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Seine Fläche ist ",[578,1139,1141,1178],{"className":1140},[581],[578,1142,1144],{"className":1143},[585],[587,1145,1146],{"xmlns":589},[591,1147,1148,1175],{},[594,1149,1150,1152,1154,1157,1159,1161,1163,1169,1171,1173],{},[597,1151,599],{},[597,1153,1022],{},[646,1155,1156],{},"⋅",[597,1158,599],{},[597,1160,1053],{},[646,1162,789],{},[878,1164,1165,1167],{},[597,1166,599],{},[650,1168,884],{},[646,1170,1156],{},[597,1172,1022],{},[597,1174,1053],{},[601,1176,1177],{"encoding":603},"ka \\cdot kb = k^2 \\cdot ab",[578,1179,1181,1204,1225,1269],{"className":1180,"ariaHidden":608},[607],[578,1182,1184,1187,1190,1193,1197,1201],{"className":1183},[612],[578,1185],{"className":1186,"style":617},[616],[578,1188,599],{"className":1189,"style":623},[621,622],[578,1191,1022],{"className":1192},[621,622],[578,1194],{"className":1195,"style":1196},[672],"margin-right:0.2222em;",[578,1198,1156],{"className":1199},[1200],"mbin",[578,1202],{"className":1203,"style":1196},[672],[578,1205,1207,1210,1213,1216,1219,1222],{"className":1206},[612],[578,1208],{"className":1209,"style":617},[616],[578,1211,599],{"className":1212,"style":623},[621,622],[578,1214,1053],{"className":1215},[621,622],[578,1217],{"className":1218,"style":673},[672],[578,1220,789],{"className":1221},[677],[578,1223],{"className":1224,"style":673},[672],[578,1226,1228,1231,1260,1263,1266],{"className":1227},[612],[578,1229],{"className":1230,"style":897},[616],[578,1232,1234,1237],{"className":1233},[621],[578,1235,599],{"className":1236,"style":623},[621,622],[578,1238,1240],{"className":1239},[907],[578,1241,1243],{"className":1242},[911],[578,1244,1246],{"className":1245},[915],[578,1247,1249],{"className":1248,"style":897},[919],[578,1250,1251,1254],{"style":922},[578,1252],{"className":1253,"style":927},[926],[578,1255,1257],{"className":1256},[931,932,933,934],[578,1258,884],{"className":1259},[621,934],[578,1261],{"className":1262,"style":1196},[672],[578,1264,1156],{"className":1265},[1200],[578,1267],{"className":1268,"style":1196},[672],[578,1270,1272,1275],{"className":1271},[612],[578,1273],{"className":1274,"style":617},[616],[578,1276,1278],{"className":1277},[621,622],"ab",". Bei einem Quader kommt die dritte Kante dazu: ",[578,1281,1283,1328],{"className":1282},[581],[578,1284,1286],{"className":1285},[585],[587,1287,1288],{"xmlns":589},[591,1289,1290,1325],{},[594,1291,1292,1294,1296,1298,1300,1302,1304,1306,1309,1311,1317,1319,1321,1323],{},[597,1293,599],{},[597,1295,1022],{},[646,1297,1156],{},[597,1299,599],{},[597,1301,1053],{},[646,1303,1156],{},[597,1305,599],{},[597,1307,1308],{},"c",[646,1310,789],{},[878,1312,1313,1315],{},[597,1314,599],{},[650,1316,959],{},[646,1318,1156],{},[597,1320,1022],{},[597,1322,1053],{},[597,1324,1308],{},[601,1326,1327],{"encoding":603},"ka \\cdot kb \\cdot kc = k^3 \\cdot abc",[578,1329,1331,1352,1373,1394,1438],{"className":1330,"ariaHidden":608},[607],[578,1332,1334,1337,1340,1343,1346,1349],{"className":1333},[612],[578,1335],{"className":1336,"style":617},[616],[578,1338,599],{"className":1339,"style":623},[621,622],[578,1341,1022],{"className":1342},[621,622],[578,1344],{"className":1345,"style":1196},[672],[578,1347,1156],{"className":1348},[1200],[578,1350],{"className":1351,"style":1196},[672],[578,1353,1355,1358,1361,1364,1367,1370],{"className":1354},[612],[578,1356],{"className":1357,"style":617},[616],[578,1359,599],{"className":1360,"style":623},[621,622],[578,1362,1053],{"className":1363},[621,622],[578,1365],{"className":1366,"style":1196},[672],[578,1368,1156],{"className":1369},[1200],[578,1371],{"className":1372,"style":1196},[672],[578,1374,1376,1379,1382,1385,1388,1391],{"className":1375},[612],[578,1377],{"className":1378,"style":617},[616],[578,1380,599],{"className":