[{"data":1,"prerenderedAt":2060},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-dreieck":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-dreieck":2047},[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":1953,"conditions":1966,"conditionsHtml":1966,"deprecated":1966,"description":1947,"extension":1967,"figure":1968,"figureCaption":1969,"figureHtml":2018,"formulaId":2019,"latex":552,"latexHtml":2020,"level":2021,"meta":2022,"navigation":2023,"path":2024,"redirectFrom":2025,"related":2026,"seo":2031,"stem":2032,"subject":598,"summary":2033,"tags":2034,"variables":2038,"__hash__":2046},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-dreieck.md","Umfang eines Dreiecks",[550,551,552,553,554],"Umfang Dreieck","Umfang Dreieck berechnen","U = a + b + c","Dreieck Umfang Formel","Umfang gleichseitiges Dreieck",{"type":556,"value":557,"toc":1946},"minimark",[558,563,572,575,578,823,827,966,1141,1187,1313,1446,1620,1624],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,571],"p",{},"Der ",[568,569,570],"strong",{},"Umfang"," ist die Länge des Randes einer Figur. Bei einem Dreieck besteht der Rand aus den drei Seiten. Man addiert also einfach alle drei Seitenlängen.",[564,573,574],{},"Die Formel gilt für jedes Dreieck, egal ob es spitz, stumpf oder rechtwinklig ist.",[564,576,577],{},"Sonderfälle:",[579,580,581,673],"ul",{},[582,583,584,587,588,672],"li",{},[568,585,586],{},"Gleichseitiges Dreieck",": Alle Seiten sind gleich lang, also ",[589,590,593,627],"span",{"className":591},[592],"katex",[589,594,597],{"className":595},[596],"katex-mathml",[598,599,601],"math",{"xmlns":600},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[602,603,604,622],"semantics",{},[605,606,607,611,615,619],"mrow",{},[608,609,610],"mi",{},"U",[612,613,614],"mo",{},"=",[616,617,618],"mn",{},"3",[608,620,621],{},"a",[623,624,626],"annotation",{"encoding":625},"application\u002Fx-tex","U = 3a",[589,628,632,659],{"className":629,"ariaHidden":631},[630],"katex-html","true",[589,633,636,641,647,652,656],{"className":634},[635],"base",[589,637],{"className":638,"style":640},[639],"strut","height:0.6833em;",[589,642,610],{"className":643,"style":646},[644,645],"mord","mathnormal","margin-right:0.109em;",[589,648],{"className":649,"style":651},[650],"mspace","margin-right:0.2778em;",[589,653,614],{"className":654},[655],"mrel",[589,657],{"className":658,"style":651},[650],[589,660,662,666,669],{"className":661},[635],[589,663],{"className":664,"style":665},[639],"height:0.6444em;",[589,667,618],{"className":668},[644],[589,670,621],{"className":671},[644,645],".",[582,674,675,678,679,764,765,793,794,822],{},[568,676,677],{},"Gleichschenkliges Dreieck",": Zwei Seiten (die Schenkel) sind gleich lang, also ",[589,680,682,709],{"className":681},[592],[589,683,685],{"className":684},[596],[598,686,687],{"xmlns":600},[602,688,689,706],{},[605,690,691,693,695,698,700,703],{},[608,692,610],{},[612,694,614],{},[616,696,697],{},"2",[608,699,621],{},[612,701,702],{},"+",[608,704,705],{},"c",[623,707,708],{"encoding":625},"U = 2a + c",[589,710,712,730,754],{"className":711,"ariaHidden":631},[630],[589,713,715,718,721,724,727],{"className":714},[635],[589,716],{"className":717,"style":640},[639],[589,719,610],{"className":720,"style":646},[644,645],[589,722],{"className":723,"style":651},[650],[589,725,614],{"className":726},[655],[589,728],{"className":729,"style":651},[650],[589,731,733,737,740,743,747,751],{"className":732},[635],[589,734],{"className":735,"style":736},[639],"height:0.7278em;vertical-align:-0.0833em;",[589,738,697],{"className":739},[644],[589,741,621],{"className":742},[644,645],[589,744],{"className":745,"style":746},[650],"margin-right:0.2222em;",[589,748,702],{"className":749},[750],"mbin",[589,752],{"className":753,"style":746},[650],[589,755,757,761],{"className":756},[635],[589,758],{"className":759,"style":760},[639],"height:0.4306em;",[589,762,705],{"className":763},[644,645],", wenn ",[589,766,768,781],{"className":767},[592],[589,769,771],{"className":770},[596],[598,772,773],{"xmlns":600},[602,774,775,779],{},[605,776,777],{},[608,778,621],{},[623,780,621],{"encoding":625},[589,782,784],{"className":783,"ariaHidden":631},[630],[589,785,787,790],{"className":786},[635],[589,788],{"className":789,"style":760},[639],[589,791,621],{"className":792},[644,645]," die Schenkel und ",[589,795,797,810],{"className":796},[592],[589,798,800],{"className":799},[596],[598,801,802],{"xmlns":600},[602,803,804,808],{},[605,805,806],{},[608,807,705],{},[623,809,705],{"encoding":625},[589,811,813],{"className":812,"ariaHidden":631},[630],[589,814,816,819],{"className":815},[635],[589,817],{"className":818,"style":760},[639],[589,820,705],{"className":821},[644,645]," die Basis ist.",[559,824,826],{"id":825},"beispiele","Beispiele",[564,828,829,830,877,878,921,922,965],{},"Ein Dreieck hat die Seiten ",[589,831,833,854],{"className":832},[592],[589,834,836],{"className":835},[596],[598,837,838],{"xmlns":600},[602,839,840,851],{},[605,841,842,844,848],{},[616,843,618],{},[845,846,847],"mtext",{}," ",[845,849,850],{},"cm",[623,852,853],{"encoding":625},"3\\,\\text{cm}",[589,855,857],{"className":856,"ariaHidden":631},[630],[589,858,860,863,866,870],{"className":859},[635],[589,861],{"className":862,"style":665},[639],[589,864,618],{"className":865},[644],[589,867],{"className":868,"style":869},[650],"margin-right:0.1667em;",[589,871,874],{"className":872},[644,873],"text",[589,875,850],{"className":876},[644],", ",[589,879,881,900],{"className":880},[592],[589,882,884],{"className":883},[596],[598,885,886],{"xmlns":600},[602,887,888,897],{},[605,889,890,893,895],{},[616,891,892],{},"4",[845,894,847],{},[845,896,850],{},[623,898,899],{"encoding":625},"4\\,\\text{cm}",[589,901,903],{"className":902,"ariaHidden":631},[630],[589,904,906,909,912,915],{"className":905},[635],[589,907],{"className":908,"style":665},[639],[589,910,892],{"className":911},[644],[589,913],{"className":914,"style":869},[650],[589,916,918],{"className":917},[644,873],[589,919,850],{"className":920},[644]," und ",[589,923,925,944],{"className":924},[592],[589,926,928],{"className":927},[596],[598,929,930],{"xmlns":600},[602,931,932,941],{},[605,933,934,937,939],{},[616,935,936],{},"5",[845,938,847],{},[845,940,850],{},[623,942,943],{"encoding":625},"5\\,\\text{cm}",[589,945,947],{"className":946,"ariaHidden":631},[630],[589,948,950,953,956,959],{"className":949},[635],[589,951],{"className":952,"style":665},[639],[589,954,936],{"className":955},[644],[589,957],{"className":958,"style":869},[650],[589,960,962],{"className":961},[644,873],[589,963,850],{"className":964},[644],":",[589,967,970],{"className":968},[969],"katex-display",[589,971,973,1021],{"className":972},[592],[589,974,976],{"className":975},[596],[598,977,979],{"xmlns":600,"display":978},"block",[602,980,981,1018],{},[605,982,983,985,987,989,991,993,995,997,999,1001,1003,1005,1007,1009,1011,1014,1016],{},[608,984,610],{},[612,986,614],{},[616,988,618],{},[845,990,847],{},[845,992,850],{},[612,994,702],{},[616,996,892],{},[845,998,847],{},[845,1000,850],{},[612,1002,702],{},[616,1004,936],{},[845,1006,847],{},[845,1008,850],{},[612,1010,614],{},[616,1012,1013],{},"12",[845,1015,847],{},[845,1017,850],{},[623,1019,1020],{"encoding":625},"U = 3\\,\\text{cm} + 4\\,\\text{cm} + 5\\,\\text{cm} = 