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den ",[568,944,945],{},"Umfang"," einer Ellipse gibt es dagegen keine geschlossene Formel aus elementaren Funktionen (er führt auf ein sogenanntes elliptisches Integral). Man rechnet mit Näherungsformeln.",[559,948,950],{"id":949},"herleitung","Herleitung",[564,952,953,954,1316,1317,1345,1346,1375,1376,1476],{},"In Koordinaten wird eine Ellipse mit Mittelpunkt im Ursprung durch ",[577,955,957,1008],{"className":956},[580],[577,958,960],{"className":959},[584],[586,961,962],{"xmlns":588},[590,963,964,1005],{},[593,965,966,982,985,1000,1002],{},[967,968,969,976],"mfrac",{},[676,970,971,974],{},[596,972,973],{},"x",[682,975,684],{},[676,977,978,980],{},[596,979,598],{},[682,981,684],{},[759,983,984],{},"+",[967,986,987,994],{},[676,988,989,992],{},[596,990,991],{},"y",[682,993,684],{},[676,995,996,998],{},[596,997,640],{},[682,999,684],{},[759,1001,761],{},[682,1003,1004],{},"1",[600,1006,1007],{"encoding":602},"\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1",[577,1009,1011,1168,1306],{"className":1010,"ariaHidden":607},[606],[577,1012,1014,1018,1159,1162,1165],{"className":1013},[611],[577,1015],{"className":1016,"style":1017},[615],"height:1.3629em;vertical-align:-0.345em;",[577,1019,1021,1026,1155],{"className":1020},[620],[577,1022],{"className":1023},[1024,1025],"mopen","nulldelimiter",[577,1027,1029],{"className":1028},[967],[577,1030,1033,1146],{"className":1031},[716,1032],"vlist-t2",[577,1034,1036,1141],{"className":1035},[720],[577,1037,1040,1087,1098],{"className":1038,"style":1039},[724],"height:1.0179em;",[577,1041,1043,1047],{"style":1042},"top:-2.655em;",[577,1044],{"className":1045,"style":1046},[731],"height:3em;",[577,1048,1050],{"className":1049},[736,737,738,739],[577,1051,1053],{"className":1052},[620,739],[577,1054,1056,1059],{"className":1055},[620,739],[577,1057,598],{"className":1058},[620,621,739],[577,1060,1062],{"className":1061},[712],[577,1063,1065],{"className":1064},[716],[577,1066,1068],{"className":1067},[720],[577,1069,1072],{"className":1070,"style":1071},[724],"height:0.7463em;",[577,1073,1075,1079],{"style":1074},"top:-2.786em;margin-right:0.0714em;",[577,1076],{"className":1077,"style":1078},[731],"height:2.5em;",[577,1080,1084],{"className":1081},[736,1082,1083,739],"reset-size3","size1",[577,1085,684],{"className":1086},[620,739],[577,1088,1090,1093],{"style":1089},"top:-3.23em;",[577,1091],{"className":1092,"style":1046},[731],[577,1094],{"className":1095,"style":1097},[1096],"frac-line","border-bottom-width:0.04em;",[577,1099,1101,1104],{"style":1100},"top:-3.394em;",[577,1102],{"className":1103,"style":1046},[731],[577,1105,1107],{"className":1106},[736,737,738,739],[577,1108,1110],{"className":1109},[620,739],[577,1111,1113,1116],{"className":1112},[620,739],[577,1114,973],{"className":1115},[620,621,739],[577,1117,1119],{"className":1118},[712],[577,1120,1122],{"className":1121},[716],[577,1123,1125],{"className":1124},[720],[577,1126,1129],{"className":1127,"style":1128},[724],"height:0.8913em;",[577,1130,1132,1135],{"style":1131},"top:-2.931em;margin-right:0.0714em;",[577,1133],{"className":1134,"style":1078},[731],[577,1136,1138],{"className":1137},[736,1082,1083,739],[577,1139,684],{"className":1140},[620,739],[577,1142,1145],{"className":1143},[1144],"vlist-s","​",[577,1147,1149],{"className":1148},[720],[577,1150,1153],{"className":1151,"style":1152},[724],"height:0.345em;",[577,1154],{},[577,1156],{"className":1157},[1158,1025],"mclose",[577,1160],{"className":1161,"style":876},[786],[577,1163,984],{"className":1164},[880],[577,1166],{"className":1167,"style":876},[786],[577,1169,1171,1175,1297,1300,1303],{"className":1170},[611],[577,1172],{"className":1173,"style":1174},[615],"height:1.415em;vertical-align:-0.345em;",[577,1176,1178,1181,1294],{"className":1177},[620],[577,1179],{"className":1180},[1024,1025],[577,1182,1184],{"className":1183},[967],[577,1185,1187,1286],{"className":1186},[716,1032],[577,1188,1190,1283],{"className":1189},[720],[577,1191,1194,1234,1242],{"className":1192,"style":1193},[724],"height:1.07em;",[577,1195,1196,1199],{"style":1042},[577,1197],{"className":1198,"style":1046},[731],[577,1200,1202],{"className":1201},[736,737,738,739],[577,1203,1205],{"className":1204},[620,739],[577,1206,1208,1211],{"className":1207},[620,739],[577,1209,640],{"className":1210},[620,621,739],[577,1212,1214],{"className":1213},[712],[577,1215,1217],{"className":1216},[716],[577,1218,1220],{"className":1219},[720],[577,1221,1223],{"className":1222,"style":1071},[724],