[{"data":1,"prerenderedAt":3458},["ShallowReactive",2],{"blog-\u002Fblog\u002Fprismoidal-vs-end-area-volumes":3,"blog-surround-\u002Fblog\u002Fprismoidal-vs-end-area-volumes":3447},{"id":4,"title":5,"author":6,"body":7,"category":3417,"description":3418,"extension":3419,"image":3420,"meta":3421,"navigation":3439,"path":3440,"publishedAt":3441,"readTime":3442,"seo":3443,"sitemap":3444,"stem":3445,"__hash__":3446},"blog\u002Fblog\u002F17.prismoidal-vs-end-area-volumes.md","Prismoidal vs. End-Area: Selecting the Right Volume Formula","Surveyiiing Expert",{"type":8,"value":9,"toc":3386},"minimark",[10,15,33,37,47,51,54,59,68,72,78,82,193,668,744,1379,1383,1983,1987,2215,2219,2264,2484,3014,3018,3180,3184,3250,3254,3367,3371,3374,3378],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,27,28,32],"p",{},"Calculating earthwork volumes is a balance between speed and accuracy. In engineering surveying, the two standard methods are the ",[20,21,22],"strong",{},"End-Area Rule"," (Trapezoidal) and the ",[20,25,26],{},"Prismoidal Rule"," (Simpson's). While the End-Area rule is simpler for field estimates, the Prismoidal rule is mathematically superior for official payment quantities ",[29,30,31],"span",{},"6, 7",".",[11,34,36],{"id":35},"why-this-matters","Why This Matters",[16,38,39,40,43,44,32],{},"Contractors are paid by the cubic meter. On a large road project, the difference between these two formulas can amount to thousands of cubic meters. Understanding the ",[20,41,42],{},"Prismoidal Excess"," allows you to quantify the error inherent in simpler methods ",[29,45,46],{},"6",[11,48,50],{"id":49},"theory","Theory",[16,52,53],{},"Most earthwork volumes are considered \"prismoids\"—solids whose end faces are parallel and whose longitudinal faces are plane or warped surfaces.",[55,56,58],"h3",{"id":57},"_1-the-end-area-rule-trapezoidal","1. The End-Area Rule (Trapezoidal)",[16,60,61,62,65,66,32],{},"This method assumes the volume is a simple average of the two end areas multiplied by the distance. It is generally easier to use but always ",[20,63,64],{},"overestimates"," volume when the solid is convex ",[29,67,46],{},[55,69,71],{"id":70},"_2-the-prismoidal-rule-simpsons","2. The Prismoidal Rule (Simpson's)",[16,73,74,75,32],{},"This rule is mathematically exact for solids that can be defined by a series of second-degree parabolas. It requires an odd number of cross-sections at equal intervals ",[29,76,77],{},"7",[11,79,81],{"id":80},"mathematical-principles","Mathematical Principles",[55,83,85,86,192],{"id":84},"the-end-area-formula-vev_eve","The End-Area Formula (",[29,87,90,119],{"className":88},[89],"katex",[29,91,94],{"className":92},[93],"katex-mathml",[95,96,98],"math",{"xmlns":97},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[99,100,101,114],"semantics",{},[102,103,104],"mrow",{},[105,106,107,111],"msub",{},[108,109,110],"mi",{},"V",[108,112,113],{},"e",[115,116,118],"annotation",{"encoding":117},"application\u002Fx-tex","V_e",[29,120,124],{"className":121,"ariaHidden":123},[122],"katex-html","true",[29,125,128,133],{"className":126},[127],"base",[29,129],{"className":130,"style":132},[131],"strut","height:0.8333em;vertical-align:-0.15em;",[29,134,137,142],{"className":135},[136],"mord",[29,138,110],{"className":139,"style":141},[136,140],"mathnormal","margin-right:0.2222em;",[29,143,146],{"className":144},[145],"msupsub",[29,147,151,183],{"className":148},[149,150],"vlist-t","vlist-t2",[29,152,155,178],{"className":153},[154],"vlist-r",[29,156,160],{"className":157,"style":159},[158],"vlist","height:0.1514em;",[29,161,163,168],{"style":162},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[29,164],{"className":165,"style":167},[166],"pstrut","height:2.7em;",[29,169,175],{"className":170},[171,172,173,174],"sizing","reset-size6","size3","mtight",[29,176,113],{"className":177},[136,140,174],[29,179,182],{"className":180},[181],"vlist-s","​",[29,184,186],{"className":185},[154],[29,187,190],{"className":188,"style":189},[158],"height:0.15em;",[29,191],{},")",[16,194,195,196,332,333,363,364,366,367],{},"For