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Alle Ecken der Grundfläche sind mit der Spitze verbunden.",[564,573,574,575,621,622,625],{},"Die Formel gilt für jede Pyramide, egal welche Form die Grundfläche hat und ob die Spitze genau über der Mitte liegt. 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Jede hat eine Würfelfläche als Grundfläche und die Kante als Höhe, also ",[576,637,639,682],{"className":638},[579],[576,640,642],{"className":641},[583],[585,643,644],{"xmlns":587},[589,645,646,679],{},[592,647,648,651,655,665,674,677],{},[595,649,650],{},"V",[652,653,654],"mo",{},"=",[656,657,658,662],"mfrac",{},[659,660,661],"mn",{},"1",[659,663,664],{},"3",[666,667,668,671],"msup",{},[595,669,670],{},"a",[659,672,673],{},"2",[652,675,676],{},"⋅",[595,678,670],{},[599,680,681],{"encoding":601},"V = \\frac{1}{3} a^2 \\cdot a",[576,683,685,708,848],{"className":684,"ariaHidden":606},[605],[576,686,688,692,696,701,705],{"className":687},[610],[576,689],{"className":690,"style":691},[614],"height:0.6833em;",[576,693,650],{"className":694,"style":695},[619,620],"margin-right:0.2222em;",[576,697],{"className":698,"style":700},[699],"mspace","margin-right:0.2778em;",[576,702,654],{"className":703},[704],"mrel",[576,706],{"className":707,"style":700},[699],[576,709,711,715,805,838,841,845],{"className":710},[610],[576,712],{"className":713,"style":714},[614],"height:1.1901em;vertical-align:-0.345em;",[576,716,718,723,801],{"className":717},[619],[576,719],{"className":720},[721,722],"mopen","nulldelimiter",[576,724,726],{"className":725},[656],[576,727,731,792],{"className":728},[729,730],"vlist-t","vlist-t2",[576,732,735,787],{"className":733},[734],"vlist-r",[576,736,740,761,772],{"className":737,"style":739},[738],"vlist","height:0.8451em;",[576,741,743,748],{"style":742},"top:-2.655em;",[576,744],{"className":745,"style":747},[746],"pstrut","height:3em;",[576,749,755],{"className":750},[751,752,753,754],"sizing","reset-size6","size3","mtight",[576,756,758],{"className":757},[619,754],[576,759,664],{"className":760},[619,754],[576,762,764,767],{"style":763},"top:-3.23em;",[576,765],{"className":766,"style":747},[746],[576,768],{"className":769,"style":771},[770],"frac-line","border-bottom-width:0.04em;",[576,773,775,778],{"style":774},"top:-3.394em;",[576,776],{"className":777,"style":747},[746],[576,779,781],{"className":780},[751,752,753,754],[576,782,784],{"className":783},[619,754],[576,785,661],{"className":786},[619,754],[576,788,791],{"className":789},[790],"vlist-s","​",[576,793,795],{"className":794},[734],[576,796,799],{"className":797,"style":798},[738],"height:0.345em;",[576,800],{},[576,802],{"className":803},[804,722],"mclose",[576,806,808,811],{"className":807},[619],[576,809,670],{"className":810},[619,620],[576,812,815],{"className":813},[814],"msupsub",[576,816,818],{"className":817},[729],[576,819,821],{"className":820},[734],[576,822,825],{"className":823,"style":824},[738],"height:0.8141em;",[576,826,828,832],{"style":827},"top:-3.063em;margin-right:0.05em;",[576,829],{"className":830,"style":831},[746],"height:2.7em;",[576,833,835],{"className":834},[751,752,753,754],[576,836,673],{"className":837},[619,754],[576,839],{"className":840,"style":695},[699],[576,842,676],{"className":843},[844],"mbin",[576,846],{"className":847,"style":695},[699],[576,849,851,855],{"className":850},[610],[576,852],{"className":853,"style":854},[614],"height:0.4306em;",[576,856,670],{"className":857},[619,620],". 