12\\,\\text{cm}",[589,1022,1024,1042,1069,1096,1123],{"className":1023,"ariaHidden":631},[630],[589,1025,1027,1030,1033,1036,1039],{"className":1026},[635],[589,1028],{"className":1029,"style":640},[639],[589,1031,610],{"className":1032,"style":646},[644,645],[589,1034],{"className":1035,"style":651},[650],[589,1037,614],{"className":1038},[655],[589,1040],{"className":1041,"style":651},[650],[589,1043,1045,1048,1051,1054,1060,1063,1066],{"className":1044},[635],[589,1046],{"className":1047,"style":736},[639],[589,1049,618],{"className":1050},[644],[589,1052],{"className":1053,"style":869},[650],[589,1055,1057],{"className":1056},[644,873],[589,1058,850],{"className":1059},[644],[589,1061],{"className":1062,"style":746},[650],[589,1064,702],{"className":1065},[750],[589,1067],{"className":1068,"style":746},[650],[589,1070,1072,1075,1078,1081,1087,1090,1093],{"className":1071},[635],[589,1073],{"className":1074,"style":736},[639],[589,1076,892],{"className":1077},[644],[589,1079],{"className":1080,"style":869},[650],[589,1082,1084],{"className":1083},[644,873],[589,1085,850],{"className":1086},[644],[589,1088],{"className":1089,"style":746},[650],[589,1091,702],{"className":1092},[750],[589,1094],{"className":1095,"style":746},[650],[589,1097,1099,1102,1105,1108,1114,1117,1120],{"className":1098},[635],[589,1100],{"className":1101,"style":665},[639],[589,1103,936],{"className":1104},[644],[589,1106],{"className":1107,"style":869},[650],[589,1109,1111],{"className":1110},[644,873],[589,1112,850],{"className":1113},[644],[589,1115],{"className":1116,"style":651},[650],[589,1118,614],{"className":1119},[655],[589,1121],{"className":1122,"style":651},[650],[589,1124,1126,1129,1132,1135],{"className":1125},[635],[589,1127],{"className":1128,"style":665},[639],[589,1130,1013],{"className":1131},[644],[589,1133],{"className":1134,"style":869},[650],[589,1136,1138],{"className":1137},[644,873],[589,1139,850],{"className":1140},[644],[564,1142,1143,1144,965],{},"Ein gleichseitiges Dreieck hat die Seitenlänge ",[589,1145,1147,1166],{"className":1146},[592],[589,1148,1150],{"className":1149},[596],[598,1151,1152],{"xmlns":600},[602,1153,1154,1163],{},[605,1155,1156,1159,1161],{},[616,1157,1158],{},"7",[845,1160,847],{},[845,1162,850],{},[623,1164,1165],{"encoding":625},"7\\,\\text{cm}",[589,1167,1169],{"className":1168,"ariaHidden":631},[630],[589,1170,1172,1175,1178,1181],{"className":1171},[635],[589,1173],{"className":1174,"style":665},[639],[589,1176,1158],{"className":1177},[644],[589,1179],{"className":1180,"style":869},[650],[589,1182,1184],{"className":1183},[644,873],[589,1185,850],{"className":1186},[644],[589,1188,1190],{"className":1189},[969],[589,1191,1193,1229],{"className":1192},[592],[589,1194,1196],{"className":1195},[596],[598,1197,1198],{"xmlns":600,"display":978},[602,1199,1200,1226],{},[605,1201,1202,1204,1206,1208,1211,1213,1215,1217,1219,1222,1224],{},[608,1203,610],{},[612,1205,614],{},[616,1207,618],{},[612,1209,1210],{},"⋅",[616,1212,1158],{},[845,1214,847],{},[845,1216,850],{},[612,1218,614],{},[616,1220,1221],{},"21",[845,1223,847],{},[845,1225,850],{},[623,1227,1228],{"encoding":625},"U = 3 \\cdot 7\\,\\text{cm} = 21\\,\\text{cm}",[589,1230,1232,1250,1268,1295],{"className":1231,"ariaHidden":631},[630],[589,1233,1235,1238,1241,1244,1247],{"className":1234},[635],[589,1236],{"className":1237,"style":640},[639],[589,1239,610],{"className":1240,"style":646},[644,645],[589,1242],{"className":1243,"style":651},[650],[589,1245,614],{"className":1246},[655],[589,1248],{"className":1249,"style":651},[650],[589,1251,1253,1256,1259,1262,1265],{"className":1252},[635],[589,1254],{"className":1255,"style":665},[639],[589,1257,618],{"className":1258},[644],[589,1260],{"className