
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259:line segment
256:
243:circumcircle
133:
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107:× (0,1)
104:
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84:
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79:
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63:
59:
51:
45:
1187:Compactness
654:defined as
652:open region
405:integrating
48:mathematics
1284:Categories
1238:Operations
1220:components
1216:Number of
1196:smoothness
1175:Properties
1123:Semisphere
1038:Orientable
932:References
125:of radius
30:An annulus
1265:Immersion
1260:cross-cap
1258:Gluing a
1252:Gluing a
1149:Cross-cap
1094:(genus 2)
1078:genus 1;
1053:(genus 1)
1047:(genus 0)
821:−
812:↦
679:−
585:−
562:θ
498:−
480:π
474:ρ
467:ρ
464:π
445:∫
403:and then
396:and area
354:π
333:−
315:π
204:−
186:π
170:π
167:−
154:π
64:annuluses
1218:boundary
1137:Cylinder
896:See also
384:calculus
247:incircle
109:and the
101:cylinder
1290:Circles
1168:notions
1166:Related
1132:Annulus
1128:Ribbon
890:ellipse
867:
851:
794:can be
775:
759:
729:with a
646:in the
629:annulus
537:, with
278:tangent
88:(as in
85:annular
80:annulus
52:annulus
21:Annulus
1254:handle
1045:Sphere
955:
869:< 1
650:is an
390:width
123:circle
76:anulus
60:annuli
1224:Genus
1051:Torus
964:9 May
925:Torus
750:plane
731:point
407:from
263:chord
74:word
72:Latin
50:, an
1119:Disk
966:2017
953:ISBN
881:The
874:The
780:ann(
727:disk
690:<
668:<
632:ann(
400:ρ dρ
286:and
245:and
117:Area
1194:or
1158:...
725:(a
717:is
711:If
627:an
623:In
417:to
92:).
78:or
62:or
56:pl.
46:In
1286::
951:.
871:.
788:,
784:;
745:.
640:,
636:;
427::
422:=
412:=
398:2π
393:dρ
253:/4
137::
113:.
58::
1018:e
1011:t
1004:v
968:.
863:R
859:/
855:r
833:.
828:R
824:a
818:z
809:z
792:)
790:R
786:r
782:a
771:R
767:/
763:r
742:a
736:R
719:0
714:r
696:.
693:R
686:|
682:a
676:z
672:|
665:r
644:)
642:R
638:r
634:a
603:.
599:)
593:2
589:r
580:2
576:R
571:(
565:2
557:=
554:A
540:θ
534:θ
516:.
512:)
506:2
502:r
493:2
489:R
484:(
477:=
471:d
461:2
454:R
449:r
441:=
438:A
424:R
420:ρ
414:r
410:ρ
367:.
362:2
358:d
351:=
347:)
341:2
337:r
328:2
324:R
319:(
312:=
309:A
295:R
289:r
283:d
269:d
267:2
251:π
222:.
218:)
212:2
208:r
199:2
195:R
190:(
183:=
178:2
174:r
162:2
158:R
151:=
148:A
134:r
128:R
105:S
54:(
23:.
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