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Mit Wasser lässt es sich auch ausprobieren: Den Inhalt einer hohlen Pyramide muss man dreimal umfüllen, bis ein Prisma mit gleicher Grundfläche und Höhe voll ist.",[559,959,961],{"id":960},"beispiele","Beispiele",[564,963,964,965,1014,1015,1058],{},"Eine Pyramide hat eine quadratische Grundfläche mit der Seite ",[576,966,968,990],{"className":967},[579],[576,969,971],{"className":970},[583],[585,972,973],{"xmlns":587},[589,974,975,987],{},[592,976,977,980,984],{},[659,978,979],{},"6",[981,982,983],"mtext",{}," ",[981,985,986],{},"cm",[599,988,989],{"encoding":601},"6\\,\\text{cm}",[576,991,993],{"className":992,"ariaHidden":606},[605],[576,994,996,1000,1003,1007],{"className":995},[610],[576,997],{"className":998,"style":999},[614],"height:0.6444em;",[576,1001,979],{"className":1002},[619],[576,1004],{"className":1005,"style":1006},[699],"margin-right:0.1667em;",[576,1008,1011],{"className":1009},[619,1010],"text",[576,1012,986],{"className":1013},[619]," und ist 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= \\frac{1}{3} \\cdot (6\\,\\text{cm})^2 \\cdot 5\\,\\text{cm} = \\frac{1}{3} \\cdot 36\\,\\text{cm}^2 \\cdot 5\\,\\text{cm} = 60\\,\\text{cm}^3",[576,1159,1161,1179,1261,1323,1350,1427,1480,1507],{"className":1160,"ariaHidden":606},[605],[576,1162,1164,1167,1170,1173,1176],{"className":1163},[610],[576,1165],{"className":1166,"style":691},[614],[576,1168,650],{"className":1169,"style":695},[619,620],[576,1171],{"className":1172,"style":700},[699],[576,1174,654],{"className":1175},[704],[576,1177],{"className":1178,"style":700},[699],[576,1180,1182,1186,1252,1255,1258],{"className":1181},[610],[576,1183],{"className":1184,"style":1185},[614],"height:2.0074em;vertical-align:-0.686em;",[576,1187,1189,1192,1249],{"className":1188},[619],[576,1190],{"className":1191},[721,722],[576,1193,1195],{"className":1194},[656],[576,1196,1198,1240],{"className":1197},[729,730],[576,1199,1201,1237],{"className":1200},[734],[576,1202,1205,1217,1225],{"className":1203,"style":1204},[738],"height:1.3214em;",[576,1206,1208,1211],{"style":1207},"top:-2.314em;",[576,1209],{"className":1210,"style":747},[746],[576,1212,1214],{"className":1213},[619],[576,1215,664],{"className":1216},[619],[576,1218,1219,1222],{"style":763},[576,1220],{"className":1221,"style":747},[746],[576,1223],{"className":1224,"style":771},[770],[576,1226,1228,1231],{"style":1227},"top:-3.677em;",[576,1229],{"className":1230,"style":747},[746],[576,1232,1234],{"className":1233},[619],[576,1235,661],{"className":1236},[619],[576,1238,791],{"className":1239},[790],[576,1241,1243],{"className":1242},[734],[576,1244,1247],{"className":1245,"style":1246},[738],"height:0.686em;",[576,1248],{},[576,1250],{"className":1251},[804,722],[576,1253],{"className":1254,"style":695},[699],[576,1256,676],{"className":1257},[844],[576,1259],{"className":1260,"style":695},[699],[576,1262,1264,1268,1271,1274,1277,1283,1314,1317,1320],{"className":1263},[610],[576,1265],{"className":1266,"style":1267},[614],"height:1.1141em;vertical-align:-0.25em;",[576,1269,1091],{"className":1270},[721],[576,1272,979],{"className":1273},[619],[576,1275],{"className":1276,"style":1006