":1261,"style":746},[650],[589,1263,1210],{"className":1264},[750],[589,1266],{"className":1267,"style":746},[650],[589,1269,1271,1274,1277,1280,1286,1289,1292],{"className":1270},[635],[589,1272],{"className":1273,"style":665},[639],[589,1275,1158],{"className":1276},[644],[589,1278],{"className":1279,"style":869},[650],[589,1281,1283],{"className":1282},[644,873],[589,1284,850],{"className":1285},[644],[589,1287],{"className":1288,"style":651},[650],[589,1290,614],{"className":1291},[655],[589,1293],{"className":1294,"style":651},[650],[589,1296,1298,1301,1304,1307],{"className":1297},[635],[589,1299],{"className":1300,"style":665},[639],[589,1302,1221],{"className":1303},[644],[589,1305],{"className":1306,"style":869},[650],[589,1308,1310],{"className":1309},[644,873],[589,1311,850],{"className":1312},[644],[564,1314,1315,1316,1359,1360,921,1403,1445],{},"Ein Dreieck hat den Umfang ",[589,1317,1319,1338],{"className":1318},[592],[589,1320,1322],{"className":1321},[596],[598,1323,1324],{"xmlns":600},[602,1325,1326,1335],{},[605,1327,1328,1331,1333],{},[616,1329,1330],{},"30",[845,1332,847],{},[845,1334,850],{},[623,1336,1337],{"encoding":625},"30\\,\\text{cm}",[589,1339,1341],{"className":1340,"ariaHidden":631},[630],[589,1342,1344,1347,1350,1353],{"className":1343},[635],[589,1345],{"className":1346,"style":665},[639],[589,1348,1330],{"className":1349},[644],[589,1351],{"className":1352,"style":869},[650],[589,1354,1356],{"className":1355},[644,873],[589,1357,850],{"className":1358},[644],". Zwei Seiten sind ",[589,1361,1363,1382],{"className":1362},[592],[589,1364,1366],{"className":1365},[596],[598,1367,1368],{"xmlns":600},[602,1369,1370,1379],{},[605,1371,1372,1375,1377],{},[616,1373,1374],{},"8",[845,1376,847],{},[845,1378,850],{},[623,1380,1381],{"encoding":625},"8\\,\\text{cm}",[589,1383,1385],{"className":1384,"ariaHidden":631},[630],[589,1386,1388,1391,1394,1397],{"className":1387},[635],[589,1389],{"className":1390,"style":665},[639],[589,1392,1374],{"className":1393},[644],[589,1395],{"className":1396,"style":869},[650],[589,1398,1400],{"className":1399},[644,873],[589,1401,850],{"className":1402},[644],[589,1404,1406,1424],{"className":1405},[592],[589,1407,1409],{"className":1408},[596],[598,1410,1411],{"xmlns":600},[602,1412,1413,1421],{},[605,1414,1415,1417,1419],{},[616,1416,1013],{},[845,1418,847],{},[845,1420,850],{},[623,1422,1423],{"encoding":625},"12\\,\\text{cm}",[589,1425,1427],{"className":1426,"ariaHidden":631},[630],[589,1428,1430,1433,1436,1439],{"className":1429},[635],[589,1431],{"className":1432,"style":665},[639],[589,1434,1013],{"className":1435},[644],[589,1437],{"className":1438,"style":869},[650],[589,1440,1442],{"className":1441},[644,873],[589,1443,850],{"className":1444},[644]," lang. Die dritte Seite ist:",[589,1447,1449],{"className":1448},[969],[589,1450,1452,1500],{"className":1451},[592],[589,1453,1455],{"className":1454},[596],[598,1456,1457],{"xmlns":600,"display":978},[602,1458,1459,1497],{},[605,1460,1461,1463,1465,1467,1469,1471,1474,1476,1478,1480,1482,1484,1486,1488,1490,1493,1495],{},[608,1462,705],{},[612,1464,614],{},[616,1466,1330],{},[845,1468,847],{},[845,1470,850],{},[612,1472,1473],{},"−",[616,1475,1374],{},[845,1477,847],{},[845,1479,850],{},[612,1481,1473],{},[616,1483,1013],{},[845,1485,847],{},[845,1487,850],{},[612,1489,614],{},[616,1491,1492],{},"10",[845,1494,847],{},[845,1496,850],{},[623,1498,1499],{"encoding":625},"c = 30\\,\\text{cm} - 8\\,\\text{cm} - 12\\,\\text{cm} = 