[{"data":1,"prerenderedAt":2078},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-kreis":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-kreis":2068},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":555,"category":247,"checks":1984,"conditions":2000,"conditionsHtml":2000,"deprecated":2000,"description":1978,"extension":2001,"figure":2002,"figureCaption":2003,"figureHtml":2034,"formulaId":2035,"latex":2036,"latexHtml":2037,"level":2038,"meta":2039,"navigation":2040,"path":2041,"redirectFrom":2042,"related":2043,"seo":2047,"stem":2048,"subject":586,"summary":2049,"tags":2050,"variables":2056,"__hash__":2067},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-kreis.md","Umfang eines Kreises",[550,551,552,553,554],"Kreisumfang","Kreisumfang berechnen","Umfang Kreis Formel","U = 2πr","U = π · d",{"type":556,"value":557,"toc":1977},"minimark",[558,563,694,792,867,871,944,1147,1194,1330,1377,1662,1666],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,571,572,575,576,623,624,575,627,693],"p",{},"Der ",[568,569,570],"strong",{},"Umfang"," ist die Länge der Kreislinie. Man braucht dafür den ",[568,573,574],{},"Radius"," ",[577,578,581,603],"span",{"className":579},[580],"katex",[577,582,585],{"className":583},[584],"katex-mathml",[586,587,589],"math",{"xmlns":588},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[590,591,592,599],"semantics",{},[593,594,595],"mrow",{},[596,597,598],"mi",{},"r",[600,601,598],"annotation",{"encoding":602},"application\u002Fx-tex",[577,604,608],{"className":605,"ariaHidden":607},[606],"katex-html","true",[577,609,612,617],{"className":610},[611],"base",[577,613],{"className":614,"style":616},[615],"strut","height:0.4306em;",[577,618,598],{"className":619,"style":622},[620,621],"mord","mathnormal","margin-right:0.0278em;"," oder den ",[568,625,626],{},"Durchmesser",[577,628,630,655],{"className":629},[580],[577,631,633],{"className":632},[584],[586,634,635],{"xmlns":588},[590,636,637,652],{},[593,638,639,642,646,650],{},[596,640,641],{},"d",[643,644,645],"mo",{},"=",[647,648,649],"mn",{},"2",[596,651,598],{},[600,653,654],{"encoding":602},"d = 2r",[577,656,658,680],{"className":657,"ariaHidden":607},[606],[577,659,661,665,668,673,677],{"className":660},[611],[577,662],{"className":663,"style":664},[615],"height:0.6944em;",[577,666,641],{"className":667},[620,621],[577,669],{"className":670,"style":672},[671],"mspace","margin-right:0.2778em;",[577,674,645],{"className":675},[676],"mrel",[577,678],{"className":679,"style":672},[671],[577,681,683,687,690],{"className":682},[611],[577,684],{"className":685,"style":686},[615],"height:0.6444em;",[577,688,649],{"className":689},[620],[577,691,598],{"className":692,"style":622},[620,621],":",[577,695,698],{"className":696},[697],"katex-display",[577,699,701,732],{"className":700},[580],[577,702,704],{"className":703},[584],[586,705,707],{"xmlns":588,"display":706},"block",[590,708,709,729],{},[593,710,711,714,716,718,721,723,725,727],{},[596,712,713],{},"U",[643,715,645],{},[647,717,649],{},[596,719,720],{},"π",[596,722,598],{},[643,724,645],{},[596,726,720],{},[596,728,641],{},[600,730,731],{"encoding":602},"U = 2\\pi r = \\pi