90:
1763:
27:
1312:
1346:
1299:
The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Under these circumstances, the circumference of a circle may be defined as the
2424:
2193:
1955:
1859:
1604:
2092:
2302:
2519:
2028:
2275:
1541:
2096:
1780:
of the ellipse that uses only elementary functions. However, there are approximate formulas in terms of these parameters. One such approximation, due to Euler (1773), for the
1981:
1308:
as the number of sides increases without bound. The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.
2221:
1381:
1864:
1652:
1629:
1435:
1340:
1708:
1456:
1506:
2464:
2444:
1480:
1776:
Circumference is used by some authors to denote the perimeter of an ellipse. There is no general formula for the circumference of an ellipse in terms of the
1750:
was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by
1787:
2296:
1552:
1218:
2673:
2587:
2032:
1986:
2705:
2624:
1746:
by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides. This method for approximating
318:
1510:
2648:
284:
2835:
1777:
1211:
1165:
771:
230:
2469:
2233:
20:
2763:
Almkvist, Gert; Berndt, Bruce (1988), "Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses,
1608:
The ratio of the circle's circumference to its radius is called the circle constant, and is equivalent to
2553:
2583:
1186:
796:
1204:
2565: ā Geometric inequality which sets a lower bound on the surface area of a set given its volume
2562:
2550: ā Geometric inequality which sets a lower bound on the surface area of a set given its volume
2547:
173:
2419:{\displaystyle C_{\rm {ellipse}}=4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta ,}
1671:
599:
279:
136:
1403:
675:
386:
264:
149:
2697:
2690:
1960:
1751:
1399:
447:
408:
367:
362:
215:
2200:
1360:
2796:
2224:
1634:
1611:
1417:
1322:
1267:
around any closed figure. Circumference may also refer to the circle itself, that is, the
1115:
1038:
886:
791:
313:
208:
122:
1682:
1441:
8:
2556: ā Radius of a circle or sphere equivalent to a non-circular or non-spherical object
2188:{\displaystyle 4{\sqrt {a^{2}+b^{2}}}\leq C\leq \pi {\sqrt {2\left(a^{2}+b^{2}\right)}}.}
1301:
1120:
1064:
977:
831:
811:
736:
626:
497:
487:
350:
225:
220:
203:
178:
166:
118:
113:
94:
2820:
1488:
2800:
2784:
2745:
2737:
2598:
2449:
2429:
1465:
1079:
806:
646:
274:
198:
188:
159:
144:
2804:
2749:
2701:
2644:
2620:
2289:
2227:
1150:
938:
916:
841:
700:
426:
355:
247:
193:
154:
1140:
1069:
866:
776:
2840:
2776:
2729:
2282:
1268:
1130:
871:
581:
459:
394:
252:
237:
102:
2792:
1305:
1276:
1272:
553:
416:
259:
242:
183:
89:
1125:
1094:
1028:
876:
821:
756:
2594:
2230:
passing through the endpoints of the ellipse's major axis, and the lower bound
1957:
Some lower and upper bounds on the circumference of the canonical ellipse with
1781:
1771:
1181:
1089:
1033:
998:
906:
816:
786:
746:
651:
1950:{\displaystyle C_{\rm {ellipse}}\sim \pi {\sqrt {2\left(a^{2}+b^{2}\right)}}.}
1762:
1155:
766:
2829:
1160:
1145:
1074:
891:
851:
801:
539:
506:
344:
340:
1659:
1411:
1288:
1264:
1260:
1099:
1048:
861:
716:
631:
421:
2663:
2295:
The circumference of an ellipse can be expressed exactly in terms of the
1414:
1354:
1135:
1008:
826:
761:
689:
661:
636:
2720:
Jameson, G.J.O. (2014). "Inequalities for the perimeter of an ellipse".