10\\,\\text{cm}",[589,1501,1503,1521,1548,1575,1602],{"className":1502,"ariaHidden":631},[630],[589,1504,1506,1509,1512,1515,1518],{"className":1505},[635],[589,1507],{"className":1508,"style":760},[639],[589,1510,705],{"className":1511},[644,645],[589,1513],{"className":1514,"style":651},[650],[589,1516,614],{"className":1517},[655],[589,1519],{"className":1520,"style":651},[650],[589,1522,1524,1527,1530,1533,1539,1542,1545],{"className":1523},[635],[589,1525],{"className":1526,"style":736},[639],[589,1528,1330],{"className":1529},[644],[589,1531],{"className":1532,"style":869},[650],[589,1534,1536],{"className":1535},[644,873],[589,1537,850],{"className":1538},[644],[589,1540],{"className":1541,"style":746},[650],[589,1543,1473],{"className":1544},[750],[589,1546],{"className":1547,"style":746},[650],[589,1549,1551,1554,1557,1560,1566,1569,1572],{"className":1550},[635],[589,1552],{"className":1553,"style":736},[639],[589,1555,1374],{"className":1556},[644],[589,1558],{"className":1559,"style":869},[650],[589,1561,1563],{"className":1562},[644,873],[589,1564,850],{"className":1565},[644],[589,1567],{"className":1568,"style":746},[650],[589,1570,1473],{"className":1571},[750],[589,1573],{"className":1574,"style":746},[650],[589,1576,1578,1581,1584,1587,1593,1596,1599],{"className":1577},[635],[589,1579],{"className":1580,"style":665},[639],[589,1582,1013],{"className":1583},[644],[589,1585],{"className":1586,"style":869},[650],[589,1588,1590],{"className":1589},[644,873],[589,1591,850],{"className":1592},[644],[589,1594],{"className":1595,"style":651},[650],[589,1597,614],{"className":1598},[655],[589,1600],{"className":1601,"style":651},[650],[589,1603,1605,1608,1611,1614],{"className":1604},[635],[589,1606],{"className":1607,"style":665},[639],[589,1609,1492],{"className":1610},[644],[589,1612],{"className":1613,"style":869},[650],[589,1615,1617],{"className":1616},[644,873],[589,1618,850],{"className":1619},[644],[559,1621,1623],{"id":1622},"typische-fehler","Typische Fehler",[579,1625,1626,1629,1632],{},[582,1627,1628],{},"Eine Seite vergessen, weil sie in der Zeichnung nicht beschriftet ist. Ein Dreieck hat immer drei Seiten.",[582,1630,1631],{},"Die Höhe zum Umfang addieren. Die Höhe liegt im Inneren des Dreiecks und gehört nicht zum Rand.",[582,1633,1634,1635,672],{},"Verschiedene Einheiten mischen. Erst umrechnen: ",[589,1636,1638,1711],{"className":1637},[592],[589,1639,1641],{"className":1640},[596],[598,1642,1643],{"xmlns":600},[602,1644,1645,1708],{},[605,1646,1647,1650,1652,1654,1656,1659,1661,1664,1666,1669,1671,1673,1675,1678,1680,1682,1684,1686,1688,1690,1692,1695,1697,1699,1701,1704,1706],{},[616,1648,1649],{},"50",[845,1651,847],{},[845,1653,850],{},[612,1655,702],{},[616,1657,1658],{},"0,8",[845,1660,847],{},[845,1662,1663],{},"m",[612,1665,702],{},[616,1667,1668],{},"60",[845,1670,847],{},[845,1672,850],{},[612,1674,614],{},[616,1676,1677],{},"0,5",[845,1679,847],{},[845,1681,1663],{},[612,1683,702],{},[616,1685,1658],{},[845,1687,847],{},[845,1689,1663],{},[612,1691,702],{},[616,1693,1694],{},"0,6",[845,1696,847],{},[845,1698,1663],{},[612,1700,614],{},[616,1702,1703],{},"1,9",[845,1705,847],{},[845,1707,1663],{},[623,1709,1710],{"encoding":625},"50\\,\\text{cm} + 0{,}8\\,\\text{m} + 60\\,\\text{cm} = 0{,}5\\,\\text{m} + 0{,}8\\,\\text{m} + 0{,}6\\,\\text{m} = 