d",[577,733,735,755,780],{"className":734,"ariaHidden":607},[606],[577,736,738,742,746,749,752],{"className":737},[611],[577,739],{"className":740,"style":741},[615],"height:0.6833em;",[577,743,713],{"className":744,"style":745},[620,621],"margin-right:0.109em;",[577,747],{"className":748,"style":672},[671],[577,750,645],{"className":751},[676],[577,753],{"className":754,"style":672},[671],[577,756,758,761,764,768,771,774,777],{"className":757},[611],[577,759],{"className":760,"style":686},[615],[577,762,649],{"className":763},[620],[577,765,720],{"className":766,"style":767},[620,621],"margin-right:0.0359em;",[577,769,598],{"className":770,"style":622},[620,621],[577,772],{"className":773,"style":672},[671],[577,775,645],{"className":776},[676],[577,778],{"className":779,"style":672},[671],[577,781,783,786,789],{"className":782},[611],[577,784],{"className":785,"style":664},[615],[577,787,720],{"className":788,"style":767},[620,621],[577,790,641],{"className":791},[620,621],[564,793,794,795,575,798,866],{},"Die ",[568,796,797],{},"Kreiszahl",[577,799,801,821],{"className":800},[580],[577,802,804],{"className":803},[584],[586,805,806],{"xmlns":588},[590,807,808,818],{},[593,809,810,812,815],{},[596,811,720],{},[643,813,814],{},"≈",[647,816,817],{},"3,14159",[600,819,820],{"encoding":602},"\\pi \\approx 3{,}14159",[577,822,824,843],{"className":823,"ariaHidden":607},[606],[577,825,827,831,834,837,840],{"className":826},[611],[577,828],{"className":829,"style":830},[615],"height:0.4831em;",[577,832,720],{"className":833,"style":767},[620,621],[577,835],{"className":836,"style":672},[671],[577,838,814],{"className":839},[676],[577,841],{"className":842,"style":672},[671],[577,844,846,850,854,862],{"className":845},[611],[577,847],{"className":848,"style":849},[615],"height:0.8389em;vertical-align:-0.1944em;",[577,851,853],{"className":852},[620],"3",[577,855,857],{"className":856},[620],[577,858,861],{"className":859},[860],"mpunct",",",[577,863,865],{"className":864},[620],"14159"," gibt genau an, wie oft der Durchmesser in den Umfang passt: bei jedem Kreis etwas mehr als dreimal.",[559,868,870],{"id":869},"beispiele","Beispiele",[564,872,873,874,693],{},"Ein Kreis hat den Radius ",[577,875,877,903],{"className":876},[580],[577,878,880],{"className":879},[584],[586,881,882],{"xmlns":588},[590,883,884,900],{},[593,885,886,888,890,893,897],{},[596,887,598],{},[643,889,645],{},[647,891,892],{},"4",[894,895,896],"mtext",{}," ",[894,898,899],{},"cm",[600,901,902],{"encoding":602},"r = 4\\,\\text{cm}",[577,904,906,924],{"className":905,"ariaHidden":607},[606],[577,907,909,912,915,918,921],{"className":908},[611],[577,910],{"className":911,"style":616},[615],[577,913,598],{"className":914,"style":622},[620,621],[577,916],{"className":917,"style":672},[671],[577,919,645],{"className":920},[676],[577,922],{"className":923,"style":672},[671],[577,925,927,930,933,937],{"className":926},[611],[577,928],{"className":929,"style":686},[615],[577,931,892],{"className":932},[620],[577,934],{"className":935,"style":936},[671],"margin-right:0.1667em;",[577,938,941],{"className":939},[620,940],"text",[577,942,899],{"className":943},[620],[577,945,947],{"className":946},[697],[577,948,950,1001],{"className":949},[580],[577,951,953],{"className":952},[584],[586,954,955],{"xmlns":588,"display":706},[590,956,957,998],{},[593,958,959,961,963,965,968,970,972,974,976,978,980,983,985,987,989,991,994,996],{},[596,960,713],{},[643,962,645],{},[647,964,649],{},[643,966,967],{},"⋅",[596,969,720],{},[643,971,967],{},[647,973,892],{},[894,975,896],{},[894,977,899],{},[643,979,645],{},[647,981,982],{},"8",[596,984,720],{},[894,986,896],{},[894,988,899],{},[643,990,814],{},[647,992,993],{},"25,13",[894,995,896],{},[894,997,899],{},[600,999,1000],{"encoding":602},"U = 2 \\cdot \\pi \\cdot 4\\,\\text{cm} = 8\\pi\\,\\text{cm} \\approx 25{,}13\\,\\text{cm}",[577,1002,1004,1022,1042,1061,1088,1118],{"className":1003,"ariaHidden":607},[606],[577,1005,1007,1010,1013,1016,1019],{"className":1006},[611],[577,1008],{"className":1009,"style":741},[615],[577,1011,713],{"className":1012,"style":745},[620,621],[577,1014],{"className":1015,"style":672},[671],[577,1017,645],{"className":1018},[676],[577,1020],{"className":1021,"style":672},[671],[577,1023,1025,1028,1031,1035,1039],{"className":1024},[611],[577,1026],{"className":1027,"style":686},[615],[577,1029,649],{"className":1030},[620],[577,1032],{"className":1033,"style":1034},[671],"margin-right:0.2222em;",[577,1036,967],{"className":1037},[1038],"mbin",[577,1040],{"className":1041,"style":1034},[671],[577,1043,1045,1049,1052,1055,1058],{"className":1044},[611],[577,1046],{"className":1047,"style":1048},[615],"height:0.4445em;",[577,1050,720],{"className":1051,"style":767},[620,621],[577,1053],{"className":1054,"style":1034},[671],[577,1056,967],{"className":1057},[1038],[577,1059],{"className":1060,"style":1034},[671],[577,1062,1064,1067,1070,1073,1079,1082,1085],{"className":1063},[611],[577,1065],{"className":1066,"style":686},[615],[577,1068,892],{"className":1069},[620],[577,1071],{"className":1072,"style":936},[671],[577,1074,1076],{"className":1075},[620,940],[577,1077,899],{"className":1078},[620],[577,1080],{"className":1081,"style":672},[671],[577,1083,645],{"className":1084},[676],[577,1086],{"className":1087,"style":672},[671],[577,1089,1091,1094,1097,1100,1103,1109,1112,1115],{"className":1090},[611],[577,1092],{"className":1093,"style":686},[615],[577,1095,982],{"className":1096},[620],[577,1098,720],{"className":1099,"style":767},[620,621],[577,1101],{"className":1102,"style":936},[671],[577,1104,1106],{"className":1105},[620,940],[577,1107,899],{"className":1108},[620],[577,1110],{"className":1111,"style":672},[671],[577,1113,814],{"className":1114},[676],[577,1116],{"className":1117,"style":672},[671],[577,1119,1121,1124,1128,1134,1138,1141],{"className":1120},[611],[577,1122],{"className":1123,"style":849},[615],[577,1125,1127],{"className":1126},[620],"25",[577,1129,1131],{"className":1130},[620],[577,1132,861],{"className":1133},[860],[577,1135,1137],{"className":1136},[620],"13",[577,1139],{"className":1140,"style":936},[671],[577,1142,1144],{"className":1143},[620,940],[577,1145,899],{"className":1146},[620],[564,1148,1149,1150,1193],{},"Ein Fahrradreifen hat den Durchmesser ",[577,1151,1153,1172],{"className":1152},[580],[577,1154,1156],{"className":1155},[584],[586,1157,1158],{"xmlns":588},[590,1159,1160,1169],{},[593,1161,1162,1165,1167],{},[647,1163,1164],{},"60",[894,1166,896],{},[894,1168,899],{},[600,1170,1171],{"encoding":602},"60\\,\\text{cm}",[577,1173,1175],{"className":1174,"ariaHidden":607},[606],[577,1176,1178,1181,1184,1187],{"className":1177},[611],[577,1179],{"className":1180,"style":686},[615],[577,1182,1164],{"className":1183},[620],[577,1185],{"className":1186,"style":936},[671],[577,1188,1190],{"className":1189},[620,940],[577,1191,899],{"className":1192},[620],". Bei