[577,1224,1225,1228],{"style":1074},[577,1226],{"className":1227,"style":1078},[731],[577,1229,1231],{"className":1230},[736,1082,1083,739],[577,1232,684],{"className":1233},[620,739],[577,1235,1236,1239],{"style":1089},[577,1237],{"className":1238,"style":1046},[731],[577,1240],{"className":1241,"style":1097},[1096],[577,1243,1245,1248],{"style":1244},"top:-3.4461em;",[577,1246],{"className":1247,"style":1046},[731],[577,1249,1251],{"className":1250},[736,737,738,739],[577,1252,1254],{"className":1253},[620,739],[577,1255,1257,1260],{"className":1256},[620,739],[577,1258,991],{"className":1259,"style":701},[620,621,739],[577,1261,1263],{"className":1262},[712],[577,1264,1266],{"className":1265},[716],[577,1267,1269],{"className":1268},[720],[577,1270,1272],{"className":1271,"style":1128},[724],[577,1273,1274,1277],{"style":1131},[577,1275],{"className":1276,"style":1078},[731],[577,1278,1280],{"className":1279},[736,1082,1083,739],[577,1281,684],{"className":1282},[620,739],[577,1284,1145],{"className":1285},[1144],[577,1287,1289],{"className":1288},[720],[577,1290,1292],{"className":1291,"style":1152},[724],[577,1293],{},[577,1295],{"className":1296},[1158,1025],[577,1298],{"className":1299,"style":787},[786],[577,1301,761],{"className":1302},[791],[577,1304],{"className":1305,"style":787},[786],[577,1307,1309,1313],{"className":1308},[611],[577,1310],{"className":1311,"style":1312},[615],"height:0.6444em;",[577,1314,1004],{"className":1315},[620]," beschrieben. Man erhält sie aus dem Kreis mit Radius ",[577,1318,1320,1333],{"className":1319},[580],[577,1321,1323],{"className":1322},[584],[586,1324,1325],{"xmlns":588},[590,1326,1327,1331],{},[593,1328,1329],{},[596,1330,598],{},[600,1332,598],{"encoding":602},[577,1334,1336],{"className":1335,"ariaHidden":607},[606],[577,1337,1339,1342],{"className":1338},[611],[577,1340],{"className":1341,"style":616},[615],[577,1343,598],{"className":1344},[620,621],", indem man alle ",[577,1347,1349,1362],{"className":1348},[580],[577,1350,1352],{"className":1351},[584],[586,1353,1354],{"xmlns":588},[590,1355,1356,1360],{},[593,1357,1358],{},[596,1359,991],{},[600,1361,991],{"encoding":602},[577,1363,1365],{"className":1364,"ariaHidden":607},[606],[577,1366,1368,1372],{"className":1367},[611],[577,1369],{"className":1370,"style":1371},[615],"height:0.625em;vertical-align:-0.1944em;",[577,1373,991],{"className":1374,"style":701},[620,621],"-Werte mit dem Faktor ",[577,1377,1379,1397],{"className":1378},[580],[577,1380,1382],{"className":1381},[584],[586,1383,1384],{"xmlns":588},[590,1385,1386,1394],{},[593,1387,1388],{},[967,1389,1390,1392],{},[596,1391,640],{},[596,1393,598],{},[600,1395,1396],{"encoding":602},"\\frac{b}{a}",[577,1398,1400],{"className":1399,"ariaHidden":607},[606],[577,1401,1403,1407],{"className":1402},[611],[577,1404],{"className":1405,"style":1406},[615],"height:1.2251em;vertical-align:-0.345em;",[577,1408,1410,1413,1473],{"className":1409},[620],[577,1411],{"className":1412},[1024,1025],[577,1414,1416],{"className":1415},[967],[577,1417,1419,1465],{"className":1418},[716,1032],[577,1420,1422,1462],{"className":1421},[720],[577,1423,1426,1440,1448],{"className":1424,"style":1425},[724],"height:0.8801em;",[577,1427,1428,1431],{"style":1042},[577,1429],{"className":1430,"style":1046},[731],[577,1432,1434],{"className":1433},[736,737,738,739],[577,1435,1437],{"className":1436},[620,739],[577,1438,598],{"className":1439},[620,621,739],[577,1441,1442,1445],{"style":1089},[577,1443],{"className":1444,"style":1046},[731],[577,1446],{"className":1447,"style":1097},[1096],[577,1449,1450,1453],{"style":1100},[577,1451],{"className":1452,"style":1046},[731],[577,1454,1456],{"className":1455},[736,737,738,739],[577,1457,1459],{"className":1458},[620,739],[577,1460,640],{"className":1461},[620,621,739],[577,1463,1145],{"className":1464},[1144],[577,1466,1468],{"className":1467},[720],[577,1469,1471],{"className":1470,"style":1152},[724],[577,1472],{},[577,1474],{"className":1475},[1158,1025]," multipliziert. Dabei ändert sich jede Fläche um denselben Faktor:",[577,1478,1481],{"className":1479},[1480],"katex-display",[577,1482,1484,1526],{"className":1483},[580],[577,1485,1487],{"className":1486},[584],[586,1488,1490],{"xmlns":588,"display":1489},"block",[590,1491,1492,1523],{},[593,1493,1494,1497,1499,1501,1507,1509,1515,1517,1519,1521],{},[596,1495,1496],{},"A",[759,1498,761],{},[596,1500,674],{},[676,1502,1503,1505],{},[596,1504,598],{},[682,1506,684],{},[759,1508,841],{},[967,1510,1511,1513],{},[596,1512,640],{},[596,1514,598],{},[759,1516,761],{},[596,1518,674],{},[596,1520,598],{},[596,1522,640],{},[600,1524,1525],{"encoding":602},"A = \\pi a^2 \\cdot \\frac{b}{a} = \\pi a b",[577,1527,1529,1548,1597,1679],{"className":1528,"ariaHidden":607},[606],[577,1530,1532,1536,1539,1542,1545],{"className":1531},[611],[577,1533],{"className":1534,"style":1535},[615],"height:0.6833em;",[577,1537,1496],{"className":1538},[620,621],[577,1540],{"className":1541,"style":787},[786],[577,1543,761],{"className":1544},[791],[577,1546],{"className":1547,"style":787},[786],[577,1549,1551,1555,1558,1588,1591,1594],{"className":1550},[611],[577,1552],{"className":1553,"style":1554},[615],"height:0.8641em;",[577,1556,674],{"className":1557,"style":701},[620,621],[577,1559,1561,1564],{"className":1560},[620],[577,1562,598],{"className":1563},[620,621],[577,1565,1567],{"className":1566},[712],[577,1568,1570],{"className":1569},[716],[577,1571,1573],{"className":1572},[720],[577,1574,1576],{"className":1575,"style":1554},[724],[577,1577,1579,1582],{"style":1578},"top:-3.113em;margin-right:0.05em;",[577,1580],{"className":1581,"style":732},[731],[577,1583,1585],{"className":1584},[736,737,738,739],[577,1586,684],{"className":1587},[620,739],[577,1589],{"className":1590,"style":876},[786],[577,1592,841],{"className":1593},[880],[577,1595],{"className":1596,"style":876},[786],[577,1598,1600,1604,1670,1673,1676],{"className":1599},[611],[577,1601],{"className":1602,"style":1603},[615],"height:2.0574em;vertical-align:-0.686em;",[577,1605,1607,1610,1667],{"className":1606},[620],[577,1608],{"className":1609},[1024,1025],[577,1611,1613],{"className":1612},[967],[577,1614,1616,1658],{"className":1615},[716,1032],[577,1617,1619,1655],{"className":1618},[720],[577,1620,1623,1635,1643],{"className":1621,"style":1622},[724],"height:1.3714em;",[577,1624,1626,1629],{"style":1625},"top:-2.314em;",[577,1627],{"className":1628,"style":1046},[731],[577,1630,1632],{"className":1631},[620],[577,1633,598],{"className":1634},[620,621],[577,1636,1637,1640],{"style":1089},[577,1638],{"className":1639,"style":1046},[731],[577,1641],{"className":1642,"style":1097},[1096],[577,1644,1646,1649],{"style":1645},"top:-3.677em;",[577,1647],{"className":1648,"style":1046},[731],[577,1650,1652],{"className":1651},[620],[577,1653,640],{"className":1654},[620,621],[577,1656,1145],{"className":1657},[1144],[577,1659,1661],{"className":1660},[720],[577,1662,1665],{"className":1663,"style":1664},[724],"height:0.686em;",[577,1666],{},[577,1668],{"className":1669},[1158,1025],[577,1671],{"className":1672,"style":787},[786],[577,1674,761],{"className":1675},[791],[577,1677],{"className":1678,"style":787},[786],[577,1680,1682,1685,1688],{"className":1681},[611],[577,1683],{"className":1684,"style":652},[615],[577,1686,674],{"className":1687,"style":701},[620,621],[577,1689,1691],{"className":1690},[620,621],"ab",[559,1693,1695],{"id":1694},"beispiele","Beispiele",[564,1697,1698,1699,1769,1770,1835],{},"Eine Ellipse hat die Halbachsen ",[577,1700,1702,1728],{"className":1701},[580],[577,1703,1705],{"className":1704},[584],[586,1706,1707],{"xmlns":588},[590,1708,1709,1725],{},[593,1710,1711,1713,1715,1718,1722],{},[596,1712,598],{},[759,1714,761],{},[682,1716,1717],{},"5",[1719,1720,1721],"mtext",{}," ",[1719,1723,1724],{},"cm",[600,1726,1727],{"encoding":602},"a = 5\\,\\text{cm}",[577,1729,1731,1749],{"className":1730,"ariaHidden":607},[606],[577,1732,1734,1737,1740,1743,1746],{"className":1733},[611],[577,1735],{"className":1736,"style":616},[615],[577,1738,598],{"className":1739},[620,621],[577,1741],{"className":1742,"style":787},[786],[577,1744,761],{"className":1745},[791],[577,1747],{"className":1748,"style":787},[786],[577,1750,1752,1755,1758,1762],{"className":1751},[611],[577,1753],{"className":1754,"style":1312},[615],[577,1756,1717],{"className":1757},[620],[577,1759],{"className":1760,"style":1761},[786],"margin-right:0.1667em;",[577,1763,1766],{"className":1764},[620,1765],"text",[577,1767,1724],{"className":1768},[620]," und ",[577,1771,1773,1796],{"className":1772},[580],[577,1774,1776],{"className":1775},[584],[586,1777,1778],{"xmlns":588},[590,1779,1780,1793],{},[593,1781,1782,1784,1786,1789,1791],{},[596,1783,640],{},[759,1785,761],{},[682,1787,1788],{},"3",[1719,1790,1721],{},[1719,1792,1724],{},[600,1794,1795],{"encoding":602},"b = 