2788:
2741:
2617:
Using and
Understanding Mathematics / A Quantitative Reasoning Approach
2529:
1676:
1256:
993:
972:
962:
952:
911:
856:
751:
741:
641:
492:
1549:. The above formula can be rearranged to solve for the circumference:
26:
2541:
2278:
1244:
1003:
721:
684:
548:
520:
2780:
2733:
1483:
1316:
1232:
1084:
1043:
1013:
901:
896:
846:
571:
530:
478:
372:
335:
81:
1311:
2285:
1545:
Or, equivalently, as the ratio of the circumference to twice the
1259:
of the circle, as if it were opened up and straightened out to a
1252:
1018:
731:
525:
469:
269:
1854:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,}
1655:
1546:
1395:
1350:
1345:
1248:
967:
957:
836:
781:
656:
619:
607:
562:
515:
433:
98:
1459:
1023:
947:
881:
726:
330:
325:
2667:
2535:
614:
464:
2582:
1666:
is ubiquitous in mathematics, engineering, and science.
2558:
Pages displaying short descriptions of redirect targets
1438:
The first few decimal digits of the numerical value of
1407:
2662:
2544: ā Geometric figure which circumscribes a circle
2472:
2452:
2432:
2305:
2236:
2203:
2099:
2035:
1989:
1963:
1867:
1790:
1685:
1637:
1614:
1555:
1513:
1491:
1468:
1444:
1420:
1363:
1325:
1599:{\displaystyle {C}=\pi \cdot {d}=2\pi \cdot {r}.\!}
1287:is the circumference, or length, of any one of its
2689:
2513:
2458:
2438:
2418:
2269:
2215:
2187:
2086:
2022:
1975:
1949:
1853:
1702:
1646:
1623:
1598:
1535:
1500:
1474:
1450:
1429:
1375:
1334:
1595:
2827:
2696:(2nd ed.), Addison-Wesley Longman, p.
1766:Circle, and ellipses with the same circumference
1458:are 3.141592653589793 ... Pi is defined as the
2762:
2292:at the endpoints of the major and minor axes.
2619:(3rd ed.), Addison-Wesley, p. 580,
2614:
2297:complete elliptic integral of the second kind
1212:
2087:{\displaystyle \pi (a+b)\leq C\leq 4(a+b),}
2692:A History of Mathematics / An Introduction
2615:Bennett, Jeffrey; Briggs, William (2005),
1219:
1205:
88:
2674:On-Line Encyclopedia of Integer Sequences
2538: ā Size of a two-dimensional surface
2446:is the length of the semi-major axis and
2821:Numericana - Circumference of an ellipse
2514:{\displaystyle {\sqrt {1-b^{2}/a^{2}}}.}
2023:{\displaystyle 2\pi b\leq C\leq 2\pi a,}
1761:
1398:is related to one of the most important
1344:
1310:
25:
2719:
1662:. The use of the mathematical constant
1263:. More generally, the perimeter is the
2828:
2643:, W. H. Freeman and Co., p. 565,
2638:
2270:{\displaystyle 4{\sqrt {a^{2}+b^{2}}}}
1386:
319:Straightedge and compass constructions
19:For the circumference of a graph, see
2687:
1536:{\displaystyle \pi ={\frac {C}{d}}.}
1243:, meaning "carrying around") is the
2588:"Perimeter, Area and Circumference"
13:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
14:
2852:
2814:
1754:who used polygons with 10 sides.
285:Noncommutative algebraic geometry
16:Perimeter of a circle or ellipse
1304:of the perimeters of inscribed
2756:
2713:
2681:
2656:
2632:
2608:
2576:
2532: ā Distance along a curve
2078:
2066:
2051:
2039:
1778:semi-major and semi-minor axes
1710:since he did not use the name
678:- / other-dimensional
1:
2769:American Mathematical Monthly
2569:
1772:Ellipse Ā§ Circumference
1462:of a circle's circumference
21:Circumference (graph theory)
7:
2554:Perimeter-equivalent radius
2523:
1319:is 1, its circumference is
1255:. The circumference is the
10:
2857:
2664:Sloane, N. J. A.