einer Umdrehung rollt das Rad so weit:",[577,1195,1197],{"className":1196},[697],[577,1198,1200,1235],{"className":1199},[580],[577,1201,1203],{"className":1202},[584],[586,1204,1205],{"xmlns":588,"display":706},[590,1206,1207,1232],{},[593,1208,1209,1211,1213,1215,1217,1219,1221,1223,1225,1228,1230],{},[596,1210,713],{},[643,1212,645],{},[596,1214,720],{},[643,1216,967],{},[647,1218,1164],{},[894,1220,896],{},[894,1222,899],{},[643,1224,814],{},[647,1226,1227],{},"188,50",[894,1229,896],{},[894,1231,899],{},[600,1233,1234],{"encoding":602},"U = \\pi \\cdot 60\\,\\text{cm} \\approx 188{,}50\\,\\text{cm}",[577,1236,1238,1256,1274,1301],{"className":1237,"ariaHidden":607},[606],[577,1239,1241,1244,1247,1250,1253],{"className":1240},[611],[577,1242],{"className":1243,"style":741},[615],[577,1245,713],{"className":1246,"style":745},[620,621],[577,1248],{"className":1249,"style":672},[671],[577,1251,645],{"className":1252},[676],[577,1254],{"className":1255,"style":672},[671],[577,1257,1259,1262,1265,1268,1271],{"className":1258},[611],[577,1260],{"className":1261,"style":1048},[615],[577,1263,720],{"className":1264,"style":767},[620,621],[577,1266],{"className":1267,"style":1034},[671],[577,1269,967],{"className":1270},[1038],[577,1272],{"className":1273,"style":1034},[671],[577,1275,1277,1280,1283,1286,1292,1295,1298],{"className":1276},[611],[577,1278],{"className":1279,"style":686},[615],[577,1281,1164],{"className":1282},[620],[577,1284],{"className":1285,"style":936},[671],[577,1287,1289],{"className":1288},[620,940],[577,1290,899],{"className":1291},[620],[577,1293],{"className":1294,"style":672},[671],[577,1296,814],{"className":1297},[676],[577,1299],{"className":1300,"style":672},[671],[577,1302,1304,1307,1311,1317,1321,1324],{"className":1303},[611],[577,1305],{"className":1306,"style":849},[615],[577,1308,1310],{"className":1309},[620],"188",[577,1312,1314],{"className":1313},[620],[577,1315,861],{"className":1316},[860],[577,1318,1320],{"className":1319},[620],"50",[577,1322],{"className":1323,"style":936},[671],[577,1325,1327],{"className":1326},[620,940],[577,1328,899],{"className":1329},[620],[564,1331,1332,1333,1376],{},"Ein Baumstamm hat den Umfang ",[577,1334,1336,1355],{"className":1335},[580],[577,1337,1339],{"className":1338},[584],[586,1340,1341],{"xmlns":588},[590,1342,1343,1352],{},[593,1344,1345,1348,1350],{},[647,1346,1347],{},"100",[894,1349,896],{},[894,1351,899],{},[600,1353,1354],{"encoding":602},"100\\,\\text{cm}",[577,1356,1358],{"className":1357,"ariaHidden":607},[606],[577,1359,1361,1364,1367,1370],{"className":1360},[611],[577,1362],{"className":1363,"style":686},[615],[577,1365,1347],{"className":1366},[620],[577,1368],{"className":1369,"style":936},[671],[577,1371,1373],{"className":1372},[620,940],[577,1374,899],{"className":1375},[620],". Gesucht ist der