1381,"style":623},[621,622],[578,1383,1308],{"className":1384},[621,622],[578,1386],{"className":1387,"style":673},[672],[578,1389,789],{"className":1390},[677],[578,1392],{"className":1393,"style":673},[672],[578,1395,1397,1400,1429,1432,1435],{"className":1396},[612],[578,1398],{"className":1399,"style":897},[616],[578,1401,1403,1406],{"className":1402},[621],[578,1404,599],{"className":1405,"style":623},[621,622],[578,1407,1409],{"className":1408},[907],[578,1410,1412],{"className":1411},[911],[578,1413,1415],{"className":1414},[915],[578,1416,1418],{"className":1417,"style":897},[919],[578,1419,1420,1423],{"style":922},[578,1421],{"className":1422,"style":927},[926],[578,1424,1426],{"className":1425},[931,932,933,934],[578,1427,959],{"className":1428},[621,934],[578,1430],{"className":1431,"style":1196},[672],[578,1433,1156],{"className":1434},[1200],[578,1436],{"className":1437,"style":1196},[672],[578,1439,1441,1444,1447],{"className":1440},[612],[578,1442],{"className":1443,"style":617},[616],[578,1445,1278],{"className":1446},[621,622],[578,1448,1308],{"className":1449},[621,622],". Jede andere Figur lässt sich beliebig genau aus kleinen Rechtecken oder Quadern zusammensetzen, deshalb gilt das allgemein.",[560,1452,1454],{"id":1453},"beispiele","Beispiele",[565,1456,1457,1458,1536,1537,1588],{},"Ein Dreieck mit ",[578,1459,1461,1487],{"className":1460},[581],[578,1462,1464],{"className":1463},[585],[587,1465,1466],{"xmlns":589},[591,1467,1468,1484],{},[594,1469,1470,1473,1477],{},[650,1471,1472],{},"5",[1474,1475,1476],"mtext",{}," ",[878,1478,1479,1482],{},[1474,1480,1481],{},"cm",[650,1483,884],{},[601,1485,1486],{"encoding":603},"5\\,\\text{cm}^2",[578,1488,1490],{"className":1489,"ariaHidden":608},[607],[578,1491,1493,1496,1499,1503],{"className":1492},[612],[578,1494],{"className":1495,"style":897},[616],[578,1497,1472],{"className":1498},[621],[578,1500],{"className":1501,"style":1502},[672],"margin-right:0.1667em;",[578,1504,1506,1513],{"className":1505},[621],[578,1507,1510],{"className":1508},[621,1509],"text",[578,1511,1481],{"className":1512},[621],[578,1514,1516],{"className":1515},[907],[578,1517,1519],{"className":1518},[911],[578,1520,1522],{"className":1521},[915],[578,1523,1525],{"className":1524,"style":897},[919],[578,1526,1527,1530],{"style":922},[578,1528],{"className":1529,"style":927},[926],[578,1531,1533],{"className":1532},[931,932,933,934],[578,1534,884],{"className":1535},[621,934]," Fläche wird mit ",[578,1538,1540,1558],{"className":1539},[581],[578,1541,1543],{"className":1542},[585],[587,1544,1545],{"xmlns":589},[591,1546,1547,1555],{},[594,1548,1549,1551,1553],{},[597,1550,599],{},[646,1552,789],{},[650,1554,959],{},[601,1556,1557],{"encoding":603},"k = 3",[578,1559,1561,1579],{"className":1560,"ariaHidden":608},[607],[578,1562,1564,1567,1570,1573,1576],{"className":1563},[612],[578,1565],{"className":1566,"style":617},[616],[578,1568,599],{"className":1569,"style":623},[621,622],[578,1571],{"className":1572,"style":673},[672],[578,1574,789],{"className":1575},[677],[578,1577],{"className":1578,"style":673},[672],[578,1580,1582,1585],{"className":1581},[612],[578,1583],{"className":1584,"style":687},[616],[578,1586,959],{"className":1587},[621]," vergrößert:",[578,1590,1593],{"className":1591},[1592],"katex-display",[578,1594,1596,1669],{"className":1595},[581],[578,1597,1599],{"className":1598},[585],[587,1600,1602],{"xmlns":589,"display":1601},"block",[591,1603,1604,1666],{},[594,1605,1606,1616,1618,1624,1626,1628,1630,1636,1638,1641,1643,1645,1647,1653,1655,1658,1660],{},[878,1607,1608,1611],{},[597,1609,1610],{},"A",[646,1612,1615],{"mathvariant":1613,"lspace":1614,"rspace":1614},"normal","0em","′",[646,1617,789],{},[878,1619,1620,1622],{},[650,1621,959],{},[650,1623,884],{},[646,1625,1156],{},[650,1627,1472],{},[1474,1629,1476],{},[878,1631,1632,1634],{},[1474,1633,1481],{},[650,1635,884],{},[646,1637,789],{},[650,1639,1640],{},"9",[646,1642,1156],{},[650,1644,1472],{},[1474,1646,1476],{},[878,1648,1649,1651],{},[1474,1650,1481],{},[650,1652,884],{},[646,1654,789],{},[650,1656,1657],{},"45",[1474,1659,1476],{},[878,1661,1662,1664],{},[1474,1663,1481],{},[650,1665,884],{},[601,1667,1668],{"encoding":603},"A' = 