27:
569:
359:
256:
564:{\displaystyle A=\int _{0}^{\theta }\int _{0}^{r}dS=\int _{0}^{\theta }\int _{0}^{r}{\tilde {r}}\,d{\tilde {r}}\,d{\tilde {\theta }}=\int _{0}^{\theta }{\frac {1}{2}}r^{2}\,d{\tilde {\theta }}={\frac {r^{2}\theta }{2}}}
635:
352:
720:
814:
870:
1015:
761:
184:
896:– the part of the sector which remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
580:
101:
The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle.
284:
1073:
1031:
982:
951:
651:
1112:
777:
830:
1104:
1171:
737:
135:(45°), which come from the sector being one quarter, sixth or eighth part of a full circle, respectively. The
914:
1167:
163:. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle
1181:
1175:
1153:
1100:
Mathematics
Standard Level for the International Baccalaureate : a text for the new syllabus
55:
1118:
1108:
1079:
1069:
1037:
1027:
988:
978:
947:
941:
909:
1198:
968:
919:
893:
824:
20:
774:
If the value of angle is given in degrees, then we can also use the following formula by:
111:
71:
904:
574:
136:
67:
1023:
1019:
1192:
899:
87:
59:
356:
Another approach is to consider this area as the result of the following integral:
152:
140:
251:{\displaystyle A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}}
123:. Sectors with other central angles are sometimes given special names, such as
1055:
1041:
120:
1083:
977:. Kathleen McKenzie (3rd ed.). New York: Industrial Press. p. 376.
771:
represents the angle in radians made by the arc at the centre of the circle.
1122:
1011:
992:
645:
174:(because the area of the sector is directly proportional to its angle, and
30:
The minor sector is shaded in green while the major sector is shaded white.
1059:
1155:
The
Elements of Geometry, in Eight Books; or, First Step in Applied Logic
1098:
972:
116:
1159:
1065:
26:
767:
represents the arc length, r represents the radius of the circle and
63:
630:{\displaystyle A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}}
347:{\displaystyle A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}}
946:. New Delhi: New Saraswati House India Pvt Ltd. p. 234.
648:
of a sector is the sum of the arc length and the two radii:
884:
represents the angular width of the sector in radians.
827:
formed with the extremal points of the arc is given by
1016:
National
Council of Educational Research and Training
833:
780:
740:
654:
583:
362:
287:
187:
109:
A sector with the central angle of 180° is called a
864:
808:
755:
714:
629:
563:
346:
250:
16:Portion of a disk enclosed by two radii and an arc
181:is the angle for the whole circle, in radians):
715:{\displaystyle P=L+2r=\theta r+2r=r(\theta +2)}
809:{\displaystyle L=2\pi r{\frac {\theta }{360}}}
264:can be obtained by multiplying the total area
865:{\displaystyle C=2R\sin {\frac {\theta }{2}}}
8:
880:represents the radius of the circle, and
852:
832:
796:
779:
739:
734:The formula for the length of an arc is:
653:
619:
609:
603:
597:
582:
546:
539:
525:
524:
520:
514:
500:
494:
489:
471:
470:
466:
455:
454:
450:
439:
438:
432:
427:
417:
412:
393:
388:
378:
373:
361:
329:
308:
307:
301:
286:
233:
226:
208:
207:
201:
186:
25:
1107:: Infinity Publishing.com. p. 79.
932:
98:is the arc length of the minor sector.
1177:Elements of Geometry and Trigonometry
1135:
62:bounded by a circle) enclosed by two
7:
573:Converting the central angle into
14:
1008:Mathematics: Textbook for class X
260:The area of a sector in terms of
143:) can also be termed a quadrant.
1160:Longmans, Green, Reader and Dyer
153:Circular arc § Sector area
709:
697:
530:
476:
460:
444:
157:The total area of a circle is
94:the radius of the circle, and
1:
1064:(3rd ed.). Boston, MA.:
876:represents the chord length,
1058:; Edwards, Bruce H. (2002).
1061:Calculus I with Precalculus
167:(expressed in radians) and
1215:
974:Technical shop mathematics
756:{\displaystyle L=r\theta }
150:
18:
1152:Gerard, L. J. V. (1874).
940:Dewan, Rajesh K. (2016).
922:– the analogous 3D figure
78:and the larger being the
19:Not to be confused with
915:Sector of (mathematics)
274:to the total perimeter
54:), is the portion of a
1182:A. S. Barnes & Co.
1168:Legendre, Adrien-Marie
1006:Uppal, Shveta (2019).