1{,}9\\,\\text{m}",[589,1712,1714,1741,1781,1808,1844,1880,1917],{"className":1713,"ariaHidden":631},[630],[589,1715,1717,1720,1723,1726,1732,1735,1738],{"className":1716},[635],[589,1718],{"className":1719,"style":736},[639],[589,1721,1649],{"className":1722},[644],[589,1724],{"className":1725,"style":869},[650],[589,1727,1729],{"className":1728},[644,873],[589,1730,850],{"className":1731},[644],[589,1733],{"className":1734,"style":746},[650],[589,1736,702],{"className":1737},[750],[589,1739],{"className":1740,"style":746},[650],[589,1742,1744,1748,1752,1760,1763,1766,1772,1775,1778],{"className":1743},[635],[589,1745],{"className":1746,"style":1747},[639],"height:0.8389em;vertical-align:-0.1944em;",[589,1749,1751],{"className":1750},[644],"0",[589,1753,1755],{"className":1754},[644],[589,1756,1759],{"className":1757},[1758],"mpunct",",",[589,1761,1374],{"className":1762},[644],[589,1764],{"className":1765,"style":869},[650],[589,1767,1769],{"className":1768},[644,873],[589,1770,1663],{"className":1771},[644],[589,1773],{"className":1774,"style":746},[650],[589,1776,702],{"className":1777},[750],[589,1779],{"className":1780,"style":746},[650],[589,1782,1784,1787,1790,1793,1799,1802,1805],{"className":1783},[635],[589,1785],{"className":1786,"style":665},[639],[589,1788,1668],{"className":1789},[644],[589,1791],{"className":1792,"style":869},[650],[589,1794,1796],{"className":1795},[644,873],[589,1797,850],{"className":1798},[644],[589,1800],{"className":1801,"style":651},[650],[589,1803,614],{"className":1804},[655],[589,1806],{"className":1807,"style":651},[650],[589,1809,1811,1814,1817,1823,1826,1829,1835,1838,1841],{"className":1810},[635],[589,1812],{"className":1813,"style":1747},[639],[589,1815,1751],{"className":1816},[644],[589,1818,1820],{"className":1819},[644],[589,1821,1759],{"className":1822},[1758],[589,1824,936],{"className":1825},[644],[589,1827],{"className":1828,"style":869},[650],[589,1830,1832],{"className":1831},[644,873],[589,1833,1663],{"className":1834},[644],[589,1836],{"className":1837,"style":746},[650],[589,1839,702],{"className":1840},[750],[589,1842],{"className":1843,"style":746},[650],[589,1845,1847,1850,1853,1859,1862,1865,1871,1874,1877],{"className":1846},[635],[589,1848],{"className":1849,"style":1747},[639],[589,1851,1751],{"className":1852},[644],[589,1854,1856],{"className":1855},[644],[589,1857,1759],{"className":1858},[1758],[589,1860,1374],{"className":1861},[644],[589,1863],{"className":1864,"style":869},[650],[589,1866,1868],{"className":1867},[644,873],[589,1869,1663],{"className":1870},[644],[589,1872],{"className":1873,"style":746},[650],[589,1875,702],{"className":1876},[750],[589,1878],{"className":1879,"style":746},[650],[589,1881,1883,1886,1889,1895,1899,1902,1908,1911,1914],{"className":1882},[635],[589,1884],{"className":1885,"style":1747},[639],[589,1887,1751],{"className":1888},[644],[589,1890,1892],{"className":1891},[644],[589,1893,1759],{"className":1894},[1758],[589,1896,1898],{"className":1897},[644],"6",[589,1900],{"className":1901,"style":869},[650],[589,1903,1905],{"className":1904},[644,873],[589,1906,1663],{"className":1907},[644],[589,1909],{"className":1910,"style":651},[650],[589,1912,614],{"className":1913},[655],[589,1915],{"className":1916,"style":651},[650],[589,1918,1920,1923,1927,1933,1937,1940],{"className":1919},[635],[589,1921],{"className":1922,"style":1747},[639],[589,1924,1926],{"className":1925},[644],"1",[589,1928,1930],{"className":1929},[644],[589,1931,1759],{"className":1932},[1758],[589,1934,1936],{"className":1935},[644],"9",[589,1938],{"className":1939,"style":869},[650],[589,1941,1943],{"className":1942},[644,873],[589,1944,1663],{"className":1945},[644],{"title":1947,"searchDepth":1948,"depth":1948,"links":1949},"",2,[1950,1951,1952],{"id":561,"depth":1948,"text":562},{"id":825,"depth":1948,"text":826},{"id":1622,"depth":1948,"text":1623},[1954,1957,1959,1961,1963],{"lhs":1955,"rhs":1956},"a + b + c","c + b + a",{"lhs":1958,"rhs":1013},"3 + 4 + 5",{"lhs":1960,"rhs":1221},"3 * 7",{"lhs":1962,"rhs":1492},"30 - 8 - 12",{"lhs":1964,"rhs":1965},"0.5 + 0.8 + 0.6","1.9",null,"md",{"caption":1969,"x":1970,"y":1973,"items":1976},"Dreieck mit den Seitenlängen a, b und c; der Umfang