},[699],[576,1278,1280],{"className":1279},[619,1010],[576,1281,986],{"className":1282},[619],[576,1284,1286,1289],{"className":1285},[804],[576,1287,1102],{"className":1288},[804],[576,1290,1292],{"className":1291},[814],[576,1293,1295],{"className":1294},[729],[576,1296,1298],{"className":1297},[734],[576,1299,1302],{"className":1300,"style":1301},[738],"height:0.8641em;",[576,1303,1305,1308],{"style":1304},"top:-3.113em;margin-right:0.05em;",[576,1306],{"className":1307,"style":831},[746],[576,1309,1311],{"className":1310},[751,752,753,754],[576,1312,673],{"className":1313},[619,754],[576,1315],{"className":1316,"style":695},[699],[576,1318,676],{"className":1319},[844],[576,1321],{"className":1322,"style":695},[699],[576,1324,1326,1329,1332,1335,1341,1344,1347],{"className":1325},[610],[576,1327],{"className":1328,"style":999},[614],[576,1330,1029],{"className":1331},[619],[576,1333],{"className":1334,"style":1006},[699],[576,1336,1338],{"className":1337},[619,1010],[576,1339,986],{"className":1340},[619],[576,1342],{"className":1343,"style":700},[699],[576,1345,654],{"className":1346},[704],[576,1348],{"className":1349,"style":700},[699],[576,1351,1353,1356,1418,1421,1424],{"className":1352},[610],[576,1354],{"className":1355,"style":1185},[614],[576,1357,1359,1362,1415],{"className":1358},[619],[576,1360],{"className":1361},[721,722],[576,1363,1365],{"className":1364},[656],[576,1366,1368,1407],{"className":1367},[729,730],[576,1369,1371,1404],{"className":1370},[734],[576,1372,1374,1385,1393],{"className":1373,"style":1204},[738],[576,1375,1376,1379],{"style":1207},[576,1377],{"className":1378,"style":747},[746],[576,1380,1382],{"className":1381},[619],[576,1383,664],{"className":1384},[619],[576,1386,1387,1390],{"style":763},[576,1388],{"className":1389,"style":747},[746],[576,1391],{"className":1392,"style":771},[770],[576,1394,1395,1398],{"style":1227},[576,1396],{"className":1397,"style":747},[746],[576,1399,1401],{"className":1400},[619],[576,1402,661],{"className":1403},[619],[576,1405,791],{"className":1406},[790],[576,1408,1410],{"className":1409},[734],[576,1411,1413],{"className":1412,"style":1246},[738],[576,1414],{},[576,1416],{"className":1417},[804,722],[576,1419],{"className":1420,"style":695},[699],[576,1422,676],{"className":1423},[844],[576,1425],{"className":1426,"style":695},[699],[576,1428,1430,1433,1436,1439,1471,1474,1477],{"className":1429},[610],[576,1431],{"className":1432,"style":1301},[614],[576,1434,1125],{"className":1435},[619],[576,1437],{"className":1438,"style":1006},[699],[576,1440,1442,1448],{"className":1441},[619],[576,1443,1445],{"className":1444},[619,1010],[576,1446,986],{"className":1447},[619],[576,1449,1451],{"className":1450},[814],[576,1452,1454],{"className":1453},[729],[576,1455,1457],{"className":1456},[734],[576,1458,1460],{"className":1459,"style":1301},[738],[576,1461,1462,1465],{"style":1304},[576,1463],{"className":1464,"style":831},[746],[576,1466,1468],{"className":1467},[751,752,753,754],[576,1469,673],{"className":1470},[619,754],[576,1472],{"className":1473,"style":695},[699],[576,1475,676],{"className":1476},[844],[576,1478],{"className":1479,"style":695},[699],[576,1481,1483,1486,1489,1492,1498,1501,1504],{"className":1482},[610],[576,1484],{"className":1485,"style":999},[614],[576,1487,1029],{"className":1488},[619],[576