Durchmesser:",[577,1378,1380],{"className":1379},[697],[577,1381,1383,1429],{"className":1382},[580],[577,1384,1386],{"className":1385},[584],[586,1387,1388],{"xmlns":588,"display":706},[590,1389,1390,1426],{},[593,1391,1392,1394,1396,1403,1405,1417,1419,1422,1424],{},[596,1393,641],{},[643,1395,645],{},[1397,1398,1399,1401],"mfrac",{},[596,1400,713],{},[596,1402,720],{},[643,1404,645],{},[1397,1406,1407,1415],{},[593,1408,1409,1411,1413],{},[647,1410,1347],{},[894,1412,896],{},[894,1414,899],{},[596,1416,720],{},[643,1418,814],{},[647,1420,1421],{},"31,8",[894,1423,896],{},[894,1425,899],{},[600,1427,1428],{"encoding":602},"d = \\frac{U}{\\pi} = \\frac{100\\,\\text{cm}}{\\pi} \\approx 31{,}8\\,\\text{cm}",[577,1430,1432,1450,1546,1634],{"className":1431,"ariaHidden":607},[606],[577,1433,1435,1438,1441,1444,1447],{"className":1434},[611],[577,1436],{"className":1437,"style":664},[615],[577,1439,641],{"className":1440},[620,621],[577,1442],{"className":1443,"style":672},[671],[577,1445,645],{"className":1446},[676],[577,1448],{"className":1449,"style":672},[671],[577,1451,1453,1457,1537,1540,1543],{"className":1452},[611],[577,1454],{"className":1455,"style":1456},[615],"height:2.0463em;vertical-align:-0.686em;",[577,1458,1460,1465,1533],{"className":1459},[620],[577,1461],{"className":1462},[1463,1464],"mopen","nulldelimiter",[577,1466,1468],{"className":1467},[1397],[577,1469,1473,1524],{"className":1470},[1471,1472],"vlist-t","vlist-t2",[577,1474,1477,1519],{"className":1475},[1476],"vlist-r",[577,1478,1482,1496,1507],{"className":1479,"style":1481},[1480],"vlist","height:1.3603em;",[577,1483,1485,1490],{"style":1484},"top:-2.314em;",[577,1486],{"className":1487,"style":1489},[1488],"pstrut","height:3em;",[577,1491,1493],{"className":1492},[620],[577,1494,720],{"className":1495,"style":767},[620,621],[577,1497,1499,1502],{"style":1498},"top:-3.23em;",[577,1500],{"className":1501,"style":1489},[1488],[577,1503],{"className":1504,"style":1506},[1505],"frac-line","border-bottom-width:0.04em;",[577,1508,1510,1513],{"style":1509},"top:-3.677em;",[577,1511],{"className":1512,"style":1489},[1488],[577,1514,1516],{"className":1515},[620],[577,1517,713],{"className":1518,"style":745},[620,621],[577,1520,1523],{"className":1521},[1522],"vlist-s","​",[577,1525,1527],{"className":1526},[1476],[577,1528,1531],{"className":1529,"style":1530},[1480],"height:0.686em;",[577,1532],{},[577,1534],{"className":1535},[1536,1464],"mclose",[577,1538],{"className":1539,"style":672},[671],[577,1541,645],{"className":1542},[676],[577,1544],{"className":1545,"style":672},[671],[577,1547,1549,1553,1625,1628,1631],{"className":1548},[611],[577,1550],{"className":1551,"style":1552},[615],"height:2.0074em;vertical-align:-0.686em;",[577,1554,1556,1559,1622],{"className":1555},[620],[577,1557],{"className":1558},[1463,1464],[577,1560,1562],{"className":1561},[1397],[577,1563,1565,1614],{"className":1564},[1471,1472],[577,1566,1568,1611],{"className":1567},[1476],[577,1569,1572,1583,1591],{"className":1570,"style":1571},[1480],"height:1.3214em;",[577,1573,1574,1577],{"style":1484},[577,1575],{"className":1576,"style":1489},[1488],[577,1578,1580],{"className":1579},[620],[577,1581,720],{"className":1582,"style":767},[620,621],[577,1584,1585,1588],{"style":1498},[577,1586],{"className":1587,"style":1489},[1488],[577,1589],{"className":1590,"style":1506},[1505],[577,1592,1593,1596],{"style":1509},[577,1594],{"className":1595,"style":1489},[1488],[577,1597,1599,1602,1605],{"className":1598},[620],[577,1