3\\,\\text{cm}",[577,1797,1799,1817],{"className":1798,"ariaHidden":607},[606],[577,1800,1802,1805,1808,1811,1814],{"className":1801},[611],[577,1803],{"className":1804,"style":652},[615],[577,1806,640],{"className":1807},[620,621],[577,1809],{"className":1810,"style":787},[786],[577,1812,761],{"className":1813},[791],[577,1815],{"className":1816,"style":787},[786],[577,1818,1820,1823,1826,1829],{"className":1819},[611],[577,1821],{"className":1822,"style":1312},[615],[577,1824,1788],{"className":1825},[620],[577,1827],{"className":1828,"style":1761},[786],[577,1830,1832],{"className":1831},[620,1765],[577,1833,1724],{"className":1834},[620],":",[577,1837,1839],{"className":1838},[1480],[577,1840,1842,1905],{"className":1841},[580],[577,1843,1845],{"className":1844},[584],[586,1846,1847],{"xmlns":588,"display":1489},[590,1848,1849,1902],{},[593,1850,1851,1853,1855,1857,1859,1861,1863,1865,1867,1873,1875,1878,1880,1882,1888,1891,1894,1896],{},[596,1852,1496],{},[759,1854,761],{},[596,1856,674],{},[759,1858,841],{},[682,1860,1717],{},[759,1862,841],{},[682,1864,1788],{},[1719,1866,1721],{},[676,1868,1869,1871],{},[1719,1870,1724],{},[682,1872,684],{},[759,1874,761],{},[682,1876,1877],{},"15",[596,1879,674],{},[1719,1881,1721],{},[676,1883,1884,1886],{},[1719,1885,1724],{},[682,1887,684],{},[759,1889,1890],{},"≈",[682,1892,1893],{},"47,12",[1719,1895,1721],{},[676,1897,1898,1900],{},[1719,1899,1724],{},[682,1901,684],{},[600,1903,1904],{"encoding":602},"A = \\pi \\cdot 5 \\cdot 3\\,\\text{cm}^2 = 15\\pi\\,\\text{cm}^2 \\approx 47{,}12\\,\\text{cm}^2",[577,1906,1908,1926,1944,1962,2015,2071],{"className":1907,"ariaHidden":607},[606],[577,1909,1911,1914,1917,1920,1923],{"className":1910},[611],[577,1912],{"className":1913,"style":1535},[615],[577,1915,1496],{"className":1916},[620,621],[577,1918],{"className":1919,"style":787},[786],[577,1921,761],{"className":1922},[791],[577,1924],{"className":1925,"style":787},[786],[577,1927,1929,1932,1935,1938,1941],{"className":1928},[611],[577,1930],{"className":1931,"style":866},[615],[577,1933,674],{"className":1934,"style":701},[620,621],[577,1936],{"className":1937,"style":876},[786],[577,1939,841],{"className":1940},[880],[577,1942],{"className":1943,"style":876},[786],[577,1945,1947,1950,1953,1956,1959],{"className":1946},[611],[577,1948],{"className":1949,"style":1312},[615],[577,1951,1717],{"className":1952},[620],[577,1954],{"className":1955,"style":876},[786],[577,1957,841],{"className":1958},[880],[577,1960],{"className":1961,"style":876},[786],[577,1963,1965,1968,1971,1974,2006,2009,2012],{"className":1964},[611],[577,1966],{"className":1967,"style":1554},[615],[577,1969,1788],{"className":1970},[620],[577,1972],{"className":1973,"style":1761},[786],[577,1975,1977,1983],{"className":1976},[620],[577,1978,1980],{"className":1979},[620,1765],[577,1981,1724],{"className":1982},[620],[577,1984,1986],{"className":1985},[712],[577,1987,1989],{"className":1988},[716],[577,1990,1992],{"className":1991},[720],[577,1993,1995],{"className":1994,"style":1554},[724],[577,1996,1997,2000],{"style":1578},[577,1998],{"className":1999,"style":732},[731],[577,2001,2003],{"className":2002},[736,737,738,739],[577,2004,684],{"className":2005},[620,739],[577,2007],{"className":2008,"style":787},[786],[577,2010,761],{"className":2011},[791],[577,2013],{"className":2014,"style":787},[786],[577,2016,2018,2021,2024,2027,2030,2062,2065,2068],{"className":2017},[611],[577,2019],{"className":2020,"style":1554},[615],[577,2022,1877],{"className":2023},[620],[577,2025,674],{"className":2026,"style":701},[620,621],[577,2028],{"className":2029,"style":1761},[786],[577,2031,2033,2039],{"className":2032},[620],[577,2034,2036],{"className":2035},[620,1765],[577,2037,1724],{"className":2038},[620],[577,2040,2042],{"className":2041},[712],[577,2043,2045],{"className":2044},[716],[577,2046,2048],{"className":2047},[720],[577,2049,2051],{"className":2050,"style":1554},[724],[577,2052,2053,2056],{"style":1578},[577,2054],{"className":2055,"style":732},[731],[577,2057,2059],{"className":2058},[736,737,738,739],[577,2060,684],{"className":2061},[620,739],[577,2063],{"className":2064,"style":787},[786],[577,2066,1890],{"className":2067},[791],[577,2069],{"className":2070,"style":787},[786],[577,2072,2074,2078,2082,2090,2094,2097],{"className":2073},[611],[577,2075],{"className":2076,"style":2077},[615],"height:1.0585em;vertical-align:-0.1944em;",[577,2079,2081],{"className":2080},[620],"47",[577,2083,2085],{"className":2084},[620],[577,2086,2089],{"className":2087},[2088],"mpunct",",",[577,2091,2093],{"className":2092},[620],"12",[577,2095],{"className":2096,"style":1761},[786],[577,2098,2100,2106],{"className":2099},[620],[577,2101,2103],{"className":2102},[620,1765],[577,2104,1724],{"className":2105},[620],[577,2107,2109],{"className":2108},[712],[577,2110,2112],{"className":2111},[716],[577,2113,2115],{"className":2114},[720],[577,2116,2118],{"className":2117,"style":1554},[724],[577,2119,2120,2123],{"style":1578},[577,2121],{"className":2122,"style":732},[731],[577,2124,2126],{"className":2125},[736,737,738,739],[577,2127,684],{"className":2128},[620,739],[564,2130,2131,2132,2187,2188,2242,2243,1769,2296,1835],{},"Ein