,1490],{"className":1491,"style":1006},[699],[576,1493,1495],{"className":1494},[619,1010],[576,1496,986],{"className":1497},[619],[576,1499],{"className":1500,"style":700},[699],[576,1502,654],{"className":1503},[704],[576,1505],{"className":1506,"style":700},[699],[576,1508,1510,1513,1516,1519],{"className":1509},[610],[576,1511],{"className":1512,"style":1301},[614],[576,1514,1146],{"className":1515},[619],[576,1517],{"className":1518,"style":1006},[699],[576,1520,1522,1528],{"className":1521},[619],[576,1523,1525],{"className":1524},[619,1010],[576,1526,986],{"className":1527},[619],[576,1529,1531],{"className":1530},[814],[576,1532,1534],{"className":1533},[729],[576,1535,1537],{"className":1536},[734],[576,1538,1540],{"className":1539,"style":1301},[738],[576,1541,1542,1545],{"style":1304},[576,1543],{"className":1544,"style":831},[746],[576,1546,1548],{"className":1547},[751,752,753,754],[576,1549,664],{"className":1550},[619,754],[564,1552,1553,1554,1598,1599,1641,1642,1058],{},"Eine Pyramide hat ein Rechteck mit ",[576,1555,1557,1577],{"className":1556},[579],[576,1558,1560],{"className":1559},[583],[585,1561,1562],{"xmlns":587},[589,1563,1564,1574],{},[592,1565,1566,1569,1571],{},[659,1567,1568],{},"4",[981,1570,983],{},[981,1572,1573],{},"m",[599,1575,1576],{"encoding":601},"4\\,\\text{m}",[576,1578,1580],{"className":1579,"ariaHidden":606},[605],[576,1581,1583,1586,1589,1592],{"className":1582},[610],[576,1584],{"className":1585,"style":999},[614],[576,1587,1568],{"className":1588},[619],[576,1590],{"className":1591,"style":1006},[699],[576,1593,1595],{"className":1594},[619,1010],[576,1596,1573],{"className":1597},[619]," und ",[576,1600,1602,1620],{"className":1601},[579],[576,1603,1605],{"className":1604},[583],[585,1606,1607],{"xmlns":587},[589,1608,1609,1617],{},[592,1610,1611,1613,1615],{},[659,1612,664],{},[981,1614,983],{},[981,1616,1573],{},[599,1618,1619],{"encoding":601},"3\\,\\text{m}",[576,1621,1623],{"className":1622,"ariaHidden":606},[605],[576,1624,1626,1629,1632,1635],{"className":1625},[610],[576,1627],{"className":1628,"style":999},[614],[576,1630,664],{"className":1631},[619],[576,1633],{"className":1634,"style":1006},[699],[576,1636,1638],{"className":1637},[619,1010],[576,1639,1573],{"className":1640},[619]," als Grundfläche und ist ",[576,1643,1645,1664],{"className":1644},[579],[576,1646,1648],{"className":1647},[583],[585,1649,1650],{"xmlns":587},[589,1651,1652,1661],{},[592,1653,1654,1657,1659],{},[659,1655,1656],{},"10",[981,1658,983],{},[981,1660,1573],{},[599,1662,1663],{"encoding":601},"10\\,\\text{m}",[576,1665,1667],{"className":1666,"ariaHidden":606},[605],[576,1668,1670,1673,1676,1679],{"className":1669},[610],[576,1671],{"className":1672,"style":999},[614],[576,1674,1656],{"className":1675},[619],[576,1677],{"className":1678,"style":1006},[699],[576,1680,1682],{"className":1681},[619,1010],[576,1683,1573],{"className":1684},[619],[576,1686,1688],{"className":1687},[1062],[576,1689,1691,1747],{"className":1690},[579],[576,1692,1694],{"className":1693},[583],[585,1695,1696],{"xmlns":587,"display":1071},[589,1697,1698,1744],{},[592,1699,1700,1702,1704,1710,1712,1715,1717,1723,1725,1727,1729,1731,1733,1736,1738],{},[595,1701,650],{},[652,1703,654],{},[656,1705,1706,1708],{},[659,1707,661],{},[659,1709,664],{},[652,1711,676],{},[659,1713,1714],{},"12",[981,1716,983],{},[666,1718,1719,1721],{},[981,1720,1573],{},[659,1722,673],{},[652,1724,676],{},[659,1726,1656],{},[981,1728,983],{},[981,1730,1573],{},[652,1732,654],{},[659,1734,1735],{},"40",[981,1737,983],{},[666,1739,1740,1742],{},[981,1741,1573],{},[659,1743,664],{},[599,1745,1746],{"encoding":601},"V = \\frac{1}{3} \\cdot 12\\,\\text{m}^2 \\cdot 10\\,\\text{m} = 40\\,\\text{m}^3",[576,1748,1750,1768,1845,1898,1925],{"className":1749,"ariaHidden":606},[605],[576,1751,1753,1756,1759,1762,1765],{"className":1752},[610],[576,1754],{"className":1755,"style":691},[614],[576,1757,650],{"className":1758,"style":695},[619,620],[576,1760],{"className":1761,"style":700},[699],[576,1763,654],{"className":1764},[704],[576,1766],{"className":1767,"style":700},[699],[576,1769,1771,1774,1836,1839,1842],{"className":1770},[610],[576,1772],{"className":1773,"style":1185},[614],[576,1775,1777,1780,1833],{"className":1776},[619],[576,1778],{"className":1779},[721,722],[576,1781,1783],{"className":1782},[656],[576,1784,1786,1825],{"className":1785},[729,730],[576,1787,1789,1822],{"className":1788},[734],[576,1790,1792,1803,1811],{"className":1791,"style":1204},[738],[576,1793,1794,1797],{"style":1207},[576,1795],{"className":1796,"style":747},[746],[576,1798,1800],{"className":1799},[619],[576,1801,664],{"className":1802},[619],[576,1804,1805,1808],{"style":763},[576,1806],{"className":1807,"style":747},[746],[576,1809],{"className":1810,"style":771},[770],[576,1812,1813,1816],{"style":1227},[576,1814],{"className":1815,"style":747},[746],[576,1817,1819],{"className":1818},[619],[576,1820,661],{"className":1821},[619],[576,1823,791],{"className":1824},[790],[576,1826,1828],{"className":1827},[734],[576,1829,1831],{"className":1830,"style":1246},[738],[576,1832],{},[576,1834],{"className":1835},[804,722],[576,1837],{"className":1838,"style":695},[699],[576,1840,676],{"className":1841},[844],[576,1843],{"className":1844,"style":695},[699],[576,1846,1848,1851,1854,1857,1889,1892,1895],{"className":1847},[610],[576,1849],{"className":1850,"style":1301},[614],[576,1852,1714],{"className":1853},[619],[576,1855],{"className":1856,"style":1006},[699],[576,1858,1860,1866],{"className":1859},[619],[576,1861,1863],{"className":1862},[619,1010],[576,1864,1573],{"className":1865},[619],[576,1867,1869],{"className":1868},[814],[576,1870,1872],{"className":1871},[729],[576,1873,1875],{"className":1874},[734],[576,1876,1878],{"className":1877,"style":1301},[738],[576,1879,1880,1883],{"style":1304},[576,1881],{"className":1882,"style":831},[746],[576,1884,1886],{"className":1885},[751,752,753,754],[576,1887,673],{"className":1888},[619,754],[576,1890],{"className":1891,"style":695},[699],[576,1893,676],{"className":1894},[844],[576,1896],{"className":1897,"style":695},[699],[576,1899,1901,1904,1907,1910,1916,1919,1922],{"className":1900},[610],[576,1902],{"className":1903,"style":999},[614],[576,1905,1656],{"className":1906},[619],[576,1908],{"className":1909,"style":1006},[699],[576,1911,1913],{"className":1912},[619,1010],[576,1914,1573],{"className":1915},[619],[576,1917],{"className":1918,"style":700},[699],[576,1920,654],{"className":1921},[704],[576,1923],{"className":1924,"style":700},[699],[576,1926,1928,1931,1934,1937],{"className":1927},[610],[576,1929],{"className":1930,"style":1301},[614],[576,1932,1735],