3^2 \\cdot 5\\,\\text{cm}^2 = 9 \\cdot 5\\,\\text{cm}^2 = 45\\,\\text{cm}^2",[578,1670,1672,1721,1766,1819,1837,1890],{"className":1671,"ariaHidden":608},[607],[578,1673,1675,1679,1712,1715,1718],{"className":1674},[612],[578,1676],{"className":1677,"style":1678},[616],"height:0.8019em;",[578,1680,1682,1685],{"className":1681},[621],[578,1683,1610],{"className":1684},[621,622],[578,1686,1688],{"className":1687},[907],[578,1689,1691],{"className":1690},[911],[578,1692,1694],{"className":1693},[915],[578,1695,1697],{"className":1696,"style":1678},[919],[578,1698,1700,1703],{"style":1699},"top:-3.113em;margin-right:0.05em;",[578,1701],{"className":1702,"style":927},[926],[578,1704,1706],{"className":1705},[931,932,933,934],[578,1707,1709],{"className":1708},[621,934],[578,1710,1615],{"className":1711},[621,934],[578,1713],{"className":1714,"style":673},[672],[578,1716,789],{"className":1717},[677],[578,1719],{"className":1720,"style":673},[672],[578,1722,1724,1728,1757,1760,1763],{"className":1723},[612],[578,1725],{"className":1726,"style":1727},[616],"height:0.8641em;",[578,1729,1731,1734],{"className":1730},[621],[578,1732,959],{"className":1733},[621],[578,1735,1737],{"className":1736},[907],[578,1738,1740],{"className":1739},[911],[578,1741,1743],{"className":1742},[915],[578,1744,1746],{"className":1745,"style":1727},[919],[578,1747,1748,1751],{"style":1699},[578,1749],{"className":1750,"style":927},[926],[578,1752,1754],{"className":1753},[931,932,933,934],[578,1755,884],{"className":1756},[621,934],[578,1758],{"className":1759,"style":1196},[672],[578,1761,1156],{"className":1762},[1200],[578,1764],{"className":1765,"style":1196},[672],[578,1767,1769,1772,1775,1778,1810,1813,1816],{"className":1768},[612],[578,1770],{"className":1771,"style":1727},[616],[578,1773,1472],{"className":1774},[621],[578,1776],{"className":1777,"style":1502},[672],[578,1779,1781,1787],{"className":1780},[621],[578,1782,1784],{"className":1783},[621,1509],[578,1785,1481],{"className":1786},[621],[578,1788,1790],{"className":1789},[907],[578,1791,1793],{"className":1792},[911],[578,1794,1796],{"className":1795},[915],[578,1797,1799],{"className":1798,"style":1727},[919],[578,1800,1801,1804],{"style":1699},[578,1802],{"className":1803,"style":927},[926],[578,1805,1807],{"className":1806},[931,932,933,934],[578,1808,884],{"className":1809},[621,934],[578,1811],{"className":1812,"style":673},[672],[578,1814,789],{"className":1815},[677],[578,1817],{"className":1818,"style":673},[672],[578,1820,1822,1825,1828,1831,1834],{"className":1821},[612],[578,1823],{"className":1824,"style":687},[616],[578,1826,1640],{"className":1827},[621],[578,1829],{"className":1830,"style":1196},[672],[578,1832,1156],{"className":1833},[1200],[578,1835],{"className":1836,"style":1196},[672],[578,1838,1840,1843,1846,1849,1881,1884,1887],{"className":1839},[612],[578,1841],{"className":1842,"style":1727},[616],[578,1844,1472],{"className":1845},[621],[578,1847],{"className":1848,"style":1502},[672],[578,1850,1852,1858],{"className":1851},[621],[578,1853,1855],{"className":1854},[621,1509],[578,1856,1481],{"className":1857},[621],[578,1859,1861],{"className":1860},[907],[578,1862,1864],{"className":1863},[911],[578,1865,1867],{"className":1866},[915],[578,1868,1870],{"className":1869,"style":1727},[919],[578,1871,1872,1875],{"style":1699},[578,1873],{"className":1874,"style":927},[926],[578,1876,1