38:
651:
195:
441:
337:
646:{\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}}}
717:
434:
802:
892:
948:
1096:
843:
265:
974:– 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.
139:
119:
662:
144:
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.
366:
849:
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.
213:
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
1154:
1112:
1060:
1029:
733:
1193:
855:
908:
248:. 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
1185:
1243:
819:
992:
1235:
224:" comes from the fact that it is based on 1/8th of the circle. Most commonly, octants are seen on the
178:(45°), which come from the sector being one 4th, 6th or 8th part of a full circle, respectively. The
1247:
221:
1251:
1229:
1181:
Mathematics
Standard Level for the International Baccalaureate : a text for the new syllabus
66:
1199:
1189:
1160:
1150:
1118:
1108:
1066:
1056:
1025:
1019:
987:
1267:
1046:
997:
971:
902:
31:
852:
If the value of angle is given in degrees, then we can also use the following formula by:
154:
82:
982:
656:
206:
179:
124:
104:
78:
1104:
1100:
1261:
977:
98:
70:
438:
Another approach is to consider this area as the result of the following integral:
237:
225:
210:
199:
183:
17:
332:{\displaystyle A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}}
166:. Sectors with other central angles are sometimes given special names, such as
1136:
1122:
163:
1164:
1055:. Kathleen McKenzie (3rd ed.). New York: Industrial Press. p. 376.
1203:
1092:
1070:
727:
214:
255:(because the area of the sector is directly proportional to its angle, and
41:
The minor sector is shaded in green while the major sector is shaded white.
1140:
217:
typically does not have enough accuracy to allow more precise indication.
1222:
The
Elements of Geometry, in Eight Books; or, First Step in Applied Logic
1179:
1050:
159:
1225:
1146:
37:
74:
194:
712:{\displaystyle A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}}
193:
429:{\displaystyle A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}}
1024:. New Delhi: New Saraswati House India Pvt Ltd. p. 234.
730:
of a sector is the sum of the arc length and the two radii:
962:
represents the angular width of the sector in radians.
905:
formed with the extremal points of the arc is given by
1097:
National
Council of Educational Research and Training
911:
858:
822:
736:
665:
444:
369:
268:
127:
107:
152:
A sector with the central angle of 180° is called a
942:
886:
837:
796:
711:
645:
428:
331:
133:
113:
27:Portion of a disk enclosed by two radii and an arc
262:is the angle for the whole circle, in radians):
797:{\displaystyle P=L+2r=\theta r+2r=r(\theta +2)}
887:{\displaystyle L=2\pi r{\frac {\theta }{360}}}
345:can be obtained by multiplying the total area
943:{\displaystyle C=2R\sin {\frac {\theta }{2}}}
8:
958:represents the radius of the circle, and
930:
910:
874:
857:
821:
816:The formula for the length of an arc is:
735:
701:
691:
685:
679:
664:
628:
621:
607:
606:
602:
596:
582:
576:
571:
553:
552:
548:
537:
536:
532:
521:
520:
514:
509:
499:
494:
475:
470:
460:
455:
443:
411:
390:
389:
383:
368:
314:
307:
289:
288:
282:
267:
126:
106:
36:
1188:: Infinity Publishing.com. p. 79.
1010:
141:is the arc length of the minor sector.
1240:Elements of Geometry and Trigonometry
73:bounded by a circle) enclosed by two
7:
1082:
1080:
655:Converting the central angle into
25:
1089:Mathematics: Textbook for class X
341:The area of a sector in terms of
186:) can also be termed a quadrant.
1226:Longmans, Green, Reader and Dyer
238:Circular arc § Sector area
791:
779:
612:
558:
542:
526:
242:The total area of a circle is
121:the radius of the circle, and
1:
1145:(3rd ed.). Boston, MA.:
954:represents the chord length,
1139:; Edwards, Bruce H. (2002).
220:The name of the instrument "
1142:Calculus I with Precalculus
1284:
1052:Technical shop mathematics
838:{\displaystyle L=r\theta }
235:
29:
1018:Dewan, Rajesh K. (2016).
1000:– the analogous 3D figure
89:and the larger being the
356:to the total perimeter 2
30:Not to be confused with
993:Sector of (mathematics)
65:), is the portion of a
1248:A. S. Barnes & Co.
1087:Uppal, Shveta (2019).
944:
888:
839:
798:
713:
647:
430:
333:
202:
135:
115:
42:
1186:West Conshohocken, PA
1021:Saraswati Mathematics
945:
889:
840:
799:
714:
648:
431:
334:
197:
136:
116:
40:
1178:Wicks, Alan (2004).