two areas ",[29,197,199,231],{"className":198},[89],[29,200,202],{"className":201},[93],[95,203,204],{"xmlns":97},[99,205,206,228],{},[102,207,208,217,221],{},[105,209,210,213],{},[108,211,212],{},"A",[214,215,216],"mn",{},"1",[218,219,220],"mo",{"separator":123},",",[105,222,223,225],{},[108,224,212],{},[214,226,227],{},"2",[115,229,230],{"encoding":117},"A_1, A_2",[29,232,234],{"className":233,"ariaHidden":123},[122],[29,235,237,241,283,287,292],{"className":236},[127],[29,238],{"className":239,"style":240},[131],"height:0.8778em;vertical-align:-0.1944em;",[29,242,244,247],{"className":243},[136],[29,245,212],{"className":246},[136,140],[29,248,250],{"className":249},[145],[29,251,253,275],{"className":252},[149,150],[29,254,256,272],{"className":255},[154],[29,257,260],{"className":258,"style":259},[158],"height:0.3011em;",[29,261,263,266],{"style":262},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[29,264],{"className":265,"style":167},[166],[29,267,269],{"className":268},[171,172,173,174],[29,270,216],{"className":271},[136,174],[29,273,182],{"className":274},[181],[29,276,278],{"className":277},[154],[29,279,281],{"className":280,"style":189},[158],[29,282],{},[29,284,220],{"className":285},[286],"mpunct",[29,288],{"className":289,"style":291},[290],"mspace","margin-right:0.1667em;",[29,293,295,298],{"className":294},[136],[29,296,212],{"className":297},[136,140],[29,299,301],{"className":300},[145],[29,302,304,324],{"className":303},[149,150],[29,305,307,321],{"className":306},[154],[29,308,310],{"className":309,"style":259},[158],[29,311,312,315],{"style":262},[29,313],{"className":314,"style":167},[166],[29,316,318],{"className":317},[171,172,173,174],[29,319,227],{"className":320},[136,174],[29,322,182],{"className":323},[181],[29,325,327],{"className":326},[154],[29,328,330],{"className":329,"style":189},[158],[29,331],{}," separated by distance ",[29,334,336,350],{"className":335},[89],[29,337,339],{"className":338},[93],[95,340,341],{"xmlns":97},[99,342,343,348],{},[102,344,345],{},[108,346,347],{},"L",[115,349,347],{"encoding":117},[29,351,353],{"className":352,"ariaHidden":123},[122],[29,354,356,360],{"className":355},[127],[29,357],{"className":358,"style":359},[131],"height:0.6833em;",[29,361,347],{"className":362},[136,140]," ",[29,365,46],{},":\n",[29,368,370,419],{"className":369},[89],[29,371,373],{"className":372},[93],[95,374,375],{"xmlns":97},[99,376,377,416],{},[102,378,379,385,388,395,399,405,408,414],{},[105,380,381,383],{},[108,382,110],{},[108,384,113],{},[218,386,387],{},"=",[389,390,391,393],"mfrac",{},[108,392,347],{},[214,394,227],{},[218,396,398],{"stretchy":397},"false","(",[105,400,401,403],{},[108,402,212],{},[214,404,216],{},[218,406,407],{},"+",[105,409,410,412],{},[108,411,212],{},[214,413,227],{},[218,415,192],{"stretchy":397},[115,417,418],{"encoding":117},"V_e = \\frac{L}{2}(A_1 + A_2)",[29,420,422,479,618],{"className":421,"ariaHidden":123},[122],[29,423,425,428,468,472,476],{"className":424},[127],[29,426],{"className":427,"style":132},[131],[29,429,431,434],{"className":430},[136],[29,432,110],{"className":433,"style":141},[136,140],[29,435,437],{"className":436},[145],[29,438,440,460],{"className":439},[149,150],[29,441,443,457],{"className":442},[154],[29,444,446],{"className":445,"style":159},[158],[29,447,448,451],{"style":162},[29,449],{"className":450,"style":167},[166],[29,452,454],{"className":453},[171,172,173,174],[29,455,113],{"className":456},[136,140,174],[29,458,182],{"className":459},[181],[29,461,463],{"className":462},[154],[29,464,466],{"className":465,"style":189},[158],[29,467],{},[29,469],{"className":470,"style":471},[290],"margin-right:0.2778em;",[29,473,387],{"className":474},[475],"mrel",[29,477],{"className":478,"style":471},[290],[29,480,482,486,565,568,608,611,615],{"className":481},[127],[29,483],{"className":484,"style":485},[131],"height:1.2173em;vertical-align:-0.345em;",[29,487,489,494,561],{"className":488},[136],[29,490],{"className":491},[492,493],"mopen","nulldelimiter",[29,495,497],{"className":496},[389],[29,498,500,552],{"className":499},[149,150],[29,501,503,549],{"className":502},[154],[29,504,507,523,534],{"className":505,"style":506},[158],"height:0.8723em;",[29,508,510,514],{"style":509},"top:-2.655em;",[29,511],{"className":512,"style":513},[166],"height:3em;",[29,515,517],{"className":516},[171,172,173,174],[29,518,520],{"className":519},[136,174],[29,521,227],{"className":522},[136,174],[29,524,526,529],{"style":525},"top:-3.23em;",[29,527],{"className":528,"style":513},[166],[29,530],{"className":531,"style":533},[532],"frac-line","border-bottom-width:0.04em;",[29,535,537,540],{"style":536},"top:-3.394em;",[29,538],{"className":539,"style":513},[166],[29,541,543],{"className":542},[171,172,173,174],[29,544,546],{"className":545},[136,174],[29,547,347],{"className":548},[136,140,174],[29,550,182],{"className":551},[181],[29,553,555],{"className":554},[154],[29,556,559],{"className":557,"style":558},[158],"height:0.345em;",[29,560],{},[29,562],{"className":563},[564,493],"mclose",[29,566,398],{"className":567},[492],[29,569,571,574],{"className":570},[136],[29,572,212],{"className":573},[136,140],[29,575,577],{"className":576},[145],[29,578,580,600],{"className":579},[149,150],[29,581,583,597],{"className":582},[154],[29,584,586],{"className":585,"style":259},[158],[29,587,588,591],{"style":262},[29,589],{"className":590,"style":167},[166],[29,592,594],{"className":593},[171,172,173,174],[29,595,216],{"className":596},[136,174],[29,598,182],{"className":599},[181],[29,601,603],{"className":602},[154],[29,604,606],{"className":605,"style":189},[158],[29,607],{},[29,609],{"className":610,"style":141},[290],[29,612,407],{"className":613},[614],"mbin",[29,616],{"className":617,"style":141},[290],[29,619,621,625,665],{"className":620},[127],[29,622],{"className":623,"style":624},[131],"height:1em;vertical-align:-0.25em;",[29,626,628,631],{"className":627},[136],[29,629,212],{"className":630},[136,140],[29,632,634],{"className":633},[145],[29,635,637,657],{"className":636},[149,150],[29,638,640,654],{"className":639},[154],[29,641,643],{"className":642,"style":259},[158],[29,644,645,648],{"style":262},[29,646],{"className":647,"style":167},[166],[29,649,651],{"className":650},[171,172,173,174],[29,652,227],{"className":653},[136,174],[29,655,182],{"className":656},[181],[29,658,660],{"className":659},[154],[29,661,663],{"className":662,"style":189},[158],[29,664],{},[29,666,192],{"className":667},[564],[55,669,671,672,192],{"id":670},"the-prismoidal-formula-vpv_pvp","The Prismoidal Formula (",[29,673,675,693],{"className":674},[89],[29,676,678],{"className":677},[93],[95,679,680],{"xmlns":97},[99,681,682,690],{},[102,683,684],{},[105,685,686,688],{},[108,687,110],{},[108,689,16],{},[115,691,692],{"encoding":117},"V_p",[29,694,696],{"className":695,"ariaHidden":123},[122],[29,697,699,703],{"className":698},[127],[29,700],{"className":701,"style":702},[131],"height:0.9694em;vertical-align:-0.2861em;",[29,704,706,709],{"className":705},[136],[29,707,110],{"className":708,"style":141},[136,140],[29,710,712],{"className":711},[145],[29,713,715,735],{"className":714},[149,150],[29,716,718,732],{"className":717},[154],[29,719,721],{"className":720,"style":159},[158],[29,722,723,726],{"style":162},[29,724],{"className":725,"style":167},[166],[29,727,729],{"className":728},[171,172,173,174],[29,730,16],{"className":731},[136,140,174],[29,733,182],{"className":734},[181],[29,736,738],{"className":737},[154],[29,739,742],{"className":740,"style":741},[158],"height:0.2861em;",[29,743],{},[16,745,746,747,926,927,955,956,1026,1027,366,1030],{},"For three areas ",[29,748,750,785],{"className":749},[89],[29,751,753],{"className":752},[93],[95,754,755],{"xmlns":97},[99,756,757,782],{},[102,758,759,765,767,774,776],{},[105,760,761,763],{},[108,762,212],{},[214,764,216],{},[218,766,220],{"separator":123},[105,768,769,771],{},[108,770,212],{},[108,772,773],{},"m",[218,775,220],{"separator":123},[105,777,778,780],{},[108,779,212],{},[214,781,227],{},[115,783,784],{"encoding":117},"A_1, A_m, A_2",[29,786,788],{"className":787,"ariaHidden":123},[122],[29,789,791,794,834,837,840,880,883,886],{"className":790},[127],[29,792],{"className":793,"style":240},[131],[29,795,797,800],{"className":796},[136],[29,798,212],{"className":799},[136,140],[29,801,803],{"className":802},[145],[29,804,806,826],{"className":805},[149,150],[29,807,809,823],{"className":808},[154],[29,810,812],{"className":811,"style":259},[158],[29,813,814,817],{"style":262},[29,815],{"className":816,"style":167},[166],[29,818,820],{"className":819},[171,172,173,174],[29,821,216],{"className":822},[136,174],[29,824,182],{"className":825},[181],[29,827,829],{"className":828},[154],[29,830,832],{"className":831,"style":189},[158],[29,833],{},[29,835,220],{"className":836},[286],[29,838],{"className":839,"style":291},[290],[29,841,843,846],{"className":842},[136],[29,844,212],{"className":845},[136,140],[29,847,849],{"className":848},[145],[29,850,852,872],{"className":851},[149,150],[29,853,855,869],{"className":854},[154],[29,856,858],{"className":857,"style":159},[158],[29,859,860,863],{"style":262},[29,861],{"className":862,"style":167},[166],[29,864,866],{"className":865},[171,172,173,174],[29,867,773],{"className":868},[136,140,174],[29,870,182],{"className":871},[181],[29,873,875],{"className":874},[154],[29,876,878],{"className":877,"style":189},[158],[29,879],{},[29,881,220],{"className":882},[286],[29,884],{"className":885,"style":291},[290],[29,887,889,892],{"className":888},[136],[29,890,212],{"className":891},[136,140],[29,893,895],{"className":894},[145],[29,896,898,918],{"className":897},[149,150],[29,899,901,915],{"className":900},[154],[29,902,904],{"className":903,"style":259},[158],[29,905,906,909],{"style":262},[29,907],{"className":908,"style":167},[166],[29,910,912],{"className":911},[171,172,173,174],[29,913,227],{"className":914},[136,174],[29,916,182],{"className":917},[181],[29,919,921],{"className":920},[154],[29,922,924],{"className":923,"style":189},[158],[29,925],{}," separated by total distance ",[29,928,930,943],{"className":929},[89],[29,931,933],{"className":932},[93],[95,934,935],{"xmlns":97},[99,936,937,941],{},[102,938,939],{},[108,940,347],{},[115,942,347],{"encoding":117},[29,944,946],{"className":945,"ariaHidden":123},[122],[29,947,949,952],{"className":948},[127],[29,950],{"className":951,"style":359},[131],[29,953,347],{"className":954},[136,140]," (where ",[29,957,959,977],{"className":958},[89],[29,960,962],{"className":961},[93],[95,963,964],{"xmlns":97},[99,965,966,974],{},[102,967,968],{},[105,969,970,972],{},[108,971,212],{},[108,973,773],{},[115,975,976],{"encoding":117},"A_m",[29,978,980],{"className":979,"ariaHidden":123},[122],[29,981,983,986],{"className":982},[127],[29,984],{"className":985,"style":132},[131],[29,987,989,992],{"className":988},[136],[29,990,212],{"className":991},[136,140],[29,993,995],{"className":994},[145],[29,996,998,1018],{"className":997},[149,150],[29,999,1001,1015],{"className":1000},[154],[29,1002,1004],{"className":1003,"style":159},[158],[29,1005,1006,1009],{"style":262},[29,1007],{"className":1008,"style":167},[166],[29,1010,1012],{"className":1011},[171,172,173,174],[29,1013,773],{"className":1014},[136,140,174],[29,1016,182],{"className":1017},[181],[29,1019,1021],{"className":1020},[154],[29,1022,1024],{"className":1023,"style":189},[158],[29,1025],{}," is the area at the exact midpoint) ",[29,1028,1029],{},"7, 8",[29,1031,1033,1088],{"className":1032},[89],[29,1034,1036],{"className":1035},[93],[95,1037,1038],{"xmlns":97},[99,1039,1040,1085],{},[102,1041,1042,1048,1050,1056,1058,1064,1066,1069,1075,1077,1083],{},[105,1043,1044,1046],{},[108,1045,110],{},[108,1047,16],{},[218,1049,387],{},[389,1051,1052,1054],{},[108,1053,347],{},[214,1055,46],{},[218,1057,398],{"stretchy":397},[105,1059,1060,1062],{},[108,1061,212],{},[214,1063,216],{},[218,1065,407],{},[214,1067,1068],{},"4",[105,1070,1071,1073],{},[108,1072,212],{},[108,1074,773],{},[218,1076,407],{},[105,1078,1079,1081],{},[108,1080,212],{},[214,1082,227],{},[218,1084,192],{"stretchy":397},[115,1086,1087],{"encoding":117},"V_p = \\frac{L}{6}(A_1 + 4A_m + A_2)",[29,1089,1091,1146,1272,1330],{"className":1090,"ariaHidden":123},[122],[29,1092,1094,1097,1137,1140,1143],{"className":1093},[127],[29,1095],{"className":1096,"style":702},[131],[29,1098,1100,1103],{"className":1099},[136],[29,1101,110],{"className":1102,"style":141},[136,140],[29,1104,1106],{"className":1105},[145],[29,1107,1109,1129],{"className":1108},[149,150],[29,1110,1112,1126],{"className":1111},[154],[29,1113,1115],{"className":1114,"style":159},[158],[29,1116,1117,1120],{"style":162},[29,1118],{"className":1119,"style":167},[166],[29,1121,1123],{"className":1122},[171,172,173,174],[29,1124,16],{"className":1125},[136,140,174],[29,1127,182],{"className":1128},[181],[29,1130,1132],{"className":1131},[154],[29,1133,1135],{"className":1134,"style":741},[158],[29,1136],{},[29,1138],{"className":1139,"style":471},[290],[29,1141,387],{"className":1142},[475],[29,1144],{"className":1145,"style":471},[290],[29,1147,1149,1152,1220,1223,1263,1266,1269],{"className":1148},[127],[29,1150],{"className":1151,"style":485},[131],[29,1153,1155,1158,1217],{"className":1154},[136],[29,1156],{"className":1157},[492,493],[29,1159,1161],{"className":1160},[389],[29,1162,1164,1209],{"className":1163},[149,150],[29,1165,1167,1206],{"className":1166},[154],[29,1168,1170,1184,1192],{"className":1169,"style":506},[158],[29,1171,1172,1175],{"style":509},[29,1173],{"className":1174,"style":513},[166],[29,1176,1178],{"className":1177},[171,172,173,174],[29,1179,1181],{"className":1180},[136,174],[29,1182,46],{"className":1183},[136,174],[29,1185,1186,1189],{"style":525},[29,1187],{"className":1188,"style":513},[166],[29,1190],{"className":1191,"style":533},[532],[29,1193,1194,1197],{"style":536},[29,1195],{"className":1196,"style":513},[166],[29,1198,1200],{"className":1199},[171,172,173,174],[29,1201,1203],{"className":1202},[136,174],[29,1204,347],{"className