878],{"className":1877},[931,932,933,934],[578,1879,884],{"className":1880},[621,934],[578,1882],{"className":1883,"style":673},[672],[578,1885,789],{"className":1886},[677],[578,1888],{"className":1889,"style":673},[672],[578,1891,1893,1896,1899,1902],{"className":1892},[612],[578,1894],{"className":1895,"style":1727},[616],[578,1897,1657],{"className":1898},[621],[578,1900],{"className":1901,"style":1502},[672],[578,1903,1905,1911],{"className":1904},[621],[578,1906,1908],{"className":1907},[621,1509],[578,1909,1481],{"className":1910},[621],[578,1912,1914],{"className":1913},[907],[578,1915,1917],{"className":1916},[911],[578,1918,1920],{"className":1919},[915],[578,1921,1923],{"className":1922,"style":1727},[919],[578,1924,1925,1928],{"style":1699},[578,1926],{"className":1927,"style":927},[926],[578,1929,1931],{"className":1930},[931,932,933,934],[578,1932,884],{"className":1933},[621,934],[565,1935,1936,1937,2019],{},"Verdoppelt man den Radius einer Kugel, wird ihr Volumen ",[578,1938,1940,1963],{"className":1939},[581],[578,1941,1943],{"className":1942},[585],[587,1944,1945],{"xmlns":589},[591,1946,1947,1960],{},[594,1948,1949,1955,1957],{},[878,1950,1951,1953],{},[650,1952,884],{},[650,1954,959],{},[646,1956,789],{},[650,1958,1959],{},"8",[601,1961,1962],{"encoding":603},"2^3 = 8",[578,1964,1966,2010],{"className":1965,"ariaHidden":608},[607],[578,1967,1969,1972,2001,2004,2007],{"className":1968},[612],[578,1970],{"className":1971,"style":897},[616],[578,1973,1975,1978],{"className":1974},[621],[578,1976,884],{"className":1977},[621],[578,1979,1981],{"className":1980},[907],[578,1982,1984],{"className":1983},[911],[578,1985,1987],{"className":1986},[915],[578,1988,1990],{"className":1989,"style":897},[919],[578,1991,1992,1995],{"style":922},[578,1993],{"className":1994,"style":927},[926],[578,1996,1998],{"className":1997},[931,932,933,934],[578,1999,959],{"className":2000},[621,934],[578,2002],{"className":2003,"style":673},[672],[578,2005,789],{"className":2006},[677],[578,2008],{"className":2009,"style":673},[672],[578,2011,2013,2016],{"className":2012},[612],[578,2014],{"className":2015,"style":687},[616],[578,2017,1959],{"className":2018},[621],"-mal so groß.",[565,2021,2022,2023,2075,2076,2165,2166,624],{},"Ein Modellauto im Maßstab ",[578,2024,2026,2045],{"className":2025},[581],[578,2027,2029],{"className":2028},[585],[587,2030,2031],{"xmlns":589},[591,2032,2033,2042],{},[594,2034,2035,2037,2039],{},[650,2036,652],{},[646,2038,624],{},[650,2040,2041],{},"20",[601,2043,2044],{"encoding":603},"1 : 20",[578,2046,2048,2066],{"className":2047,"ariaHidden":608},[607],[578,2049,2051,2054,2057,2060,2063],{"className":2050},[612],[578,2052],{"className":2053,"style":687},[616],[578,2055,652],{"className":2056},[621],[578,2058],{"className":2059,"style":673},[672],[578,2061,624],{"className":2062},[677],[578,2064],{"className":2065,"style":673},[672],[578,2067,2069,2072],{"className":2068},[612],[578,2070],{"className":2071,"style":687},[616],[578,2073,2041],{"className":2074},[621]," hat eine Dachfläche von ",[578,2077,2079,2103],{"className":2078},[581],[578,2080,2082],{"className":2081},[585],[587,2083,2084],{"xmlns":589},[591,2085,2086,2100],{},[594,2087,2088,2091,2093],{},[650,2089,2090],{},"0,4",[1474,2092,1476],{},[878,2094,2095,2098],{},[1474,2096,2097],{},"dm",[650,2099,884],{},[601,2101,2102],{"encoding":603},"0{,}4\\,\\text{dm}^2",[578,2104,2106],{"className":2105,"ariaHidden":608},[607],[578,2107,2109,2113,2116,2124,2128,2131],{"className":2108},[612],[578,2110],{"className":2111,"style":2112},[616],"height:1.0929em;vertical-align:-0.1944em;",[578,2114,708],{"className":2115},[621],[578,2117,2119],{"className":2118},[621],[578,2120,2123],{"className":2121},[2122],"mpunct",",",[578,2125,2127],{"className":2126},[621],"4",[578,2129],{"className