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31:
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A practical way to determine the length of an arc in a circle is to plot two lines from the arc's endpoints to the center of the circle, measure the angle where the two lines meet the center, then solve for L by cross-multiplying the statement:
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with the same endpoints as the arc. Its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is
187:
526:
This is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional.
808:
736:
The area of the sector formed by an arc and the center of a circle (bounded by the arc and the two radii drawn to its endpoints) is
1118:, we need to subtract the area of the triangle, determined by the circle's center and the two end points of the arc, from the area
279:
1161:
AP and PB equals the product of the line segments CP and PD. If the arc has a width AB and height CP, then the circle's diameter
1053:
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878:
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Using the conversion described above, we find that the area of the sector for a central angle measured in degrees is
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516:{\displaystyle {\begin{aligned}{\frac {60}{360}}&={\frac {L}{24}}\\360L&=1440\\L&=4.\end{aligned}}}
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For example, if the measure of the angle is 60 degrees and the circumference is 24 inches, then
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The area of the shape bounded by the arc and the straight line between its two end points is
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The arc, chord, and sagitta derive their names respectively from the Latin words for
1281:, and it is divided into two parts by the first chord. The length of one part is the
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1158:
17:
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of a circle. A straight line that connects the two ends of the arc is known as a
58:. If the two points are not directly opposite each other, one of these arcs, the
263:{\displaystyle {\frac {L}{\mathrm {circumference} }}={\frac {\theta }{2\pi }}.}
126:
120:
107:
1594:
1578:
1259:, and it is divided by the bisector into two equal halves, each with length
1589:
1297:. Applying the intersecting chords theorem to these two chords produces
38:
is shaded in green. Its curved boundary of length L is a circular arc.
30:
1572:
91:
radians. The arc of a circle is defined as the part or segment of the
1115:
795:
102:
72:
51:
1289:, and the other part is the remainder of the diameter, with length 2
1515:
1236:
or secant tangent theorem) it is possible to calculate the radius
1149:
420:
84:
29:
858:{\displaystyle {\frac {A}{\pi r^{2}}}={\frac {\theta }{2\pi }}.}
1579:
Math Open
Reference page on Radius of a circular arc or segment
325:{\displaystyle {\frac {L}{2\pi r}}={\frac {\theta }{2\pi }},}
67:
1104:{\displaystyle {\frac {1}{2}}r^{2}(\theta -\sin \theta ).}
105:
of an arc is exactly half of the circle, it is known as a
137:(measured in radians) with the circle center — i.e., the
1362:{\displaystyle H(2r-H)=\left({\frac {W}{2}}\right)^{2},}
918:{\displaystyle {\frac {A}{r^{2}}}={\frac {\theta }{2}}.}
66:
an angle at the center of the circle that is less than
1568:
Table of contents for Math Open
Reference Circle pages
1492:{\displaystyle r={\frac {W^{2}}{8H}}+{\frac {H}{2}}.}
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529:The upper half of a circle can be parameterized as
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1032:{\displaystyle A={\frac {\alpha }{360}}\pi r^{2}.}
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339:being the same angle measured in degrees, since
1218:{\displaystyle CD={\frac {AP\cdot PB}{CP}}+CP}
404:{\displaystyle L={\frac {\alpha \pi r}{180}}.}
977:{\displaystyle A={\frac {1}{2}}r^{2}\theta .}
8:
780:{\displaystyle A={\frac {r^{2}\theta }{2}}.}
1573:Math Open Reference page on circular arcs
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1425:{\displaystyle 2r-H={\frac {W^{2}}{4H}},}
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578:{\displaystyle y={\sqrt {r^{2}-x^{2}}}.}
1277:. The total length of the diameter is 2
7:
129:) of an arc of a circle with radius
715:{\displaystyle L=r{\Big }_{a}^{b}.}
27:Part of a circle between two points
273:Substituting in the circumference
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25:
121:Arc length § Arcs of circles
794:has the same proportion to the
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1310:
1095:
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1:
1240:of a circle given the height
928:By multiplying both sides by
125:The length (more precisely,
1230:intersecting chords theorem
932:, we get the final result:
732:Circular sector § Area
171:{\displaystyle L=\theta r.}
54:between a pair of distinct
1635:
1581:With interactive animation
1575:With interactive animation
729:
118:
79:); and the other arc, the
1504:bow, bowstring, and arrow
588:Then the arc length from
364:, the arc length equals
133:and subtending an angle
1114:To get the area of the
1531:Circular interpolation
1493:
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1363:
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1132:
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371:
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101:of a circle. If the
708:
633:{\displaystyle x=b}
607:{\displaystyle x=a}
18:Circular arc length
1587:Weisstein, Eric W.