2639:Jacobs, Harold R. (1974),
2584:San Diego State University
2223:is the circumference of a
1769:
1757:
18:
2767:, and the Ladies Diary",
1294:
1283:circumference of a sphere
2688:Katz, Victor J. (1998),
2563:Isoperimetric inequality
2548:Isoperimetric inequality
1679:showed that this ratio (
1410:, is represented by the
174:Non-Archimedean geometry
2668:"Sequence A000796"
1976:{\displaystyle a\geq b}
1675:written circa 250 BCE,
1672:Measurement of a Circle
1394:The circumference of a
280:Noncommutative geometry
61: center or origin
2515:
2460:
2440:
2420:
2271:
2217:
2216:{\displaystyle 2\pi a}
2189:
2088:
2024:
1977:
1951:
1855:
1767:
1704:
1654:is also the amount of
1648:
1625:
1600:
1537:
1502:
1476:
1452:
1431:
1400:mathematical constants
1383:
1377:
1376:{\displaystyle 2\pi .}
1357:āits circumference is
1342:
1336:
248:Discrete/Combinatorial
75:
2836:Geometric measurement
2516:
2461:
2441:
2421:
2272:
2218:
2197:Here the upper bound
2190:
2089:
2025:
1978:
1952:
1856:
1765:
1752:Christoph Grienberger
1705:
1649:
1647:{\displaystyle 2\pi }
1626:
1624:{\displaystyle 2\pi }
1601:
1538:
1503:
1477:
1453:
1432:
1430:{\displaystyle \pi .}
1378:
1348:
1337:
1335:{\displaystyle \pi .}
1314:
1271:corresponding to the
231:Discrete differential
29:
2722:Mathematical Gazette
2470:
2466:is the eccentricity
2450:
2430:
2303:
2234:
2201:
2097:
2033:
1987:
1961:
1865:
1788:
1714:) was greater than 3
1703:{\displaystyle C/d,}
1683:
1635:
1612:
1553:
1511:
1489:
1466:
1451:{\displaystyle \pi }
1442:
1418:
1361:
1323:
34: circumference
2367:
498:Pythagorean theorem
2677:. OEIS Foundation.
2604:on 6 October 2014.
2511:
2456:
2436:
2416:
2345:
2299:. More precisely,
2267:
2213:
2185:
2084:
2020:
1973:
1947:
1851:
1768:
1700:
1644:
1621:
1596:
1533:
1501:{\displaystyle d:}
1498:
1472:
1448:
1427:
1387:Relationship with
1384:
1373:
1343:
1332:
76:
2707:978-0-321-01618-8
2626:978-0-321-22773-7
2506:
2459:{\displaystyle e}
2439:{\displaystyle a}
2406:
2402:
2265:
2228:concentric circle
2180:
2128:
1942:
1840:
1813:
1528:
1475:{\displaystyle C}
1229:
1228:
1194:
1193:
917:List of geometers
600:Three-dimensional
589:
588:
2848:
2808:
2807:
2766:
2760:
2754:
2753:
2728:(499): 227ā234.
2717:
2711:
2710:
2695:
2685:
2679:
2678:
2660:
2654:
2653:
2636:
2630:
2629:
2612:
2606:
2605:
2603:
2597:. Archived from
2592:
2580:
2559:
2520:
2518:
2517:
2512:
2507:
2505:
2504:
2495:
2490:
2489:
2474:
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2457:
2445:
2443:
2442:
2437:
2425:
2423:
2422:
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2404:
2403:
2395:
2394:
2385:
2384:
2369:
2366:
2362:
2353:
2335:
2334:
2333:
2276:
2274:
2273:
2268:
2266:
2264:
2263:
2251:
2250:
2241:
2222:
2220:
2219:
2214:
2194:
2192:
2191:
2186:
2181:
2179:
2175:
2174:
2173:
2161:
2160:
2143:
2129:
2127:
2126:
2114:
2113:
2104:
2093:
2091:
2090:
2085:
2029:
2027:
2026:
2021:
1982:
1980:
1979:
1974:
1956:
1954:
1953:
1948:
1943:
1941:
1937:
1936:
1935:
1923:
1922:
1905:
1897:
1896:
1895:
1860:
1858:
1857:
1852:
1841:
1839:
1838:
1829:
1828:
1819:
1814:
1812:
1811:
1802:
1801:
1792:
1749:
1745:
1743:
1742:
1739:
1736:
1729:
1727:
1726:
1723:
1720:
1713:
1709:
1707:
1706:
1701:
1693:
1665:
1653:
1651:
1650:
1645:
1630:
1628:
1627:
1622:
1605:
1603:
1602:
1597:
1591:
1574:
1560:
1542:
1540:
1539:
1534:
1529:
1521:
1507:
1505:
1504:
1499:
1481:
1479:
1478:
1473:
1457:
1455:
1454:
1449:
1436:
1434:
1433:
1428:
1390:
1382:
1380:
1379:
1374:
1349:When a circle's
1341:
1339:
1338:
1333:
1315:When a circle's
1306:regular polygons
1285:
1284:
1221:
1214:
1207:
935:
934:
454:
453:
387:Zero-dimensional
92:
78:
77:
73:
69:
66:Circumference =
60:
51:
42:
33:
2856:
2855:
2851:
2850:
2849:
2847:
2846:
2845:
2826:
2825:
2817:
2812:
2811:
2781:10.2307/2323302
2764:
2761:
2757:
2734:10.2307/3621497
2718:
2714:
2708:
2686:
2682:
2661:
2657:
2651:
2637:
2633:
2627:
2613:
2609:
2601:
2590:
2581:
2577:
2572:
2557:
2526:
2500:
2496:
2491:
2485:
2481:
2473:
2471:
2468:
2467:
2451:
2448:
2447:
2431:
2428:
2427:
2390:
2386:
2380:
2376:
2368:
2358:
2354:
2349:
2311:
2310:
2306:
2304:
2301:
2300:
2259:
2255:
2246:
2242:
2240:
2235:
2232:
2231:
2202:
2199:
2198:
2169:
2165:
2156:
2152:
2151:
2147:
2142:
2122:
2118:
2109:
2105:
2103:
2098:
2095:
2094:
2034:
2031:
2030:
1988:
1985:
1984:
1962:
1959:
1958:
1931:
1927:
1918:
1914:
1913:
1909:
1904:
1873:
1872:
1868:
1866:
1863:
1862:
1834:
1830:
1824:
1820:
1818:
1807:
1803:
1797:
1793:
1791:
1789:
1786:
1785:
1774:
1760:
1747:
1740:
1737:
1734:
1733:
1731:
1730:but less than 3
1724:
1721:
1718:
1717:
1715:
1711:
1689:
1684:
1681:
1680:
1663:
1636:
1633:
1632:
1613:
1610:
1609:
1587:
1570:
1556:
1554:
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1550:
1520:
1512:
1509:
1508:
1490:
1487:
1486:
1467:
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1463:
1443:
1440:
1439:
1419:
1416:
1415:
1392:
1388:
1362:
1359:
1358:
1324:
1321:
1320:
1297:
1282:
1281:
1225:
1196:
1195:
932:
931:
922:
921:
712:
711:
695:
694:
680:
679:
667:
666:
603:
602:
591:
590:
451:
450:
448:Two-dimensional
439:
438:
412:
411:
409:One-dimensional
400:
399:
390:
389:
378:
377:
311:
310:
309:
292:
291:
140:
139:
128:
105:
71:
67:
65:
58:
56:
49:
47:
43: diameter
40:
38:
31:
24:
17:
12:
11:
5:
2854:
2844:
2843:
2838:
2824:
2823:
2816:
2815:External links
2813:
2810:
2809:
2775:(7): 585ā608,
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2712:
2706:
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2649:
2631:
2625:
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2595:Addison-Wesley
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1770:Main article:
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1353:is 1ācalled a
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927:
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923:
920:
919:
914:
909:
904:
899:
894:
889:
884:
879:
874:
869:
864:
859:
854:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
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739:
734:
729:
724:
719:
713:
709:
708:
707:
704:
703:
697:
696:
693:
692:
687:
681:
674:
673:
672:
669:
668:
665:
664:
659:
654:
652:Platonic Solid
649:
644:
639:
634:
629:
624:
623:
622:
611:
610:
604:
598:
597:
596:
593:
592:
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584:
579:
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559:
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543:
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398:
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385:
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370:
365:
359:
358:
353:
348:
338:
333:
328:
322:
321:
312:
308:
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300:
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196:
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147:
141:
135:
134:
133:
130:
129:
127:
126:
116:
110:
107:
106:
93:
85:
84:
70:Ć diameter = 2
57:
48:
39:
30:
15:
9:
6:
4:
3:
2:
2853:
2842:
2839:
2837:
2834:
2833:
2831:
2822:
2819:
2818:
2806:
2802:
2798:
2794:
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2786:
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2759:
2751:
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2731:
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2723:
2716:
2709:
2703:
2699:
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2693:
2684:
2676:
2675:
2669:
2665:
2659:
2652:
2650:0-7167-0456-0
2646:
2642:
2635:
2628:
2622:
2618:
2611:
2600:
2596:
2589:
2585:
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2497:
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2478:
2475:
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2413:
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2407:
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2363:
2359:
2355:
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2307:
2298:
2293:
2291:
2287:
2284:
2280:
2260:
2256:
2252:
2247:
2243:
2237:
2229:
2226:
2225:circumscribed
2210:
2207:
2204:
2195:
2182:
2176:
2170:
2166:
2162:
2157:
2153:
2148:
2144:
2139:
2136:
2133:
2130:
2123:
2119:
2115:
2110:
2106:
2100:
2081:
2075:
2072:
2069:
2063:
2060:
2057:
2054:
2048:
2045:
2042:
2036:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1970:
1967:
1964:
1944:
1938:
1932:
1928:
1924:
1919:
1915:
1910:
1906:
1901:
1898:
1869:
1848:
1845:
1842:
1835:
1831:
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1821:
1815:
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1783:
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1421:
1413:
1409:
1405:
1401:
1397:
1370:
1367:
1364:
1356:
1352:
1347:
1329:
1326:
1318:
1313:
1309:
1307:
1303:
1292:
1290:
1289:great circles
1280:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1237:circumference
1234:
1222:
1217:
1215:
1210:
1208:
1203:
1202:
1200:
1199:
1188:
1185:
1183:
1180:
1179:
1178:
1177:
1173:
1172:
1167:
1164:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1134:
1132:
1129:
1127:
1124:
1122:
1119:
1117:
1114:
1113:
1112:
1111:
1107:
1106:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
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1063:
1062:
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1060:
1056:
1055:
1050:
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1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
991:
990:
989:
985:
984:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
945:
944:
943:
940:
937:
936:
926:
925:
918:
915:
913:
910:
908:
905:
903:
900:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
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773:
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768:
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763:
760:
758:
755:
753:
750:
748:
745:
743:
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738:
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733:
730:
728:
725:
723:
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718:
715:
714:
706:
705:
702:
699:
698:
691:
688:
686:
683:
682:
677:
671:
670:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
621:
618:
617:
616:
613:
612:
609:
606:
605:
601:
595:
594:
583:
580:
578:
577:Circumference
575:
573:
570:
569:
568:
567:
564:
561:
560:
555:
552:
550:
547:
546:
545:
544:
541:
540:Quadrilateral
538:
537:
532:
529:
527:
524:
522:
519:
517:
514:
513:
512:
511:
508:
507:Parallelogram
505:
504:
499:
496:
494:
491:
489:
486:
485:
484:
483:
480:
477:
476:
471:
468:
466:
463:
461:
458:
457:
456:
455:
449:
443:
442:
435:
432:
428:
425:
423:
420:
419:
418:
415:
414:
410:
404:
403:
396:
393:
392:
388:
382:
381:
374:
371:
369:
366:
364:
361:
360:
357:
354:
352:
349:
346:
345:Perpendicular
342:
341:Orthogonality
339:
337:
334:
332:
329:
327:
324:
323:
320:
317:
316:
315:
305:
302:
301:
296:
295:
286:
283:
282:
281:
278:
276:
273:
271:
268:
266:
265:Computational
263:
261:
258:
254:
251:
250:
249:
246:
244:
241:
239:
236:
232:
229:
227:
224:
222:
219:
218:
217:
214:
210:
207:
205:
202:
201:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
168:
165:
161:
158:
157:
156:
153:
152:
151:
150:Non-Euclidean
148:
146:
143:
142:
138:
132:
131:
124:
120:
117:
115:
112:
111:
109:
108:
104:
100:
96:
91:
87:
86:
83:
80:
79:
64:
55:
52: radius
46:
37:
28:
22:
2772:
2768:
2758:
2725:
2721:
2715:
2691:
2683:
2671:
2658:
2640:
2634:
2616:
2610:
2599:the original
2578:
2294:
2196:
1775:
1670:
1668:
1631:. The value
1607:
1544:
1412:Greek letter
1393:
1298:
1265:curve length
1261:line segment
1241:circumferens
1240:
1239:(from Latin
1236:
1230:
1049:Parameshvara
862:Parameshvara
632:Dodecahedron
576:
216:Differential
62:
53:
44:
35:
1355:unit circle
1174:Present day
1121:Lobachevsky
1108:1700sā1900s
1065:Jyeį¹£į¹hadeva
1057:1400sā1700s
1009:Brahmagupta
832:Lobachevsky
812:Jyeį¹£į¹hadeva
762:Brahmagupta
690:Hypersphere
662:Tetrahedron
637:Icosahedron
209:Diophantine
2830:Categories
2570:References
2530:Arc length
1784:ellipse,
1677:Archimedes
1257:arc length
1034:al-Yasamin
978:Apollonius
973:Archimedes
963:Pythagoras
953:Baudhayana
907:al-Yasamin
857:Pythagoras
752:Baudhayana
742:Archimedes
737:Apollonius
642:Octahedron
493:Hypotenuse
368:Similarity
363:Congruence
275:Incidence
226:Symplectic
221:Riemannian
204:Arithmetic
179:Projective
167:Hyperbolic
95:Projecting
2805:119810884
2750:126427943
2542:Circumgon
2479:−
2411:θ
2400:θ
2397:
2374:−
2356:π
2347:∫
2283:inscribed
2279:perimeter
2208:π
2140:π
2137:≤
2131:≤
2061:≤
2055:≤
2037:π
2012:π
2006:≤
2000:≤
1994:π
1968:≥
1902:π
1899:∼
1782:canonical
1642:π
1619:π
1585:⋅
1582:π
1568:⋅
1565:π
1515:π
1446:π
1422:π
1368:π
1327:π
1245:perimeter
1151:Minkowski
1070:Descartes
1004:Aryabhata
999:KÄtyÄyana
930:by period
842:Minkowski
817:KÄtyÄyana
777:Descartes
722:Aryabhata
701:Geometers
685:Tesseract
549:Trapezoid
521:Rectangle
314:Dimension
199:Algebraic
189:Synthetic
160:Spherical
145:Euclidean
74:Ć radius.
2641:Geometry
2586:(2004).
2524:See also
2290:vertices
1484:diameter
1404:constant
1317:diameter
1233:geometry
1141:PoincarƩ
1085:Minggatu
1044:Yang Hui
1014:Virasena
902:Yang Hui
897:Virasena
867:PoincarƩ
847:Minggatu
627:Cylinder
572:Diameter
531:Rhomboid
488:Altitude
479:Triangle
373:Symmetry
351:Parallel
336:Diagonal
306:Features
303:Concepts
194:Analytic
155:Elliptic
137:Branches
123:Timeline
82:Geometry
2841:Circles
2797:0966232
2789:2323302
2742:3621497
2666:(ed.).
