27:
184:
640:
430:
326:
635:{\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}}}
706:
423:
791:
881:
937:
832:
254:
963:– 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.
128:
108:
1085:
651:
133:
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.
355:
838:
represents the arc length, r represents the radius of the circle and θ represents the angle in radians made by the arc at the centre of the circle.
202:
are given as one of the 8 octants (N, NE, E, SE, S, SW, W, NW) because that is more precise than merely giving one of the 4 quadrants, and the
722:
844:
897:
237:. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle θ (expressed in radians) and
1143:
1101:
1049:
1018:
1182:
1174:
1236:
808:
1232:
213:" comes from the fact that it is based on 1/8th of the circle. Most commonly, octants are seen on the
167:(45°), which come from the sector being one 4th, 6th or 8th part of a full circle, respectively. The
1240:
1218:
1170:
Mathematics
Standard Level for the International Baccalaureate : a text for the new syllabus
981:
1008:
1224:
841:
If the value of angle is given in degrees, then we can also use the following formula by:
8:
210:
226:
113:
93:
55:
1093:
1089:
1188:
1178:
1149:
1139:
1107:
1097:
1055:
1045:
1014:
976:
1256:
1035:
986:
960:
891:
427:
Another approach is to consider this area as the result of the following integral:
20:
143:
71:
971:
645:
321:{\displaystyle A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}}
195:
168:
155:. Sectors with other central angles are sometimes given special names, such as
67:
1111:
1250:
1153:
1044:. Kathleen McKenzie (3rd ed.). New York: Industrial Press. p. 376.
966:
87:
59:
1192:
1059:
244:(because the area of the sector is directly proportional to its angle, and
214:
199:
188:
172:
30:
The minor sector is shaded in green while the major sector is shaded white.
1129:
206:
typically does not have enough accuracy to allow more precise indication.
1211:
The
Elements of Geometry, in Eight Books; or, First Step in Applied Logic
1168:
1039:
1125:
152:
1081:
716:
203:
701:{\displaystyle A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}}
148:
1214:
1135:
418:{\displaystyle A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}}
63:
1013:. New Delhi: New Saraswati House India Pvt Ltd. p. 234.
26:
719:
of a sector is the sum of the arc length and the two radii:
183:
951:
represents the angular width of the sector in radians.
894:
formed with the extremal points of the arc is given by
1086:
National
Council of Educational Research and Training
900:
847:
811:
725:
654:
433:
358:
257:
116:
96:
141:
A sector with the central angle of 180° is called a
931:
875:
826:
785:
700:
634:
417:
320:
122:
102:
16:Portion of a disk enclosed by two radii and an arc
1248:
251:is the angle for the whole circle, in radians):
786:{\displaystyle P=L+2r=\theta r+2r=r(\theta +2)}
1033:
876:{\displaystyle L=2\pi r{\frac {\theta }{360}}}
334:can be obtained by multiplying the total area
1124:
932:{\displaystyle C=2R\sin {\frac {\theta }{2}}}
947:represents the radius of the circle, and
805:The formula for the length of an arc is:
591:
537:
521:
378:
277:
182:
25:
1177:: Infinity Publishing.com. p. 79.
130:is the arc length of the minor sector.
1249:
1229:Elements of Geometry and Trigonometry
1166:
1075:
1006:
62:bounded by a circle) enclosed by two
1071:
1069:
13:
644:Converting the central angle into
14:
1268:
1078:Mathematics: Textbook for class X
1066:
330:The area of a sector in terms of
175:) can also be termed a quadrant.
1215:Longmans, Green, Reader and Dyer
885:
227:Circular arc § Sector area
1160:
1118:
1027:
1000:
780:
768:
601:
547:
531:
515:
231:The total area of a circle is
110:the radius of the circle, and
1:
1134:(3rd ed.). Boston, MA.:
993:
943:represents the chord length,
800:
1128:; Edwards, Bruce H. (2002).
710:
209:The name of the instrument "
7:
1131:Calculus I with Precalculus
954:
10:
1275:
1203:
1041:Technical shop mathematics
827:{\displaystyle L=r\theta }
224:
178:
18:
1007:Dewan, Rajesh K. (2016).
989:– the analogous 3D figure
78:and the larger being the
345:to the total perimeter 2
136:
19:Not to be confused with
982:Sector of (mathematics)
220:
54:), is the portion of a
1237:A. S. Barnes & Co.
1076:Uppal, Shveta (2019).
933:
877:
828:
787:
702:
636:
419:
322:
191:
124:
104:
31:
1175:West Conshohocken, PA
1010:Saraswati Mathematics
934:
878:
829:
788:
703:
637:
420:
323:
186:
125:
105:
29:
1167:Wicks, Alan (2004).