":2130,"style":1502},[672],[578,2132,2134,2140],{"className":2133},[621],[578,2135,2137],{"className":2136},[621,1509],[578,2138,2097],{"className":2139},[621],[578,2141,2143],{"className":2142},[907],[578,2144,2146],{"className":2145},[911],[578,2147,2149],{"className":2148},[915],[578,2150,2153],{"className":2151,"style":2152},[919],"height:0.8984em;",[578,2154,2156,2159],{"style":2155},"top:-3.1473em;margin-right:0.05em;",[578,2157],{"className":2158,"style":927},[926],[578,2160,2162],{"className":2161},[931,932,933,934],[578,2163,884],{"className":2164},[621,934],". Für das echte Auto ist ",[578,2167,2169,2187],{"className":2168},[581],[578,2170,2172],{"className":2171},[585],[587,2173,2174],{"xmlns":589},[591,2175,2176,2184],{},[594,2177,2178,2180,2182],{},[597,2179,599],{},[646,2181,789],{},[650,2183,2041],{},[601,2185,2186],{"encoding":603},"k = 20",[578,2188,2190,2208],{"className":2189,"ariaHidden":608},[607],[578,2191,2193,2196,2199,2202,2205],{"className":2192},[612],[578,2194],{"className":2195,"style":617},[616],[578,2197,599],{"className":2198,"style":623},[621,622],[578,2200],{"className":2201,"style":673},[672],[578,2203,789],{"className":2204},[677],[578,2206],{"className":2207,"style":673},[672],[578,2209,2211,2214],{"className":2210},[612],[578,2212],{"className":2213,"style":687},[616],[578,2215,2041],{"className":2216},[621],[578,2218,2220],{"className":2219},[1592],[578,2221,2223,2274],{"className":2222},[581],[578,2224,2226],{"className":2225},[585],[587,2227,2228],{"xmlns":589,"display":1601},[591,2229,2230,2271],{},[594,2231,2232,2238,2240,2246,2248,2250,2252,2258,2260,2263,2265],{},[878,2233,2234,2236],{},[597,2235,1610],{},[646,2237,1615],{"mathvariant":1613,"lspace":1614,"rspace":1614},[646,2239,789],{},[878,2241,2242,2244],{},[650,2243,2041],{},[650,2245,884],{},[646,2247,1156],{},[650,2249,2090],{},[1474,2251,1476],{},[878,2253,2254,2256],{},[1474,2255,2097],{},[650,2257,884],{},[646,2259,789],{},[650,2261,2262],{},"160",[1474,2264,1476],{},[878,2266,2267,2269],{},[1474,2268,2097],{},[650,2270,884],{},[601,2272,2273],{"encoding":603},"A' = 20^2 \\cdot 0{,}4\\,\\text{dm}^2 = 160\\,\\text{dm}^2",[578,2275,2277,2324,2371,2433],{"className":2276,"ariaHidden":608},[607],[578,2278,2280,2283,2315,2318,2321],{"className":2279},[612],[578,2281],{"className":2282,"style":1678},[616],[578,2284,2286,2289],{"className":2285},[621],[578,2287,1610],{"className":2288},[621,622],[578,2290,2292],{"className":2291},[907],[578,2293,2295],{"className":2294},[911],[578,2296,2298],{"className":2297},[915],[578,2299,2301],{"className":2300,"style":1678},[919],[578,2302,2303,2306],{"style":1699},[578,2304],{"className":2305,"style":927},[926],[578,2307,2309],{"className":2308},[931,932,933,934],[578,2310,2312],{"className":2311},[621,934],[578,2313,1615],{"className":2314},[621,934],[578,2316],{"className":2317,"style":673},[672],[578,2319,789],{"className":2320},[677],[578,2322],{"className":2323,"style":673},[672],[578,2325,2327,2330,2333,2362,2365,2368],{"className":2326},[612],[578,2328],{"className":2329,"style":1727},[616],[578,2331,884],{"className":2332},[621],[578,2334,2336,2339],{"className":2335},[621],[578,2337,708],{"className":2338},[621],[578,2340,2342],{"className":2341},[907],[578,2343,2345],{"className":2344},[911],[578,2346,2348],{"className":2347},[915],[578,2349,2351],{"className":2350,"style":1727},[919],[578,2352,2353,2356],{"style":1699},[578,2354],{"className":2355,"style":927},[926],[578,2357,2359],{"className":2358},[931,932,933,934],[578,2360,884],{"className":2361},[621,934],[578,2363],{"className":2364,"style":1196},[672],[578,2366,1156],{"className":2367},[1200],[578,2369],{"className":2370,"style":1196},[672],[578,2372,2374,2377,2380,2386,2389,2392,2424,2427,2430],{"className":2373},[612],[578,2375],{"className