1526:Circular-arc graph
1521:Circle of a sphere
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1422:
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802:to a full circle:
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423:in degrees/360° =
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322:
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168:
40:
1484:
1471:
1417:
1344:
1204:
1131:{\displaystyle A}
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16:(Redirected from
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1556:Tangential speed
1536:Lemon (geometry)
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181:This is because
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108:semicircular arc
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1551:Circular motion
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872:on both sides:
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36:circular sector
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1562:External links
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1244:and the width
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868:We can cancel
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1546:Circumference
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1251:Consider the
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1159:line segments
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1142:for details.
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822:
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798:as the angle
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140:central angle
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93:circumference
87:greater than
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1541:Meridian arc
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1371:
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1285:of the arc,
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1043:Segment area
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44:circular arc
43:
41:
1248:of an arc:
1155:The product
1116:arc segment
796:circle area
726:Sector area
419:measure of
1608:Categories
1228:Using the
335:and, with
127:arc length
119:See also:
1595:MathWorld
1389:−
1320:−
1187:⋅
1093:θ
1090:
1084:−
1081:θ
1014:π
1006:α
969:θ
905:θ
847:π
840:θ
819:π
790:The area
766:θ
670:
558:−
387:π
384:α
314:π
307:θ
293:π
252:π
245:θ
160:θ
81:major arc
60:minor arc
1510:See also
64:subtends
1614:Circles
1372:whence
1283:sagitta
1275:
1261:
1157:of the
359:
345:
77:degrees
73:radians
46:is the
1619:Curves
1146:Radius
1138:. See
667:arcsin
115:Length
103:length
56:points
52:circle
1590:"Arc"
1516:Biarc
1253:chord
421:angle
143:— is
98:chord
85:angle
75:(180
50:of a
490:1440
1435:so
1087:sin
1009:360
640:is
614:to
477:360
451:360
394:180
356:180
48:arc
1610::
1592:.
1506:.
507:4.
468:24
448:60
111:.
62:,
42:A
34:A
1598:.
1487:.
1482:2
1479:H
1474:+
1468:H
1465:8
1459:2
1455:W
1449:=
1446:r
1420:,
1414:H
1411:4
1405:2
1401:W
1395:=
1392:H
1386:r
1383:2
1357:,
1352:2
1347:)
1342:2
1339:W
1334:(
1329:=
1326:)
1323:H
1317:r
1314:2
1311:(
1308:H
1295:H
1291:r
1287:H
1279:r
1272:2
1269:/
1265:W
1257:W
1246:W
1242:H
1238:r
1213:P
1210:C
1207:+
1201:P
1198:C
1193:B
1190:P
1184:P
1181:A
1175:=
1172:D
1169:C
1126:A
1099:.
1096:)
1078:(
1073:2
1069:r
1063:2
1060:1
1027:.
1022:2
1018:r
1001:=
998:A
972:.
964:2
960:r
954:2
951:1
946:=
943:A
930:r
913:.
908:2
900:=
893:2
889:r
885:A
870:π
853:.
844:2
835:=
827:2
823:r
815:A
800:θ
792:A
775:.
770:2
761:2
757:r
750:=
747:A
710:.
705:b
700:a
694:]
687:)
682:r
679:x
674:(
662:[
657:r
654:=
651:L
628:b
625:=
622:x
602:a
599:=
596:x
573:.
566:2
562:x
553:2
549:r
543:=
540:y
504:=
497:L
487:=
480:L
465:L
460:=
425:L
399:.
390:r
378:=
375:L
362:π
353:/
349:α
341:θ
337:α
320:,
311:2
302:=
296:r
290:2
286:L
258:.
249:2
240:=
234:e
231:c
228:n
225:e
222:r
219:e
216:f
213:m
210:u
207:c
204:r
201:i
198:c
194:L
166:.
163:r
157:=
154:L
135:θ
131:r
89:π
69:π
20:)
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