909:
856:
820:
734:
663:
442:
367:
266:
158:and is bounded by a
125:
105:
581:
519:
504:
480:
465:
85:being known as the
81:, with the smaller
1220:Gerard, L. J. V.,
940:
884:
835:
794:
726:The length of the
709:
643:
567:
505:
490:
466:
451:
426:
329:
203:
131:
111:
93:. In the diagram,
43:
1246:, ed. (New York:
1156:978-0-8400-6833-0
1114:978-81-7450-634-4
1047:Anderson, John G.
988:Hyperbolic sector
938:
882:
707:
641:
615:
590:
561:
545:
529:
424:
406:
327:
302:
182:of a quadrant (a
134:{\displaystyle L}
114:{\displaystyle r}
18:Quadrant (circle)
16:(Redirected from
1275:
1208:
1207:
1175:
1169:
1168:
1133:
1127:
1126:
1084:
1075:
1074:
1045:Achatz, Thomas;
1042:
1036:
1035:
1015:
998:Spherical sector
972:Circular segment
961:
957:
953:
949:
947:
946:
941:
939:
931:
901:The length of a
893:
891:
890:
885:
883:
875:
848:
844:
842:
841:
836:
807:
803:
801:
800:
795:
718:
716:
715:
710:
708:
706:
705:
696:
695:
686:
684:
683:
652:
650:
649:
644:
642:
637:
633:
632:
622:
617:
616:
608:
601:
600:
591:
583:
580:
575:
563:
562:
554:
547:
546:
538:
531:
530:
522:
518:
513:
503:
498:
479:
474:
464:
459:
435:
433:
432:
427:
425:
420:
412:
407:
405:
391:
388:
387:
359:
352:by the ratio of
348:
338:
336:
335:
330:
328:
323:
319:
318:
308:
303:
301:
290:
287:
286:
261:
254:
247:
140:
138:
137:
132:
120:
118:
117:
112:
96:
49:, also known as
32:circular section
21:
1283:
1282:
1278:
1277:
1276:
1274:
1273:
1272:
1258:
1257:
1236:Legendre, A. M.
1217:
1212:
1211:
1196:
1177:
1176:
1172:
1157:
1149:. p. 570.
1135:
1134:
1130:
1115:
1086:
1085:
1078:
1063:
1044:
1043:
1039:
1032:
1017:
1016:
1012:
1007:
968:
959:
955:
951:
907:
906:
899:
854:
853:
846:
818:
817:
814:
808:is in radians.
805:
732:
731:
724:
697:
687:
675:
661:
660:
624:
623:
592:
440:
439:
413:
395:
379:
365:
364:
357:
346:
310:
309:
294:
278:
264:
263:
256:
249:
243:
240:
234:
207:wind directions
192:
150:
123:
122:
103:
102:
94:
47:circular sector
35:
28:
23:
22:
15:
12:
11:
5:
1281:
1279:
1271:
1270:
1260:
1259:
1256:
1255:
1244:Charles Davies
1233:
1216:
1213:
1210:
1209:
1194:
1170:
1155:
1128:
1113:
1076:
1062:978-0831130862
1061:
1037:
1031:978-8173358371
1030:
1009:
1008:
1006:
1003:
1002:
1001:
995:
990:
985:
983:Earth quadrant
980:
975:
967:
964:
937:
934:
929:
926:
923:
920:
917:
914:
898:
895:
881:
878:
873:
870:
867:
864:
861:
834:
831:
828:
825:
813:
810:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
723:
720:
704:
700:
694:
690:
682:
678:
674:
671:
668:
640:
636:
631:
627:
620:
614:
611:
605:
599:
595:
589:
586:
579:
574:
570:
566:
560:
557:
551:
544:
541:
535:
528:
525:
517:
512:
508:
502:
497:
493:
489:
486:
483:
478:
473:
469:
463:
458:
454:
450:
447:
423:
419:
416:
410:
404:
401:
398:
394:
386:
382:
378:
375:
372:
326:
322:
317:
313:
306:
300:
297:
293:
285:
281:
277:
274:
271:
233:
230:
205:Traditionally
191:
188:
149:
146:
130:
110:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1280:
1269:
1266:
1265:
1263:
1253:
1249:
1245:
1241:
1237:
1234:
1231:
1227:
1223:
1219:
1218:
1214:
1205:
1201:
1197:
1195:0-7414-2141-0
1191:
1187:
1183:
1182:
1174:
1171:
1166:
1162:
1158:
1152:
1148:
1144:
1143:
1138:
1132:
1129:
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1120:
1116:
1110:
1106:
1102:
1098:
1094:
1090:
1083:
1081:
1077:
1072:
1068:
1064:
1058:
1054:
1053:
1048:
1041:
1038:
1033:
1027:
1023:
1022:
1014:
1011:
1004:
999:
996:
994:
991:
989:
986:
984:
981:
979:
978:Conic section
976:
973:
970:
969:
965:
963:
935:
932:
927:
924:
921:
918:
915:
912:
904:
896:
894:
879:
876:
871:
868:
865:
862:
859:
850:
832:
829:
826:
823:
811:
809:
788:
785:
782:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
729:
721:
719:
702:
698:
692:
688:
680:
676:
672:
669:
666:
658:
653:
638:
634:
629:
625:
618:
609:
603:
597:
593:
587:
584:
577:
572:
568:
564:
555:
549:
539:
533:
523:
515:
510:
506:
500:
495:
491:
487:
484:
481:
476:
471:
467:
461:
456:
452:
448:
445:
436:
421:
417:
414:
408:
402:
399:
396:
392:
384:
380:
376:
373:
370:
362:
355:
351:
344:
339:
324:
320:
315:
311:
304:
298:
295:
291:
283:
279:
275:
272:
269:
260:
253:
246:
239:
231:
229:
227:
223:
218:
216:
212:
208:
201:
196:
189:
187:
185:
181:
177:
173:
169:
165:
161:
157:
156:
147:
145:
142:
128:
108:
100:
99:central angle
92:
88:
84:
80:
76:
72:
71:closed region
68:
64:
60:
56:
52:
51:circle sector
48:
39:
33:
19:
1239:
1221:
1180:
1173:
1141:
1131:
1088:
1051:
1040:
1020:
1013:
900:
897:Chord length
851:
815:
725:
654:
437:
360:
353:
349:
342:
340:
258:
251:
244:
241:
226:compass rose
219:
211:compass rose
204:
184:circular arc
175:
171:
167:
153:
151:
143:
91:major sector
90:
87:minor sector
86:
62:
58:
57:or simply a
54:
50:
46:
44:
1147:Brooks/Cole
1137:Larson, Ron
1099:. pp.
198:An 8-point
174:(60°), and
55:disk sector
1123:1145113954
1005:References
812:Arc length
236:See also:
164:semicircle
1250:, 1858),
1228:, 1874),
1224:(London,
1165:706621772
1093:New Delhi
933:θ
928:
877:θ
869:π
833:θ
783:θ
759:θ
728:perimeter
722:Perimeter
703:∘
693:∘
689:θ
673:π
635:θ
613:~
610:θ
578:θ
569:∫
559:~
556:θ
543:~
527:~
507:∫
501:θ
492:∫
468:∫
462:θ
453:∫
400:π
377:π
321:θ
299:π
292:θ
276:π
215:wind vane
168:quadrants
155:half-disk
61:(symbol:
1262:Category
1204:58869667
1071:56559272
1049:(2005).
966:See also
200:windrose
172:sextants
160:diameter
1268:Circles
1215:Sources
657:degrees
209:on the
190:Compass
176:octants
170:(90°),
97:is the
77:and an
1252:p. 119
1230:p. 285
1202:
1192:
1163:
1153:
1121:
1111:
1069:
1059:
1028:
950:where
845:where
804:where
659:gives
222:octant
162:and a
59:sector
903:chord
148:Types
75:radii
1200:OCLC
1190:ISBN
1161:OCLC
1151:ISBN
1119:OCLC
1109:ISBN
1067:OCLC
1057:ISBN
1026:ISBN
232:Area
83:area
67:disk
1105:227
1101:226
925:sin
880:360
699:360
180:arc
79:arc
69:(a
53:or
1264::
1242:,
1238:,
1198:.
1184:.
1159:.
1117:.
1107:.
1103:,
1095::
1091:.
1079:^
1065:.
363:.
245:πr
228:.
101:,
45:A
1254:.
1232:.
1206:.
1167:.
1125:.
1073:.
1034:.
960:θ
956:R
952:C
936:2
922:R
919:2
916:=
913:C
872:r
866:2
863:=
860:L
847:L
830:r
827:=
824:L
806:θ
792:)
789:2
786:+
780:(
777:r
774:=
771:r
768:2
765:+
762:r
756:=
753:r
750:2
747:+
744:L
741:=
738:P
681:2
677:r
670:=
667:A
639:2
630:2
626:r
619:=
604:d
598:2
594:r
588:2
585:1
573:0
565:=
550:d
540:r
534:d
524:r
516:r
511:0
496:0
488:=
485:S
482:d
477:r
472:0
457:0
449:=
446:A
422:2
418:L
415:r
409:=
403:r
397:2
393:L
385:2
381:r
374:=
371:A
361:r
358:π
354:L
350:r
347:π
343:L
325:2
316:2
312:r
305:=
296:2
284:2
280:r
273:=
270:A
259:π
257:2
252:π
250:2
129:L
109:r
95:θ
63:⌔
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