":2376,"style":2112},[616],[578,2378,708],{"className":2379},[621],[578,2381,2383],{"className":2382},[621],[578,2384,2123],{"className":2385},[2122],[578,2387,2127],{"className":2388},[621],[578,2390],{"className":2391,"style":1502},[672],[578,2393,2395,2401],{"className":2394},[621],[578,2396,2398],{"className":2397},[621,1509],[578,2399,2097],{"className":2400},[621],[578,2402,2404],{"className":2403},[907],[578,2405,2407],{"className":2406},[911],[578,2408,2410],{"className":2409},[915],[578,2411,2413],{"className":2412,"style":2152},[919],[578,2414,2415,2418],{"style":2155},[578,2416],{"className":2417,"style":927},[926],[578,2419,2421],{"className":2420},[931,932,933,934],[578,2422,884],{"className":2423},[621,934],[578,2425],{"className":2426,"style":673},[672],[578,2428,789],{"className":2429},[677],[578,2431],{"className":2432,"style":673},[672],[578,2434,2436,2439,2442,2445],{"className":2435},[612],[578,2437],{"className":2438,"style":2152},[616],[578,2440,2262],{"className":2441},[621],[578,2443],{"className":2444,"style":1502},[672],[578,2446,2448,2454],{"className":2447},[621],[578,2449,2451],{"className":2450},[621,1509],[578,2452,2097],{"className":2453},[621],[578,2455,2457],{"className":2456},[907],[578,2458,2460],{"className":2459},[911],[578,2461,2463],{"className":2462},[915],[578,2464,2466],{"className":2465,"style":2152},[919],[578,2467,2468,2471],{"style":2155},[578,2469],{"className":2470,"style":927},[926],[578,2472,2474],{"className":2473},[931,932,933,934],[578,2475,884],{"className":2476},[621,934],[565,2478,2479,2480,2565],{},"Sein Volumen ist ",[578,2481,2483,2506],{"className":2482},[581],[578,2484,2486],{"className":2485},[585],[587,2487,2488],{"xmlns":589},[591,2489,2490,2503],{},[594,2491,2492,2498,2500],{},[878,2493,2494,2496],{},[650,2495,2041],{},[650,2497,959],{},[646,2499,789],{},[650,2501,2502],{},"8000",[601,2504,2505],{"encoding":603},"20^3 = 8000",[578,2507,2509,2556],{"className":2508,"ariaHidden":608},[607],[578,2510,2512,2515,2518,2547,2550,2553],{"className":2511},[612],[578,2513],{"className":2514,"style":897},[616],[578,2516,884],{"className":2517},[621],[578,2519,2521,2524],{"className":2520},[621],[578,2522,708],{"className":2523},[621],[578,2525,2527],{"className":2526},[907],[578,2528,2530],{"className":2529},[911],[578,2531,2533],{"className":2532},[915],[578,2534,2536],{"className":2535,"style":897},[919],[578,2537,2538,2541],{"style":922},[578,2539],{"className":2540,"style":927},[926],[578,2542,2544],{"className":2543},[931,932,933,934],[578,2545,959],{"className":2546},[621,934],[578,2548],{"className":2549,"style":673},[672],[578,2551,789],{"className":2552},[677],[578,2554],{"className":2555,"style":673},[672],[578,2557,2559,2562],{"className":2558},[612],[578,2560],{"className":2561,"style":687},[616],[578,2563,2502],{"className":2564},[621],"-mal so groß wie das des Modells.",[560,2567,2569],{"id":2568},"typische-fehler","Typische Fehler",[626,2571,2572,2810,2893],{},[629,2573,2574,2575,2603,2604,2662,2663,2736,2737,2809],{},"Die Fläche mit ",[578,2576,2578,2591],{"className":2577},[581],[578,2579,2581],{"className":2580},[585],[587,2582,2583],{"xmlns":589},[591,2584,2585,2589],{},[594,2586,2587],{},[597,2588,599],{},[601,2590,599],{"encoding":603},[578,2592,2594],{"className":2593,"ariaHidden":608},[607],[578,2595,2597,2600],{"className":2596},[612],[578,2598],{"className":2599,"style":617},[616],[578,2601,599],{"className":2602,"style":623},[621,622]," statt mit ",[578,2605,2607,2624],{"className":2606},[581],[578,2608,2610],{"className":2609},[585],[587,2611,2612],{"xmlns":589},[591,2613,2614,2622],{},[594,2615,2616],{},[878,2617,2618,2620],{},[597,2619,599],{},[650,2621,884],{},[601,2623,887],{"encoding":603},[578,2625,2627],{"className":2626,"ariaHidden":608},[607],[578,2628,2630,2633],{"className":2629},[612],[578,2631],{"className":2632,"style":897},[616],[578,2634,2636,2639],{"className":2635},[621],[578,2637,599],{"className":2638,"style":623},[621,622],[578,2640,2642],{"className":2641},[907],[578,2643,2645],{"className":2644},[911],[578,2646,2648],{"className":2647},[915],[578,2649,2651],{"className":2650,"style":897},[919],[578,2652,2653,2656],{"style":922},[578,2654],{"className":2655,"style":927},[926],[578,2657,2659],{"className":2658},[931,932,933,934],[578,2660,884],{"className":2661},[621,934]," umrechnen. Im ersten Beispiel ergäbe das ",[578,2664,2666,2689],{"className":2665},[581],[578,2667,2669],{"className":2668},[585],[587,2670,2671],{"xmlns":589},[591,2672,2673,2686],{},[594,2674,2675,2678,2680],{},[650,2676,2677],{},"15",[1474,2679,1476],{},[878,2681,2682,2684],{},[1474,2683,1481],{},[650,2685,884],{},[601,2687,2688],{"encoding":603},"15\\,\\text{cm}^2",[578,2690,2692],{"className":2691,"ariaHidden":608},[607],[578,2693,2695,2698,2701,2704],{"className":2694},[612],[578,2696],{"className":2697,"style":897},[616],[578,2699,2677],{"className":2700},[621],[578,2702],{"className":2703,"style":1502},[672],[578,2705,2707,2713],{"className":2706},[621],[578,2708,2710],{"className":2709},[621,1509],[578,2711,1481],{"className":2712},[621],[578,2714,2716],{"className":2715},[907],[578,2717,2719],{"className":2718},[911],[578,2720,2722],{"className":2721},[915],[578,2723,2725],{"className":2724,"style":897},[919],[578,2726,2727,2730],{"style":922},[578,2728],{"className":2729,"style":927},[926],[578,2731,2733],{"className":2732},[931,932,933,934],[578,2734,884],{"className":2735},[621,934]," statt ",[578,2738,2740,2762],{"className":2739},[581],[578,2741,2743],{"className":2742},[585],[587,2744,2745],{"xmlns":589},[591,2746,2747,2759],{},[594,2748,2749,2751,2753],{},[650,2750,1657],{},[1474,2752,1476],{},[878,2754,2755,2757],{},[1474,2756,1481],{},[650,2758,884],{},[601,2760,2761],{"encoding":603},"45\\,\\text{cm}^2",[578,2763,2765],{"className":2764,"ariaHidden":608},[607],[578,2766,2768,2771,2774,2777],{"className":2767},[612],[578,2769],{"className":2770,"style":897},[616],[578,2772,1657],{"className":2773},[621],[578,2775],{"className":2776,"style":1502},[672],[578,2778,2780,2786],{"className":2779},[621],[578,2781,2783],{"className":2782},[621,1509],[578,2784,1481],{"className":2785},[621],[578,2787,2789],{"className":2788},[907],[578,2790,2792],{"className":2791},[911],[578,2793,2795],{"className":2794},[915],[578,2796,2798],{"className":2797,"style":897},[919],[578,2799,2800,2803],{"style":922},[578,2801],{"className":2802,"style":927},[926],[578,2804,2806],{"className":2805},[931,932,933,934],[578,2807,884],{"className":2808},[621,934],".",[629,2811,2812,2813,2841,2842,2809],{},"Den Streckfaktor falsch herum bilden. ",[578,2814,2816,2829],{"className":2815},[581],[578,2817,2819],{"className":2818},[585],[587,2820,2821],{"xmlns":589},[591,2822,2823,2827],{},[594,2824,2825],{},[597,2826,599],{},[601,2828,599],{"encoding":603},[578,2830,2832],{"className":2831,"ariaHidden":608},[607],[578,2833,2835,2838],{"className":2834},[612],[578,2836],{"className":2837,"style":617},[616],[578,2839,599],{"className":2840,"style":623},[621,622]," ist immer Bildlänge durch Originallänge; bei einer Verkleinerung ist ",[578,2843,2845,2863],{"className":2844},[581],[578,2846,2848],{"className":2847},[585],[587,2849,2850],{"xmlns":589},[591,2851,2852,2860],{},[594,2853,2854,2856,2858],{},[597,2855,599],{},[646,2857,711],{},[650,2859,652],{},[601,2861,2862],{"encoding":603},"k \u003C 