600,1347],{"className":1601},[620],[577,1603],{"className":1604,"style":936},[671],[577,1606,1608],{"className":1607},[620,940],[577,1609,899],{"className":1610},[620],[577,1612,1523],{"className":1613},[1522],[577,1615,1617],{"className":1616},[1476],[577,1618,1620],{"className":1619,"style":1530},[1480],[577,1621],{},[577,1623],{"className":1624},[1536,1464],[577,1626],{"className":1627,"style":672},[671],[577,1629,814],{"className":1630},[676],[577,1632],{"className":1633,"style":672},[671],[577,1635,1637,1640,1644,1650,1653,1656],{"className":1636},[611],[577,1638],{"className":1639,"style":849},[615],[577,1641,1643],{"className":1642},[620],"31",[577,1645,1647],{"className":1646},[620],[577,1648,861],{"className":1649},[860],[577,1651,982],{"className":1652},[620],[577,1654],{"className":1655,"style":936},[671],[577,1657,1659],{"className":1658},[620,940],[577,1660,899],{"className":1661},[620],[559,1663,1665],{"id":1664},"typische-fehler","Typische Fehler",[1667,1668,1669,1822,1922],"ul",{},[1670,1671,1672,1706,1707,1746,1747,1781,1782,1821],"li",{},[577,1673,1675,1691],{"className":1674},[580],[577,1676,1678],{"className":1677},[584],[586,1679,1680],{"xmlns":588},[590,1681,1682,1688],{},[593,1683,1684,1686],{},[647,1685,649],{},[596,1687,720],{},[600,1689,1690],{"encoding":602},"2\\pi",[577,1692,1694],{"className":1693,"ariaHidden":607},[606],[577,1695,1697,1700,1703],{"className":1696},[611],[577,1698],{"className":1699,"style":686},[615],[577,1701,649],{"className":1702},[620],[577,1704,720],{"className":1705,"style":767},[620,621]," mit dem Durchmesser multiplizieren. Entweder ",[577,1708,1710,1728],{"className":1709},[580],[577,1711,1713],{"className":1712},[584],[586,1714,1715],{"xmlns":588},[590,1716,1717,1725],{},[593,1718,1719,1721,1723],{},[647,1720,649],{},[596,1722,720],{},[596,1724,598],{},[600,1726,1727],{"encoding":602},"2\\pi r",[577,1729,1731],{"className":1730,"ariaHidden":607},[606],[577,1732,1734,1737,1740,1743],{"className":1733},[611],[577,1735],{"className":1736,"style":686},[615],[577,1738,649],{"className":1739},[620],[577,1741,720],{"className":1742,"style":767},[620,621],[577,1744,598],{"className":1745,"style":622},[620,621]," oder ",[577,1748,1750,1766],{"className":1749},[580],[577,1751,1753],{"className":1752},[584],[586,1754,1755],{"xmlns":588},[590,1756,1757,1763],{},[593,1758,1759,1761],{},[596,1760,720],{},[596,1762,641],{},[600,1764,1765],{"encoding":602},"\\pi d",[577,1767,1769],{"className":1768,"ariaHidden":607},[606],[577,1770,1772,1775,1778],{"className":1771},[611],[577,1773],{"className":1774,"style":664},[615],[577,1776,720],{"className":1777,"style":767},[620,621],[577,1779,641],{"className":1780},[620,621],", nicht ",[577,1783,1785,1803],{"className":1784},[580],[577,1786,1788],{"className":1787},[584],[586,1789,1790],{"xmlns":588},[590,1791,1792,1800],{},[593,1793,1794,1796,1798],{},[647,1795,649],{},[596,1797,720],{},[596,1799,641],{},[600,1801,1802],{"encoding":602},"2\\pi d",[577,1804,1806],{"className":1805,"ariaHidden":607},[606],[577,1807,1809,1812,1815,1818],{"className":1808},[611],[577,1810],{"className":1811,"style":664},[615],[577,1813,649],{"className":1814},[620],[577,1816,720],{"className":1817,"style":767},[620,621],[577,1819,641],{"className":1820},[620,621],".",[1670,1823,1824,1825,1921],{},"Umfang