{"className":1933},[619],[576,1935],{"className":1936,"style":1006},[699],[576,1938,1940,1946],{"className":1939},[619],[576,1941,1943],{"className":1942},[619,1010],[576,1944,1573],{"className":1945},[619],[576,1947,1949],{"className":1948},[814],[576,1950,1952],{"className":1951},[729],[576,1953,1955],{"className":1954},[734],[576,1956,1958],{"className":1957,"style":1301},[738],[576,1959,1960,1963],{"style":1304},[576,1961],{"className":1962,"style":831},[746],[576,1964,1966],{"className":1965},[751,752,753,754],[576,1967,664],{"className":1968},[619,754],[559,1970,1972],{"id":1971},"typische-fehler","Typische Fehler",[1974,1975,1976,2078],"ul",{},[1977,1978,1979,1980,2077],"li",{},"Das ",[576,1981,1983,2000],{"className":1982},[579],[576,1984,1986],{"className":1985},[583],[585,1987,1988],{"xmlns":587},[589,1989,1990,1998],{},[592,1991,1992],{},[656,1993,1994,1996],{},[659,1995,661],{},[659,1997,664],{},[599,1999,879],{"encoding":601},[576,2001,2003],{"className":2002,"ariaHidden":606},[605],[576,2004,2006,2009],{"className":2005},[610],[576,2007],{"className":2008,"style":714},[614],[576,2010,2012,2015,2074],{"className":2011},[619],[576,2013],{"className":2014},[721,722],[576,2016,2018],{"className":2017},[656],[576,2019,2021,2066],{"className":2020},[729,730],[576,2022,2024,2063],{"className":2023},[734],[576,2025,2027,2041,2049],{"className":2026,"style":739},[738],[576,2028,2029,2032],{"style":742},[576,2030],{"className":2031,"style":747},[746],[576,2033,2035],{"className":2034},[751,752,753,754],[576,2036,2038],{"className":2037},[619,754],[576,2039,664],{"className":2040},[619,754],[576,2042,2043,2046],{"style":763},[576,2044],{"className":2045,"style":747},[746],[576,2047],{"className":2048,"style":771},[770],[576,2050,2051,2054],{"style":774},[576,2052],{"className":2053,"style":747},[746],[576,2055,2057],{"className":2056},[751,752,753,754],[576,2058,2060],{"className":2059},[619,754],[576,2061,661],{"className":2062},[619,754],[576,2064,791],{"className":2065},[790],[576,2067,2069],{"className":2068},[734],[576,2070,2072],{"className":2071,"style":798},[738],[576,2073],{},[576,2075],{"className":2076},[804,722]," vergessen. Dann hat man das Volumen eines Prismas.",[1977,2079,2080,2081,2109],{},"Die Seitenkante oder die Höhe einer Seitenfläche statt der Körperhöhe einsetzen. Kennt man nur diese, berechnet man ",[576,2082,2084,2097],{"className":2083},[579],[576,2085,2087],{"className":2086},[583],[585,2088,2089],{"xmlns":587},[589,2090,2091,2095],{},[592,2092,2093],{},[595,2094,597],{},[599,2096,597],{"encoding":601},[576,2098,2100],{"className":2099,"ariaHidden":606},[605],[576,2101,2103,2106],{"className":2102},[610],[576,2104],{"className":2105,"style":615},[614],[576,2107,597],{"className":2108},[619,620]," vorher mit dem Satz des Pythagoras.",{"title":2111,"searchDepth":2112,"depth":2112,"links":2113},"",2,[2114,2115,2116,2117],{"id":561,"depth":2112,"text":562},{"id":631,"depth":2112,"text":632},{"id":960,"depth":2112,"text":961},{"id":1971,"depth":2112,"text":1972},[2119,2121],{"lhs":2120,"rhs":1146},"1\u002F3 * 6^2 * 5",{"lhs":2122,"rhs":1735},"1\u002F3 * (4 * 3) * 10",null,"md",{"caption":2126,"x":2127,"y":2130,"items":2133},"Pyramide mit quadratischer Grundfläche G und der Höhe h von der Spitze senkrecht auf die 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