1",[578,2864,2866,2884],{"className":2865,"ariaHidden":608},[607],[578,2867,2869,2872,2875,2878,2881],{"className":2868},[612],[578,2870],{"className":2871,"style":665},[616],[578,2873,599],{"className":2874,"style":623},[621,622],[578,2876],{"className":2877,"style":673},[672],[578,2879,711],{"className":2880},[677],[578,2882],{"className":2883,"style":673},[672],[578,2885,2887,2890],{"className":2886},[612],[578,2888],{"className":2889,"style":687},[616],[578,2891,652],{"className":2892},[621],[629,2894,2895],{},"Annehmen, dass Figuren mit gleichen Winkeln immer ähnlich sind. Bei Dreiecken stimmt das, bei Rechtecken nicht: Alle Rechtecke haben vier rechte Winkel, sind aber nicht alle ähnlich.",{"title":2897,"searchDepth":2898,"depth":2898,"links":2899},"",2,[2900,2901,2902,2903],{"id":562,"depth":2898,"text":563},{"id":1003,"depth":2898,"text":1004},{"id":1453,"depth":2898,"text":1454},{"id":2568,"depth":2898,"text":2569},[2905,2908,2911,2913,2915,2918,2920,2922],{"lhs":2906,"rhs":2907},"(k*a)*(k*b)","k^2*(a*b)",{"lhs":2909,"rhs":2910},"(k*a)*(k*b)*(k*c)","k^3*(a*b*c)",{"lhs":2912,"rhs":1657},"3^2*5",{"lhs":2914,"rhs":1959},"2^3",{"lhs":2916,"rhs":2917},"4\u002F3*pi*(2*r)^3","8*(4\u002F3*pi*r^3)",{"lhs":2919,"rhs":2262},"0.4 * 20^2",{"lhs":2921,"rhs":2502},"20^3",{"lhs":2923,"rhs":2677},"3*5","k > 0","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath 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class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md",{"caption":2929,"x":2930,"y":2933,"items":2936},"Dreieck mit dem Flächeninhalt A und ähnliches Bilddreieck mit doppelt so langen Seiten (k = 2) und dem Flächeninhalt A'",[2931,2932],-0.4,5.6,[2934,2935],-0.5,2.6,[2937,2948,2958,2963],{"type":2938,"points":2939,"style":2947},"polygon",[2940,2942,2944],[2941,2941],0,[2943,2941],1.5,[2945,2946],0.5,1.1,"muted",{"type":2938,"points":2949,"fill":2957},[2950,2952,2954],[2951,2941],2.4,[2953,2941],5.4,[2955,2956],3.4,2.2,true,{"type":2959,"at":2960,"text":1610},"label",[2961,2962],0.67,0.37,{"type":2959,"at":2964,"text":2967},[2965,2966],3.73,0.73,"A'","\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 288 148.8\" width=\"288\" height=\"148.8\" role=\"img\" aria-label=\"Dreieck mit dem Flächeninhalt A und ähnliches Bilddreieck mit doppelt so langen Seiten (k = 2) und dem Flächeninhalt A'\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-similarity-ratio-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-similarity-ratio-clip\">\u003Crect x=\"0\" y=\"0\" width=\"288\" height=\"148.8\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Cpolygon points=\"19.2,124.8 91.2,124.8 43.2,72\" class=\"fig-muted\"\u002F>\u003Cpolygon points=\"134.4,124.8 278.4,124.8 182.4,19.2\" class=\"fig-main fig-fill\"\u002F>\u003Ctext x=\"51.36\" y=\"107.04\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">A\u003C\u002Ftext>\u003Ctext x=\"198.24\" y=\"89.76\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">A'\u003C\u002Ftext>\u003C\u002Fsvg>","similarity-ratio","A' = k^2 \\cdot A,\\quad V' = k^3 \\cdot V","\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>\u003Cmsup>\u003Cmi>A\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmi>k\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"1em\"\u002F>\u003Cmsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmi>k\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>V\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A' = k^2 \\cdot A,\\quad V' = k^3 \\cdot 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style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","8SRG93JNSf-PW_8R6_Bt0CBVRhB_X6GS3go8p17s8vg",[3004,3007,3010,3013],{"stem":3005,"title":3006,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fflaecheninhalt-rechteck","Flächeninhalt eines Rechtecks",{"stem":3008,"title":3009,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fstrahlensatz","Strahlensatz",{"stem":3011,"title":3012,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fvolumen-kugel","Volumen einer 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