und Flächeninhalt verwechseln. Der Flächeninhalt ist ",[577,1826,1828,1854],{"className":1827},[580],[577,1829,1831],{"className":1830},[584],[586,1832,1833],{"xmlns":588},[590,1834,1835,1851],{},[593,1836,1837,1840,1842,1844],{},[596,1838,1839],{},"A",[643,1841,645],{},[596,1843,720],{},[1845,1846,1847,1849],"msup",{},[596,1848,598],{},[647,1850,649],{},[600,1852,1853],{"encoding":602},"A = \\pi r^2",[577,1855,1857,1875],{"className":1856,"ariaHidden":607},[606],[577,1858,1860,1863,1866,1869,1872],{"className":1859},[611],[577,1861],{"className":1862,"style":741},[615],[577,1864,1839],{"className":1865},[620,621],[577,1867],{"className":1868,"style":672},[671],[577,1870,645],{"className":1871},[676],[577,1873],{"className":1874,"style":672},[671],[577,1876,1878,1882,1885],{"className":1877},[611],[577,1879],{"className":1880,"style":1881},[615],"height:0.8141em;",[577,1883,720],{"className":1884,"style":767},[620,621],[577,1886,1888,1891],{"className":1887},[620],[577,1889,598],{"className":1890,"style":622},[620,621],[577,1892,1895],{"className":1893},[1894],"msupsub",[577,1896,1898],{"className":1897},[1471],[577,1899,1901],{"className":1900},[1476],[577,1902,1904],{"className":1903,"style":1881},[1480],[577,1905,1907,1911],{"style":1906},"top:-3.063em;margin-right:0.05em;",[577,1908],{"className":1909,"style":1910},[1488],"height:2.7em;",[577,1912,1918],{"className":1913},[1914,1915,1916,1917],"sizing","reset-size6","size3","mtight",[577,1919,649],{"className":1920},[620,1917]," und hat eine Flächeneinheit.",[1670,1923,1924,1925,1976],{},"Zu früh runden, z.B. mit ",[577,1926,1928,1946],{"className":1927},[580],[577,1929,1931],{"className":1930},[584],[586,1932,1933],{"xmlns":588},[590,1934,1935,1943],{},[593,1936,1937,1939,1941],{},[596,1938,720],{},[643,1940,645],{},[647,1942,853],{},[600,1944,1945],{"encoding":602},"\\pi = 3",[577,1947,1949,1967],{"className":1948,"ariaHidden":607},[606],[577,1950,1952,1955,1958,1961,1964],{"className":1951},[611],[577,1953],{"className":1954,"style":616},[615],[577,1956,720],{"className":1957,"style":767},[620,621],[577,1959],{"className":1960,"style":672},[671],[577,1962,645],{"className":1963},[676],[577,1965],{"className":1966,"style":672},[671],[577,1968,1970,1973],{"className":1969},[611],[577,1971],{"className":1972,"style":686},[615],[577,1974,853],{"className":1975},[620],". Erst am Ende runden.",{"title":1978,"searchDepth":1979,"depth":1979,"links":1980},"",2,[1981,1982,1983],{"id":561,"depth":1979,"text":562},{"id":869,"depth":1979,"text":870},{"id":1664,"depth":1979,"text":1665},[1985,1988,1991,1994,1997],{"lhs":1986,"rhs":1987},"2 * pi * r","pi * (2*r)",{"lhs":1989,"rhs":1990},"2 * pi * 4","8 * pi",{"lhs":1992,"rhs":1993},"round(8 * pi, 2)","25.13",{"lhs":1995,"rhs":1996},"round(60 * pi, 2)","188.5",{"lhs":1998,"rhs":1999},"round(100 \u002F pi, 1)","31.8",null,"md",{"caption":2003,"x":2004,"y":2007,"items":2008},"Kreis mit Mittelpunkt M und Radius r, der Umfang U ist die Länge der Kreislinie",[2005,2006],-2.6,2.6,[2005,2006],[2009,2014,2019,2024,2027],{"type":2010,"center":2011,"r":2013},"circle",[2012,2012],0,2.2,{"type":2015,"from":2016,"to":2017,"style":2018},"segment",[2012,2012],[2013,2012],"muted",{"type":2020,"at":2021,"text":598},"label",[2022,2023],1.1,0.3,{"type":2020,"at":2025,"text":713},[2026,2026],1.86,{"type":2028,"at":2029,"label":2030,"offset":2031},"point",[2012,2012],"M",[2032,2033],-6,16,"\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 