866:
810:
757:
716:
631:
565:
348:
252:
31:
1105:West Conshohocken, PA
943:Saraswati Mathematics
867:
811:
758:
717:
632:
566:
349:
253:
29:
1097:Wicks, Alan (2004).
831:
778:
738:
652:
581:
360:
285:
185:
115:and is bounded by a
499:
437:
422:
398:
383:
74:being known as the
70:, with the smaller
862:
806:
753:
712:
644:The length of the
627:
561:
485:
423:
408:
384:
369:
344:
248:
82:. In the diagram,
32:
1075:978-0-8400-6833-0
1033:978-81-7450-634-4
969:Anderson, John G.
910:Hyperbolic sector
860:
804:
625:
559:
533:
508:
479:
463:
447:
342:
324:
246:
221:
139:of a quadrant (a
1206:
1185:
1163:
1139:
1133:
1127:
1126:
1094:
1088:
1087:
1052:
1046:
1045:
1003:
997:
996:
967:Achatz, Thomas;
964:
958:
957:
937:
920:Spherical sector
894:Circular segment
883:
879:
875:
871:
869:
868:
863:
861:
853:
823:The length of a
815:
813:
812:
807:
805:
797:
770:
766:
762:
760:
759:
754:
725:
721:
719:
718:
713:
636:
634:
633:
628:
626:
624:
623:
614:
613:
604:
602:
601:
570:
568:
567:
562:
560:
555:
551:
550:
540:
535:
534:
526:
519:
518:
509:
501:
498:
493:
481:
480:
472:
465:
464:
456:
449:
448:
440:
436:
431:
421:
416:
397:
392:
382:
377:
353:
351:
350:
345:
343:
338:
330:
325:
323:
309:
306:
305:
280:
273:
270:by the ratio of
269:
263:
257:
255:
254:
249:
247:
242:
238:
237:
227:
222:
220:
209:
206:
205:
180:
173:
166:
162:
97:
93:
85:
38:, also known as
21:circular section
1214:
1213:
1209:
1208:
1207:
1205:
1204:
1203:
1189:
1188:
1172:Davies, Charles
1166:
1151:
1148:
1143:
1142:
1134:
1130:
1115:
1096:
1095:
1091:
1076:
1068:. p. 570.
1054:
1053:
1049:
1034:
1005:
1004:
1000:
985:
966:
965:
961:
954:
939:
938:
934:
929:
890:
881:
877:
873:
829:
828:
821:
776:
775:
768:
764:
736:
735:
732:
726:is in radians.
723:
650:
649:
642:
615:
605:
593:
579:
578:
542:
541:
510:
358:
357:
331:
313:
297:
283:
282:
275:
271:
265:
261:
229:
228:
213:
197:
183:
182:
175:
168:
164:
158:
155:
149:
107:
95:
91:
83:
36:circular sector
24:
17:
12:
11:
5:
1212:
1210:
1202:
1201:
1191:
1190:
1187:
1186:
1164:
1162:. p. 285.
1147:
1144:
1141:
1140:
1128:
1113:
1089:
1074:
1047:
1032:
998:
984:978-0831130862
983:
959:
953:978-8173358371
952:
931:
930:
928:
925:
924:
923:
917:
912:
907:
905:Earth quadrant
902:
897:
889:
886:
859:
856:
851:
848:
845:
842:
839:
836:
820:
817:
803:
800:
795:
792:
789:
786:
783:
752:
749:
746:
743:
731:
728:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
641:
638:
622:
618:
612:
608:
600:
596:
592:
589:
586:
558:
554:
549:
545:
538:
532:
529:
523:
517:
513:
507:
504:
497:
492:
488:
484:
478:
475:
469:
462:
459:
453:
446:
443:
435:
430:
426:
420:
415:
411:
407:
404:
401:
396:
391:
387:
381:
376:
372:
368:
365:
341:
337:
334:
328:
322:
319:
316:
312:
304:
300:
296:
293:
290:
245:
241:
236:
232:
225:
219:
216:
212:
204:
200:
196:
193:
190:
148:
145:
106:
103:
15:
13:
10:
9:
6:
4:
3:
2:
1211:
1200:
1197:
1196:
1194:
1183:
1179:
1178:
1173:
1169:
1165:
1161:
1157:
1156:
1150:
1149:
1145:
1137:
1132:
1129:
1124:
1120:
1116:
1114:0-7414-2141-0
1110:
1106:
1102:
1101:
1093:
1090:
1085:
1081:
1077:
1071:
1067:
1063:
1062:
1057:
1051:
1048:
1043:
1039:
1035:
1029:
1025:
1021:
1017:
1013:
1009:
1002:
999:
994:
990:
986:
980:
976:
975:
970:
963:
960:
955:
949:
945:
944:
936:
933:
926:
921:
918:
916:
913:
911:
908:
906:
903:
901:
900:Conic section
898:
895:
892:
891:
887:
885:
857:
854:
849:
846:
843:
840:
837:
834:
826:
818:
816:
801:
798:
793:
790:
787:
784:
781:
772:
750:
747:
744:
741:
729:
727:
706:
703:
700:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
647:
639:
637:
620:
616:
610:
606:
598:
594:
590:
587:
584:
576:
571:
556:
552:
547:
543:
536:
527:
521:
515:
511:
505:
502:
495:
490:
486:
482:
473:
467:
457:
451:
441:
433:
428:
424:
418:
413:
409:
405:
402:
399:
394:
389:
385:
379:
374:
370:
366:
363:
354:
339:
335:
332:
326:
320:
317:
314:
310:
302:
298:
294:
291:
288:
279:
268:
258:
243:
239:
234:
230:
223:
217:
214:
210:
202:
198:
194:
191:
188:
179:
172:
161:
154:
146:
144:
142:
138:
134:
130:
126:
122:
118:
114:
113:
104:
102:
99:
89:
88:central angle
81:
77:
73:
69:
65:
61:
60:closed region
57:
53:
49:
45:
41:
40:circle sector
37:
28:
22:
1184:p. 119.
1180:. New York:
1176:
1154:
1136:Uppal (2019)
1131:
1099:
1092:
1060:
1050:
1007:
1001:
973:
962:
942:
935:
822:
819:Chord length
773:
733:
643:
572:
355:
277:
266:
259:
177:
170:
159:
156:
141:circular arc
132:
128:
124:
110:
108:
100:
80:major sector
79:
76:minor sector
75:
51:
47:
46:or simply a
43:
39:
35:
33:
1066:Brooks/Cole
1056:Larson, Ron
1018:. pp.
131:(60°), and
44:disk sector
1158:. London:
1042:1145113954
927:References
730:Arc length
151:See also:
121:semicircle
1084:706621772
1012:New Delhi
855:θ
850:
799:θ
791:π
751:θ
701:θ
677:θ
646:perimeter
640:Perimeter
621:∘
611:∘
607:θ
591:π
553:θ
531:~
528:θ
496:θ
487:∫
477:~
474:θ
461:~
445:~
425:∫
419:θ
410:∫
386:∫
380:θ
371:∫
318:π
295:π
240:θ
218:π
211:θ
195:π
125:quadrants
112:half-disk
50:(symbol:
1193:Category
1170:(1858).
1123:58869667
993:56559272
971:(2005).
888:See also
129:sextants
117:diameter
1199:Circles
1174:(ed.).
1146:Sources
575:degrees
133:octants
127:(90°),
86:is the
66:and an
1121:
1111:
1082:
1072:
1040:
1030:
991:
981:
950:
872:where
763:where
722:where
577:gives
119:and a
48:sector
825:chord
105:Types
64:radii
1119:OCLC
1109:ISBN
1080:OCLC
1070:ISBN
1038:OCLC
1028:ISBN
989:OCLC
979:ISBN
948:ISBN
147:Area
72:area
56:disk
1024:227
1020:226
847:sin
802:360
617:360
137:arc
68:arc
58:(a
42:or
1195::
1117:.
1103:.
1078:.
1036:.
1026:.
1022:,
1014::
1010:.
987:.
281:.
278:πr
267:πr
160:πr
90:,
34:A
1138:.
1125:.
1086:.
1044:.
995:.
956:.
882:θ
878:R
874:C
858:2
844:R
841:2
838:=
835:C
794:r
788:2
785:=
782:L
769:θ
765:L
748:r
745:=
742:L
724:θ
710:)
707:2
704:+
698:(
695:r
692:=
689:r
686:2
683:+
680:r
674:=
671:r
668:2
665:+
662:L
659:=
656:P
599:2
595:r
588:=
585:A
557:2
548:2
544:r
537:=
522:d
516:2
512:r
506:2
503:1
491:0
483:=
468:d
458:r
452:d
442:r
434:r
429:0
414:0
406:=
403:S
400:d
395:r
390:0
375:0
367:=
364:A
340:2
336:L
333:r
327:=
321:r
315:2
311:L
303:2
299:r
292:=
289:A
276:2
272:L
262:L
244:2
235:2
231:r
224:=
215:2
203:2
199:r
192:=
189:A
178:π
176:2
171:π
169:2
165:θ
96:L
92:r
84:θ
52:⌔
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