ovaler Tisch ist ",[577,2133,2135,2155],{"className":2134},[580],[577,2136,2138],{"className":2137},[584],[586,2139,2140],{"xmlns":588},[590,2141,2142,2152],{},[593,2143,2144,2147,2149],{},[682,2145,2146],{},"1,20",[1719,2148,1721],{},[1719,2150,2151],{},"m",[600,2153,2154],{"encoding":602},"1{,}20\\,\\text{m}",[577,2156,2158],{"className":2157,"ariaHidden":607},[606],[577,2159,2161,2165,2168,2174,2178,2181],{"className":2160},[611],[577,2162],{"className":2163,"style":2164},[615],"height:0.8389em;vertical-align:-0.1944em;",[577,2166,1004],{"className":2167},[620],[577,2169,2171],{"className":2170},[620],[577,2172,2089],{"className":2173},[2088],[577,2175,2177],{"className":2176},[620],"20",[577,2179],{"className":2180,"style":1761},[786],[577,2182,2184],{"className":2183},[620,1765],[577,2185,2151],{"className":2186},[620]," lang und ",[577,2189,2191,2210],{"className":2190},[580],[577,2192,2194],{"className":2193},[584],[586,2195,2196],{"xmlns":588},[590,2197,2198,2207],{},[593,2199,2200,2203,2205],{},[682,2201,2202],{},"0,80",[1719,2204,1721],{},[1719,2206,2151],{},[600,2208,2209],{"encoding":602},"0{,}80\\,\\text{m}",[577,2211,2213],{"className":2212,"ariaHidden":607},[606],[577,2214,2216,2219,2223,2229,2233,2236],{"className":2215},[611],[577,2217],{"className":2218,"style":2164},[615],[577,2220,2222],{"className":2221},[620],"0",[577,2224,2226],{"className":2225},[620],[577,2227,2089],{"className":2228},[2088],[577,2230,2232],{"className":2231},[620],"80",[577,2234],{"className":2235,"style":1761},[786],[577,2237,2239],{"className":2238},[620,1765],[577,2240,2151],{"className":2241},[620]," breit. Die Halbachsen sind ",[577,2244,2246,2265],{"className":2245},[580],[577,2247,2249],{"className":2248},[584],[586,2250,2251],{"xmlns":588},[590,2252,2253,2262],{},[593,2254,2255,2258,2260],{},[682,2256,2257],{},"0,6",[1719,2259,1721],{},[1719,2261,2151],{},[600,2263,2264],{"encoding":602},"0{,}6\\,\\text{m}",[577,2266,2268],{"className":2267,"ariaHidden":607},[606],[577,2269,2271,2274,2277,2283,2287,2290],{"className":2270},[611],[577,2272],{"className":2273,"style":2164},[615],[577,2275,2222],{"className":2276},[620],[577,2278,2280],{"className":2279},[620],[577,2281,2089],{"className":2282},[2088],[577,2284,2286],{"className":2285},[620],"6",[577,2288],{"className":2289,"style":1761},[786],[577,2291,2293],{"className":2292},[620,1765],[577,2294,2151],{"className":2295},[620],[577,2297,2299,2318],{"className":2298},[580],[577,2300,2302],{"className":2301},[584],[586,2303,2304],{"xmlns":588},[590,2305,2306,2315],{},[593,2307,2308,2311,2313],{},[682,2309,2310],{},"0,4",[1719,2312,1721],{},[1719,2314,2151],{},[600,2316,2317],{"encoding":602},"0{,}4\\,\\text{m}",[577,2319,2321],{"className":2320,"ariaHidden":607},[606],[577,2322,2324,2327,2330,2336,2340,2343],{"className":2323},[611],[577,2325],{"className":2326,"style":2164},[615],[577,2328,2222],{"className":2329},[620],[577,2331,2333],{"className":2332},[620],[577,2334,2089],{"className":2335},[2088],[577,2337,2339],{"className":2338},[620],"4",[577,2341],{"className":2342,"style":1761},[786],[577,2344,2346],{"className":2345},[620,1765],[577,2347,2151],{"className":2348},[620],[577,2350,2352],{"className":2351},[1480],[577,2353,2355,2417],{"className":2354},[580],[577,2356,2358],{"className":2357},[584],[586,2359,2360],{"xmlns":588,"display":1489},[590,2361,2362,2414],{},[593,2363,2364,2366,2368,2370,2372,2374,2376,2378,2380,2386,2388,2391,2393,2395,2401,2403,2406,2408],{},[596,2365,1496],{},[759,2367,761],{},[596,2369,674],{},[759,2371,841],{},[682,2373,2257],{},[759,2375,841],{},[682,2377,2310],{},[1719,2379,1721],{},[676,2381,2382,2384],{},[1719,2383,2151],{},[682,2385,684],{},[759,2387,761],{},[682,2389,2390],{},"0,24",[596,2392,674],{},[1719,2394,1721],{},[676,2396,2397,2399],{},[1719,2398,2151],{},[682,2400,684],{},[759,2402,1890],{},[682,2404,2405],{},"0,754",[1719,2407,1721],{},[676,2409,2410,2412],{},[1719,2411,2151],{},[682,2413,684],{},[600,2415,2416],{"encoding":602},"A = \\pi \\cdot 0{,}6 \\cdot 0{,}4\\,\\text{m}^2 = 0{,}24\\pi\\,\\text{m}^2 \\approx 