2286:rhombus
2277:is the
1758:Ellipse
1744:
1732:
1728:
1716:
1658:in one
1656:radians
1482:to its
1402:. This
1279:. The
1253:ellipse
1166:Coxeter
1146:Hilbert
1131:Riemann
1080:Huygens
1039:al-Tusi
1029:KhayyƔm
1019:Alhazen
986:1ā1400s
887:al-Tusi
872:Riemann
822:KhayyƔm
807:Huygens
802:Hilbert
772:Coxeter
732:Alhazen
710:by name
647:Pyramid
526:Rhombus
470:Polygon
422:segment
270:Fractal
253:Digital
238:Complex
119:History
114:Outline
2803:
2795:
2787:
2748:
2740:
2704:
2647:
2623:
2426:where
2405:
2281:of an
1547:radius
1396:circle
1351:radius
1295:Circle
1249:circle
1235:, the
1187:Gromov
1182:Atiyah
1161:Veblen
1156:Cartan
1126:Bolyai
1095:Sakabe
1075:Pascal
968:Euclid
958:Manava
892:Veblen
877:Sakabe
852:Pascal
837:Manava
797:Gromov
782:Euclid
767:Cartan
757:Bolyai
747:Atiyah
657:Sphere
620:cuboid
608:Volume
563:Circle
516:Square
434:Length
356:Vertex
260:Convex
243:Finite
184:Affine
99:sphere
59:
50:
41:
32:
2801:S2CID
2785:JSTOR
2746:S2CID
2738:JSTOR
2602:(PDF)
2591:(PDF)
2288:with
1983:are:
1460:ratio
1302:limit
1275:of a
1269:locus
1247:of a
1136:Klein
1116:Gauss
1090:Euler
1024:Sijzi
994:Zhang
948:Ahmes
912:Zhang
882:Sijzi
827:Klein
792:Gauss
787:Euler
727:Ahmes
460:Plane
395:Point
331:Curve
326:Angle
103:plane
101:to a
2702:ISBN
2672:The
2645:ISBN
2621:ISBN
2536:Area
1861:is
1660:turn
1277:disk
1273:edge
1100:Aida
717:Aida
676:Four
615:Cube
582:Area
554:Kite
465:Area
417:Line
2777:doi
2730:doi
2698:109
2388:sin
1669:In
1251:or
1231:In
939:BCE
427:ray
2832::
2799:,
2793:MR
2791:,
2783:,
2773:95
2771:,
2744:.
2736:.
2726:98
2724:.
2700:,
2670:.
2593:.
1725:71
1719:10
1408:pi
1406:,
1291:.
97:a
2779::
2765:Ļ
2752:.
2732::
2509:.
2502:2
2498:a
2493:/
2487:2
2483:b
2476:1
2454:e
2434:a
2414:,
2408:d
2392:2
2382:2
2378:e
2371:1
2364:2
2360:/
2351:0
2343:a
2340:4
2337:=
2331:e
2328:s
2325:p
2322:i
2319:l
2316:l
2313:e
2308:C
2261:2
2257:b
2253:+
2248:2
2244:a
2238:4
2211:a
2205:2
2183:.
2177:)
2171:2
2167:b
2163:+
2158:2
2154:a
2149:(
2145:2
2134:C
2124:2
2120:b
2116:+
2111:2
2107:a
2101:4
2082:,
2079:)
2076:b
2073:+
2070:a
2067:(
2064:4
2058:C
2052:)
2049:b
2046:+
2043:a
2040:(
2018:,
2015:a
2009:2
2003:C
1997:b
1991:2
1971:b
1965:a
1945:.
1939:)
1933:2
1929:b
1925:+
1920:2
1916:a
1911:(
1907:2
1893:e
1890:s
1887:p
1884:i
1881:l
1878:l
1875:e
1870:C
1849:,
1846:1
1843:=
1836:2
1832:b
1826:2
1822:y
1816:+
1809:2
1805:a
1799:2
1795:x
1748:Ļ
1741:7
1738:/
1735:1
1722:/
1712:Ļ
1698:,
1695:d
1691:/
1687:C
1664:Ļ
1639:2
1616:2
1593:.
1589:r
1579:2
1576:=
1572:d
1562:=
1558:C
1531:.
1526:d
1523:C
1518:=
1496::
1493:d
1470:C
1425:.
1389:Ļ
1371:.
1365:2
1330:.
1220:e
1213:t
1206:v
347:)
343:(
125:)
121:(
72:Ļ
68:Ļ
63:O
54:R
45:D
36:C
23:.
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