898:
845:
809:
723:
652:
431:
356:
255:
147:and is bounded by a
114:
94:
570:
508:
493:
469:
454:
74:being known as the
70:, with the smaller
1209:Gerard, L. J. V.,
929:
873:
824:
783:
715:The length of the
698:
632:
556:
494:
479:
455:
440:
415:
318:
192:
120:
100:
82:. In the diagram,
32:
1235:, ed. (New York:
1145:978-0-8400-6833-0
1103:978-81-7450-634-4
1036:Anderson, John G.
977:Hyperbolic sector
927:
871:
696:
630:
604:
579:
550:
534:
518:
413:
395:
316:
291:
171:of a quadrant (a
123:{\displaystyle L}
103:{\displaystyle r}
1264:
1197:
1196:
1164:
1158:
1157:
1122:
1116:
1115:
1073:
1064:
1063:
1034:Achatz, Thomas;
1031:
1025:
1024:
1004:
987:Spherical sector
961:Circular segment
950:
946:
942:
938:
936:
935:
930:
928:
920:
890:The length of a
882:
880:
879:
874:
872:
864:
837:
833:
831:
830:
825:
796:
792:
790:
789:
784:
707:
705:
704:
699:
697:
695:
694:
685:
684:
675:
673:
672:
641:
639:
638:
633:
631:
626:
622:
621:
611:
606:
605:
597:
590:
589:
580:
572:
569:
564:
552:
551:
543:
536:
535:
527:
520:
519:
511:
507:
502:
492:
487:
468:
463:
453:
448:
424:
422:
421:
416:
414:
409:
401:
396:
394:
380:
377:
376:
348:
341:by the ratio of
337:
327:
325:
324:
319:
317:
312:
308:
307:
297:
292:
290:
279:
276:
275:
250:
243:
236:
129:
127:
126:
121:
109:
107:
106:
101:
85:
38:, also known as
21:circular section
1274:
1273:
1267:
1266:
1265:
1263:
1262:
1261:
1247:
1246:
1225:Legendre, A. M.
1206:
1201:
1200:
1185:
1165:
1161:
1146:
1138:. p. 570.
1123:
1119:
1104:
1074:
1067:
1052:
1032:
1028:
1021:
1005:
1001:
996:
957:
948:
944:
940:
919:
899:
896:
895:
888:
863:
846:
843:
842:
835:
810:
807:
806:
803:
797:is in radians.
794:
724:
721:
720:
713:
690:
686:
680:
676:
674:
668:
664:
653:
650:
649:
617:
613:
612:
610:
596:
595:
585:
581:
571:
565:
560:
542:
541:
526:
525:
510:
509:
503:
498:
488:
483:
464:
459:
449:
444:
432:
429:
428:
402:
400:
384:
379:
372:
368:
357:
354:
353:
346:
335:
303:
299:
298:
296:
283:
278:
271:
267:
256:
253:
252:
245:
238:
232:
229:
223:
196:wind directions
181:
139:
115:
112:
111:
95:
92:
91:
83:
36:circular sector
24:
17:
12:
11:
5:
1272:
1271:
1260:
1259:
1245:
1244:
1233:Charles Davies
1222:
1205:
1202:
1199:
1198:
1183:
1159:
1144:
1117:
1102:
1065:
1051:978-0831130862
1050:
1026:
1020:978-8173358371
1019:
998:
997:
995:
992:
991:
990:
984:
979:
974:
972:Earth quadrant
969:
964:
956:
953:
926:
923:
918:
915:
912:
909:
906:
903:
887:
884:
870:
867:
862:
859:
856:
853:
850:
823:
820:
817:
814:
802:
799:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
712:
709:
693:
689:
683:
679:
671:
667:
663:
660:
657:
629:
625:
620:
616:
609:
603:
600:
594:
588:
584:
578:
575:
568:
563:
559:
555:
549:
546:
540:
533:
530:
524:
517:
514:
506:
501:
497:
491:
486:
482:
478:
475:
472:
467:
462:
458:
452:
447:
443:
439:
436:
412:
408:
405:
399:
393:
390:
387:
383:
375:
371:
367:
364:
361:
315:
311:
306:
302:
295:
289:
286:
282:
274:
270:
266:
263:
260:
222:
219:
194:Traditionally
180:
177:
138:
135:
119:
99:
15:
9:
6:
4:
3:
2:
1270:
1269:
1258:
1255:
1254:
1252:
1242:
1238:
1234:
1230:
1226:
1223:
1220:
1216:
1212:
1208:
1207:
1194:
1190:
1186:
1184:0-7414-2141-0
1180:
1176:
1172:
1171:
1163:
1155:
1151:
1147:
1141:
1137:
1133:
1132:
1127:
1121:
1113:
1109:
1105:
1099:
1095:
1091:
1087:
1083:
1079:
1072:
1070:
1061:
1057:
1053:
1047:
1043:
1042:
1037:
1030:
1022:
1016:
1012:
1011:
1003:
999:
988:
985:
983:
980:
978:
975:
973:
970:
968:
967:Conic section
965:
962:
959:
958:
952:
924:
921:
916:
913:
910:
907:
904:
901:
893:
883:
868:
865:
860:
857:
854:
851:
848:
839:
821:
818:
815:
812:
798:
777:
774:
771:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
718:
708:
691:
687:
681:
677:
669:
665:
661:
658:
655:
647:
642:
627:
623:
618:
614:
607:
598:
592:
586:
582:
576:
573:
566:
561:
557:
553:
544:
538:
528:
522:
512:
504:
499:
495:
489:
484:
480:
476:
473:
470:
465:
460:
456:
450:
445:
441:
437:
434:
425:
410:
406:
403:
397:
391:
388:
385:
381:
373:
369:
365:
362:
359:
351:
344:
340:
333:
328:
313:
309:
304:
300:
293:
287:
284:
280:
272:
268:
264:
261:
258:
249:
242:
235:
228:
218:
216:
212:
207:
205:
201:
197:
190:
185:
176:
174:
170:
166:
162:
158:
154:
150:
146:
145:
134:
131:
117:
97:
89:
88:central angle
81:
77:
73:
69:
65:
61:
60:closed region
57:
53:
49:
45:
41:
40:circle sector
37:
28:
22:
1228:
1210:
1169:
1162:
1130:
1120:
1077:
1040:
1029:
1009:
1002:
889:
886:Chord length
840:
804:
714:
643:
426:
349:
342:
338:
331:
329:
247:
240:
233:
230:
215:compass rose
208:
200:compass rose
193:
173:circular arc
164:
160:
156:
142:
140:
132:
80:major sector
79:
76:minor sector
75:
51:
47:
46:or simply a
43:
39:
35:
33:
1136:Brooks/Cole
1126:Larson, Ron
1088:. pp.
187:An 8-point
163:(60°), and
44:disk sector
1112:1145113954
994:References
801:Arc length
225:See also:
153:semicircle
1239:, 1858),
1217:, 1874),
1213:(London,
1154:706621772
1082:New Delhi
922:θ
917:
866:θ
858:π
822:θ
772:θ
748:θ
717:perimeter
711:Perimeter
692:∘
682:∘
678:θ
662:π
624:θ
602:~
599:θ
567:θ
558:∫
548:~
545:θ
532:~
516:~
496:∫
490:θ
481:∫
457:∫
451:θ
442:∫
389:π
366:π
310:θ
288:π
281:θ
265:π
204:wind vane
157:quadrants
144:half-disk
50:(symbol:
1251:Category
1193:58869667
1060:56559272
1038:(2005).
955:See also
189:windrose
161:sextants
149:diameter
1257:Circles
1204:Sources
646:degrees
198:on the
179:Compass
165:octants
159:(90°),
86:is the
66:and an
1241:p. 119
1219:p. 285
1191:
1181:
1152:
1142:
1110:
1100:
1058:
1048:
1017:
939:where
834:where
793:where
648:gives
211:octant
151:and a
48:sector
892:chord
137:Types
64:radii
1189:OCLC
1179:ISBN
1150:OCLC
1140:ISBN
1108:OCLC
1098:ISBN
1056:OCLC
1046:ISBN
1015:ISBN
221:Area
72:area
56:disk
1094:227
1090:226
914:sin
869:360
688:360
169:arc
68:arc
58:(a
42:or
1253::
1231:,
1227:,
1187:.
1173:.
1148:.
1106:.
1096:.
1092:,
1084::
1080:.
1068:^
1054:.
352:.
234:πr
217:.
90:,
34:A
1243:.
1221:.
1195:.
1156:.
1114:.
1062:.
1023:.
949:θ
945:R
941:C
925:2
911:R
908:2
905:=
902:C
861:r
855:2
852:=
849:L
836:L
819:r
816:=
813:L
795:θ
781:)
778:2
775:+
769:(
766:r
763:=
760:r
757:2
754:+
751:r
745:=
742:r
739:2
736:+
733:L
730:=
727:P
670:2
666:r
659:=
656:A
628:2
619:2
615:r
608:=
593:d
587:2
583:r
577:2
574:1
562:0
554:=
539:d
529:r
523:d
513:r
505:r
500:0
485:0
477:=
474:S
471:d
466:r
461:0
446:0
438:=
435:A
411:2
407:L
404:r
398:=
392:r
386:2
382:L
374:2
370:r
363:=
360:A
350:r
347:π
343:L
339:r
336:π
332:L
314:2
305:2
301:r
294:=
285:2
273:2
269:r
262:=
259:A
248:π
246:2
241:π
239:2
118:L
98:r
84:θ
52:⌔
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