0{,}754\\,\\text{m}^2",[577,2418,2420,2438,2456,2483,2545,2611],{"className":2419,"ariaHidden":607},[606],[577,2421,2423,2426,2429,2432,2435],{"className":2422},[611],[577,2424],{"className":2425,"style":1535},[615],[577,2427,1496],{"className":2428},[620,621],[577,2430],{"className":2431,"style":787},[786],[577,2433,761],{"className":2434},[791],[577,2436],{"className":2437,"style":787},[786],[577,2439,2441,2444,2447,2450,2453],{"className":2440},[611],[577,2442],{"className":2443,"style":866},[615],[577,2445,674],{"className":2446,"style":701},[620,621],[577,2448],{"className":2449,"style":876},[786],[577,2451,841],{"className":2452},[880],[577,2454],{"className":2455,"style":876},[786],[577,2457,2459,2462,2465,2471,2474,2477,2480],{"className":2458},[611],[577,2460],{"className":2461,"style":2164},[615],[577,2463,2222],{"className":2464},[620],[577,2466,2468],{"className":2467},[620],[577,2469,2089],{"className":2470},[2088],[577,2472,2286],{"className":2473},[620],[577,2475],{"className":2476,"style":876},[786],[577,2478,841],{"className":2479},[880],[577,2481],{"className":2482,"style":876},[786],[577,2484,2486,2489,2492,2498,2501,2504,2536,2539,2542],{"className":2485},[611],[577,2487],{"className":2488,"style":2077},[615],[577,2490,2222],{"className":2491},[620],[577,2493,2495],{"className":2494},[620],[577,2496,2089],{"className":2497},[2088],[577,2499,2339],{"className":2500},[620],[577,2502],{"className":2503,"style":1761},[786],[577,2505,2507,2513],{"className":2506},[620],[577,2508,2510],{"className":2509},[620,1765],[577,2511,2151],{"className":2512},[620],[577,2514,2516],{"className":2515},[712],[577,2517,2519],{"className":2518},[716],[577,2520,2522],{"className":2521},[720],[577,2523,2525],{"className":2524,"style":1554},[724],[577,2526,2527,2530],{"style":1578},[577,2528],{"className":2529,"style":732},[731],[577,2531,2533],{"className":2532},[736,737,738,739],[577,2534,684],{"className":2535},[620,739],[577,2537],{"className":2538,"style":787},[786],[577,2540,761],{"className":2541},[791],[577,2543],{"className":2544,"style":787},[786],[577,2546,2548,2551,2554,2560,2564,2567,2570,2602,2605,2608],{"className":2547},[611],[577,2549],{"className":2550,"style":2077},[615],[577,2552,2222],{"className":2553},[620],[577,2555,2557],{"className":2556},[620],[577,2558,2089],{"className":2559},[2088],[577,2561,2563],{"className":2562},[620],"24",[577,2565,674],{"className":2566,"style":701},[620,621],[577,2568],{"className":2569,"style":1761},[786],[577,2571,2573,2579],{"className":2572},[620],[577,2574,2576],{"className":2575},[620,1765],[577,2577,2151],{"className":2578},[620],[577,2580,2582],{"className":2581},[712],[577,2583,2585],{"className":2584},[716],[577,2586,2588],{"className":2587},[720],[577,2589,2591],{"className":2590,"style":1554},[724],[577,2592,2593,2596],{"style":1578},[577,2594],{"className":2595,"style":732},[731],[577,2597,2599],{"className":2598},[736,737,738,739],[577,2600,684],{"className":2601},[620,739],[577,2603],{"className":2604,"style":787},[786],[577,2606,1890],{"className":2607},[791],[577,2609],{"className":2610,"style":787},[786],[577,2612,2614,2617,2620,2626,2630,2633],{"className":2613},[611],[577,2615],{"className":2616,"style":2077},[615],[577,2618,2222],{"className":2619},[620],[577,2621,2623],{"className":2622},[620],[577,2624,2089],{"className":2625},[2088],[577,2627,2629],{"className":2628},[620],"754",[577,2631],{"className":2632,"style":1761},[786],[577,2634,2636,2642],{"className":2635},[620],[577,2637,2639],{"className":2638},[620,1765],[577,2640,2151],{"className":2641},[620],[577,2643,2645],{"className":2644},[712],[577,2646,2648],{"className":2647},[716],[577,2649,2651],{"className":2650},[720],[577,2652,2654],{"className":2653,"style":1554},[724],[577,2655,2656,2659],{"style":1578},[577,2657],{"className":2658,"style":732},[731],[577,2660,2662],{"className":2661},[736,737,738,739],[577,2663,684],{"className":2664},[620,739],[559,2666,2668],{"id":2667},"typische-fehler","Typische Fehler",[2670,2671,2672,2852,2985],"ul",{},[2673,2674,2675,2676,2808,2809,656],"li",{},"Die ganzen Achsen statt der Halbachsen einsetzen. Das ergibt das Vierfache, beim Tisch ",[577,2677,2679,2710],{"className":2678},[580],[577,2680,2682],{"className":2681},[584],[586,2683,2684],{"xmlns":588},[590,2685,2686,2707],{},[593,2687,2688,2690,2692,2695,2697,2700,2702,2705],{},[596,2689,674],{},[759,2691,841],{},[682,2693,2694],{},"1,2",[759,2696,841],{},[682,2698,2699],{},"0,8",[759,2701,761],{},[682,2703,2704],{},"0,96",[596,2706,674],{},[600,2708,2709],{"encoding":602},"\\pi \\cdot 1{,}2 \\cdot 0{,}8 = 