ist die Länge des ganzen Randes",[1971,1972],-0.6,5.6,[1974,1975],-0.7,3.4,[1977,1987,1992,1996,2000,2007,2012],{"type":1978,"points":1979},"polygon",[1980,1982,1984],[1981,1981],0,[1983,1981],5,[1985,1986],1.6,2.8,{"type":1988,"at":1989,"text":705},"label",[1990,1991],2.5,-0.35,{"type":1988,"at":1993,"text":621},[1994,1995],3.55,1.65,{"type":1988,"at":1997,"text":1999},[1998,1985],0.5,"b",{"type":2001,"at":2002,"label":2003,"offset":2004},"point",[1981,1981],"A",[2005,2006],-18,4,{"type":2001,"at":2008,"label":2009,"offset":2010},[1983,1981],"B",[2011,2006],8,{"type":2001,"at":2013,"label":2014,"offset":2015},[1985,1986],"C",[2016,2017],-6,-14,"\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 297.6 196.8\" width=\"297.6\" height=\"196.8\" role=\"img\" aria-label=\"Dreieck mit den Seitenlängen a, b und c; der Umfang ist die Länge des ganzen Randes\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-triangle-perimeter-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-triangle-perimeter-clip\">\u003Crect x=\"0\" y=\"0\" width=\"297.6\" height=\"196.8\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Cpolygon points=\"28.8,163.2 268.8,163.2 105.6,28.8\" class=\"fig-main\"\u002F>\u003Ctext x=\"148.8\" y=\"180\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">c\u003C\u002Ftext>\u003Ctext x=\"199.2\" y=\"84\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">a\u003C\u002Ftext>\u003Ctext x=\"52.8\" y=\"86.4\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">b\u003C\u002Ftext>\u003Ccircle cx=\"28.8\" cy=\"163.2\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"10.8\" y=\"167.2\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">A\u003C\u002Ftext>\u003Ccircle cx=\"268.8\" cy=\"163.2\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"276.8\" y=\"167.2\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">B\u003C\u002Ftext>\u003Ccircle cx=\"105.6\" cy=\"28.8\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"99.6\" y=\"14.8\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">C\u003C\u002Ftext>\u003C\u002Fsvg>","triangle-perimeter","\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>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">U = a + b + c\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:0.6667em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","lower",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-dreieck",[],[2027,2028,2029,2030],"triangle-area","rectangle-perimeter","square-perimeter","heron-formula",{"title":548,"description":1947},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-dreieck","Der Umfang eines Dreiecks ist die Summe seiner drei Seitenlängen.",[2035,2036,2037],"umfang","dreieck","seitenlänge",[2039,2042],{"symbol":610,"meaning":2040,"symbolHtml":2041},"Umfang, die Länge des ganzen Randes","\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":2043,"meaning":2044,"symbolHtml":2045},"a, b, c","die drei Seitenlängen des Dreiecks","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, b, c\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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","k7jj145sIl5H4lXFRh3hNk74l6v6Egtp_mIbX3_SIfg",[2048,2051,2054,2057],{"stem":2049,"title":2050,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fflaecheninhalt-dreieck","Flächeninhalt eines Dreiecks",{"stem":2052,"title":2053,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fsatz-des-heron","Satz des Heron",{"stem":2055,"title":2056,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-quadrat","Umfang eines Quadrats",{"stem":2058,"title":2059,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfang-rechteck","Umfang eines Rechtecks",1793824270236]