249.6 249.6\" width=\"249.6\" height=\"249.6\" role=\"img\" aria-label=\"Kreis mit Mittelpunkt M und Radius r, der Umfang U ist die Länge der Kreislinie\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-circle-circumference-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-circle-circumference-clip\">\u003Crect x=\"0\" y=\"0\" width=\"249.6\" height=\"249.6\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Ccircle cx=\"124.8\" cy=\"124.8\" r=\"105.6\" class=\"fig-main\"\u002F>\u003Cline x1=\"124.8\" y1=\"124.8\" x2=\"230.4\" y2=\"124.8\" class=\"fig-muted\"\u002F>\u003Ctext x=\"177.6\" y=\"110.4\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">r\u003C\u002Ftext>\u003Ctext x=\"214.08\" y=\"35.52\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">U\u003C\u002Ftext>\u003Ccircle cx=\"124.8\" cy=\"124.8\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"118.8\" y=\"140.8\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">M\u003C\u002Ftext>\u003C\u002Fsvg>","circle-circumference","U = 2 \\pi r","\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>U\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmi>π\u003C\u002Fmi>\u003Cmi>r\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">U = 2 \\pi r\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\" style=\"margin-right:0.109em;\">U\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\">2\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","middle",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-kreis",[],[2044,2045,2046],"circle-area","arc-length","cylinder-surface",{"title":548,"description":1978},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-kreis","Der Umfang eines Kreises ist 2 mal π mal Radius, also π mal Durchmesser.",[2051,2052,2053,2054,2055],"umfang","kreis","radius","durchmesser","kreiszahl pi",[2057,2060,2063],{"symbol":713,"meaning":2058,"symbolHtml":2059},"Umfang des Kreises, die Länge der Kreislinie","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>U\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">U\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\" style=\"margin-right:0.109em;\">U\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":598,"meaning":2061,"symbolHtml":2062},"Radius, der Abstand vom Mittelpunkt zum Rand","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>r\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">r\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:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":2064,"meaning":2065,"symbolHtml":2066},"\\pi","Kreiszahl, ungefähr 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>","9R-dBSrsDgmbXQdTcz9bKT45OawYOQm2WEXUNOzV46o",[2069,2072,2075],{"stem":2070,"title":2071,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fflaecheninhalt-kreis","Flächeninhalt eines Kreises",{"stem":2073,"title":2074,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fkreisbogen","Länge eines Kreisbogens",{"stem":2076,"title":2077,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Foberflaeche-zylinder","Oberfläche eines Zylinders",1793824270253]