0{,}96\\pi",[577,2711,2713,2731,2758,2786],{"className":2712,"ariaHidden":607},[606],[577,2714,2716,2719,2722,2725,2728],{"className":2715},[611],[577,2717],{"className":2718,"style":866},[615],[577,2720,674],{"className":2721,"style":701},[620,621],[577,2723],{"className":2724,"style":876},[786],[577,2726,841],{"className":2727},[880],[577,2729],{"className":2730,"style":876},[786],[577,2732,2734,2737,2740,2746,2749,2752,2755],{"className":2733},[611],[577,2735],{"className":2736,"style":2164},[615],[577,2738,1004],{"className":2739},[620],[577,2741,2743],{"className":2742},[620],[577,2744,2089],{"className":2745},[2088],[577,2747,684],{"className":2748},[620],[577,2750],{"className":2751,"style":876},[786],[577,2753,841],{"className":2754},[880],[577,2756],{"className":2757,"style":876},[786],[577,2759,2761,2764,2767,2773,2777,2780,2783],{"className":2760},[611],[577,2762],{"className":2763,"style":2164},[615],[577,2765,2222],{"className":2766},[620],[577,2768,2770],{"className":2769},[620],[577,2771,2089],{"className":2772},[2088],[577,2774,2776],{"className":2775},[620],"8",[577,2778],{"className":2779,"style":787},[786],[577,2781,761],{"className":2782},[791],[577,2784],{"className":2785,"style":787},[786],[577,2787,2789,2792,2795,2801,2805],{"className":2788},[611],[577,2790],{"className":2791,"style":2164},[615],[577,2793,2222],{"className":2794},[620],[577,2796,2798],{"className":2797},[620],[577,2799,2089],{"className":2800},[2088],[577,2802,2804],{"className":2803},[620],"96",[577,2806,674],{"className":2807,"style":701},[620,621]," statt ",[577,2810,2812,2828],{"className":2811},[580],[577,2813,2815],{"className":2814},[584],[586,2816,2817],{"xmlns":588},[590,2818,2819,2825],{},[593,2820,2821,2823],{},[682,2822,2390],{},[596,2824,674],{},[600,2826,2827],{"encoding":602},"0{,}24\\pi",[577,2829,2831],{"className":2830,"ariaHidden":607},[606],[577,2832,2834,2837,2840,2846,2849],{"className":2833},[611],[577,2835],{"className":2836,"style":2164},[615],[577,2838,2222],{"className":2839},[620],[577,2841,2843],{"className":2842},[620],[577,2844,2089],{"className":2845},[2088],[577,2847,2563],{"className":2848},[620],[577,2850,674],{"className":2851,"style":701},[620,621],[2673,2853,2854,2855,2919,2920,2984],{},"Wie beim Kreis quadrieren: ",[577,2856,2858,2878],{"className":2857},[580],[577,2859,2861],{"className":2860},[584],[586,2862,2863],{"xmlns":588},[590,2864,2865,2875],{},[593,2866,2867,2869],{},[596,2868,674],{},[676,2870,2871,2873],{},[596,2872,598],{},[682,2874,684],{},[600,2876,2877],{"encoding":602},"\\pi a^2",[577,2879,2881],{"className":2880,"ariaHidden":607},[606],[577,2882,2884,2887,2890],{"className":2883},[611],[577,2885],{"className":2886,"style":697},[615],[577,2888,674],{"className":2889,"style":701},[620,621],[577,2891,2893,2896],{"className":2892},[620],[577,2894,598],{"className":2895},[620,621],[577,2897,2899],{"className":2898},[712],[577,2900,2902],{"className":2901},[716],[577,2903,2905],{"className":2904},[720],[577,2906,2908],{"className":2907,"style":697},[724],[577,2909,2910,2913],{"style":727},[577,2911],{"className":2912,"style":732},[731],[577,2914,2916],{"className":2915},[736,737,738,739],[577,2917,684],{"className":2918},[620,739]," oder ",[577,2921,2923,2943],{"className":2922},[580],[577,2924,2926],{"className":2925},[584],[586,2927,2928],{"xmlns":588},[590,2929,2930,2940],{},[593,2931,2932,2934],{},[596,2933,674],{},[676,2935,2936,2938],{},[596,2937,640],{},[682,2939,684],{},[600,2941,2942],{"encoding":602},"\\pi b^2",[577,2944,2946],{"className":2945,"ariaHidden":607},[606],[577,2947,2949,2952,2955],{"className":2948},[611],[577,2950],{"className":2951,"style":697},[615],[577,2953,674],{"className":2954,"style":701},[620,621],[577,2956,2958,2961],{"className":2957},[620],[577,2959,640],{"className":2960},[620,621],[577,2962,2964],{"className":2963},[712],[577,2965,2967],{"className":2966},[716],[577,2968,2970],{"className":2969},[720],[577,2971,2973],{"className":2972,"style":697},[724],[577,2974,2975,2978],{"style":727},[577,2976],{"className":2977,"style":732},[731],[577,2979,2981],{"className":2980},[736,737,738,739],[577,2982,684],{"className":2983},[620,739]," ist nur richtig, wenn die Ellipse ein Kreis ist.",[2673,2986,2987],{},"Jedes Oval für eine Ellipse halten. Ei- oder Stadionformen sind keine Ellipsen, dort gilt die Formel nicht.",{"title":2989,"searchDepth":2990,"depth":2990,"links":2991},"",2,[2992,2993,2994,2995],{"id":561,"depth":2990,"text":562},{"id":949,"depth":2990,"text":950},{"id":1694,"depth":2990,"text":1695},{"id":2667,"depth":2990,"text":2668},[2997,3000,3003,3006,3009,3012,3015],{"lhs":2998,"rhs":2999},"pi*r*r","pi*r^2",{"lhs":3001,"rhs":3002},"pi*a^2 * b\u002Fa","pi*a*b",{"lhs":3004,"rhs":3005},"pi*5*3","15*pi",{"lhs":3007,"rhs":3008},"round(15*pi, 2)","47.12",{"lhs":3010,"rhs":3011},"pi*(1.2\u002F2)*(0.8\u002F2)","0.24*pi",{"lhs":3013,"rhs":3014},"round(0.24*pi, 3)","0.754",{"lhs":3016,"rhs":3017},"pi*1.2*0.8","4*0.24*pi","a > 0,\\ b > 0","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a > 0,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b > 0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md",{"caption":3023,"x":3024,"y":3027,"items":3030},"Ellipse mit den Halbachsen a und b, die Fläche A ist hervorgehoben",[3025,3026],-2.9,2.9,[3028,3029],-1.95,1.95,[3031,3038,3042,3045,3050,3055,3059,3063],{"type":3032,"center":3033,"rx":3035,"ry":3036,"fill":3037},"ellipse",[3034,3034],0,2.6,1.6,true,{"type":3039,"from":3040,"to":3041},"segment",[3034,3034],[3035,3034],{"type":3039,"from":3043,"to":3044},[3034,3034],[3034,3036],{"type":3046,"at":3047,"from":3048,"to":3049},"rightAngle",[3034,3034],[3035,3034],[3034,3036],{"type":3051,"at":3052,"text":598},"label",[3053,3054],1.3,-0.3,{"type":3051,"at":3056,"text":640},[3057,3058],-0.28,0.8,{"type":3051,"at":3060,"text":1496},[3061,3062],-1.4,-0.55,{"type":3064,"at":3065},"point",[3034,3034],"\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 278.4 187.2\" width=\"278.4\" height=\"187.2\" role=\"img\" aria-label=\"Ellipse mit den Halbachsen a und b, die Fläche A ist hervorgehoben\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-ellipse-area-arrow\" viewBox=\"0 0 10 10\" refX=\"9\" refY=\"5\" markerWidth=\"7\" markerHeight=\"7\" orient=\"auto-start-reverse\">\u003Cpath d=\"M 0 0 L 10 5 L 0 10 z\" class=\"fig-arrowhead\"\u002F>\u003C\u002Fmarker>\u003CclipPath id=\"fig-ellipse-area-clip\">\u003Crect x=\"0\" y=\"0\" width=\"278.4\" height=\"187.2\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Cellipse cx=\"139.2\" cy=\"93.6\" rx=\"124.8\" ry=\"76.8\" class=\"fig-main fig-fill\"\u002F>\u003Cline x1=\"139.2\" y1=\"93.6\" x2=\"264\" y2=\"93.6\" class=\"fig-main\"\u002F>\u003Cline x1=\"139.2\" y1=\"93.6\" x2=\"139.2\" y2=\"16.8\" class=\"fig-main\"\u002F>\u003Cpath d=\"M 151.2 93.6 L 151.2 81.6 L 139.2 81.6\" class=\"fig-mark\"\u002F>\u003Ccircle cx=\"145.2\" cy=\"87.6\" r=\"1.5\" class=\"fig-dot\"\u002F>\u003Ctext x=\"201.6\" y=\"108\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">a\u003C\u002Ftext>\u003Ctext x=\"125.76\" y=\"55.2\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">b\u003C\u002Ftext>\u003Ctext x=\"72\" y=\"120\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">A\u003C\u002Ftext>\u003Ccircle cx=\"139.2\" cy=\"93.6\" r=\"3\" class=\"fig-dot\"\u002F>\u003C\u002Fsvg>","ellipse-area","A = \\pi a b","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>π\u003C\u002Fmi>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A = \\pi a b\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ab\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},"\u002Fde\u002Fformulas\u002Fmath\u002Fgeometry\u002Fflaecheninhalt-ellipse",[],[3075,3076,3077],"circle-area","annulus-area","circle-equation",{"title":548,"description":2989},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fflaecheninhalt-ellipse","Eine Ellipse mit den Halbachsen a und b hat den Flächeninhalt π mal a mal b.",[3032,3082,3083,3084],"flächeninhalt","halbachse","kreis",[3086,3089,3092,3095],{"symbol":1496,"meaning":3087,"symbolHtml":3088},"Flächeninhalt der Ellipse","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":598,"meaning":3090,"symbolHtml":3091},"große Halbachse, halbe Länge der Ellipse","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":640,"meaning":3093,"symbolHtml":3094},"kleine Halbachse, halbe Breite der Ellipse","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":3096,"meaning":3097,"symbolHtml":3098},"\\pi","Kreiszahl, etwa 3,14159","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>π\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\pi\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","IFYOa16gRhiNM3vJWVKtG_6Kq_gRwFEE8uNOko1qSdA",[3101,3104,3107],{"stem":3102,"title":3103,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fflaecheninhalt-kreis","Flächeninhalt eines Kreises",{"stem":3105,"title":3106,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fkreisgleichung","Kreisgleichung",{"stem":3108,"title":3109,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fkreisring","Flächeninhalt eines Kreisrings",1793824268250]