":1205},[136,140,174],[29,1207,182],{"className":1208},[181],[29,1210,1212],{"className":1211},[154],[29,1213,1215],{"className":1214,"style":558},[158],[29,1216],{},[29,1218],{"className":1219},[564,493],[29,1221,398],{"className":1222},[492],[29,1224,1226,1229],{"className":1225},[136],[29,1227,212],{"className":1228},[136,140],[29,1230,1232],{"className":1231},[145],[29,1233,1235,1255],{"className":1234},[149,150],[29,1236,1238,1252],{"className":1237},[154],[29,1239,1241],{"className":1240,"style":259},[158],[29,1242,1243,1246],{"style":262},[29,1244],{"className":1245,"style":167},[166],[29,1247,1249],{"className":1248},[171,172,173,174],[29,1250,216],{"className":1251},[136,174],[29,1253,182],{"className":1254},[181],[29,1256,1258],{"className":1257},[154],[29,1259,1261],{"className":1260,"style":189},[158],[29,1262],{},[29,1264],{"className":1265,"style":141},[290],[29,1267,407],{"className":1268},[614],[29,1270],{"className":1271,"style":141},[290],[29,1273,1275,1278,1281,1321,1324,1327],{"className":1274},[127],[29,1276],{"className":1277,"style":132},[131],[29,1279,1068],{"className":1280},[136],[29,1282,1284,1287],{"className":1283},[136],[29,1285,212],{"className":1286},[136,140],[29,1288,1290],{"className":1289},[145],[29,1291,1293,1313],{"className":1292},[149,150],[29,1294,1296,1310],{"className":1295},[154],[29,1297,1299],{"className":1298,"style":159},[158],[29,1300,1301,1304],{"style":262},[29,1302],{"className":1303,"style":167},[166],[29,1305,1307],{"className":1306},[171,172,173,174],[29,1308,773],{"className":1309},[136,140,174],[29,1311,182],{"className":1312},[181],[29,1314,1316],{"className":1315},[154],[29,1317,1319],{"className":1318,"style":189},[158],[29,1320],{},[29,1322],{"className":1323,"style":141},[290],[29,1325,407],{"className":1326},[614],[29,1328],{"className":1329,"style":141},[290],[29,1331,1333,1336,1376],{"className":1332},[127],[29,1334],{"className":1335,"style":624},[131],[29,1337,1339,1342],{"className":1338},[136],[29,1340,212],{"className":1341},[136,140],[29,1343,1345],{"className":1344},[145],[29,1346,1348,1368],{"className":1347},[149,150],[29,1349,1351,1365],{"className":1350},[154],[29,1352,1354],{"className":1353,"style":259},[158],[29,1355,1356,1359],{"style":262},[29,1357],{"className":1358,"style":167},[166],[29,1360,1362],{"className":1361},[171,172,173,174],[29,1363,227],{"className":1364},[136,174],[29,1366,182],{"className":1367},[181],[29,1369,1371],{"className":1370},[154],[29,1372,1374],{"className":1373,"style":189},[158],[29,1375],{},[29,1377,192],{"className":1378},[564],[55,1380,1382],{"id":1381},"the-prismoidal-excess","The Prismoidal Excess",[16,1384,1385,1386,1520,1521,1594,1595,1665,1666,1696,1697,366,1699],{},"The difference between the two methods (",[29,1387,1389,1416],{"className":1388},[89],[29,1390,1392],{"className":1391},[93],[95,1393,1394],{"xmlns":97},[99,1395,1396,1413],{},[102,1397,1398,1404,1407],{},[105,1399,1400,1402],{},[108,1401,110],{},[108,1403,113],{},[218,1405,1406],{},"−",[105,1408,1409,1411],{},[108,1410,110],{},[108,1412,16],{},[115,1414,1415],{"encoding":117},"V_e - V_p",[29,1417,1419,1474],{"className":1418,"ariaHidden":123},[122],[29,1420,1422,1425,1465,1468,1471],{"className":1421},[127],[29,1423],{"className":1424,"style":132},[131],[29,1426,1428,1431],{"className":1427},[136],[29,1429,110],{"className":1430,"style":141},[136,140],[29,1432,1434],{"className":1433},[145],[29,1435,1437,1457],{"className":1436},[149,150],[29,1438,1440,1454],{"className":1439},[154],[29,1441,1443],{"className":1442,"style":159},[158],[29,1444,1445,1448],{"style":162},[29,1446],{"className":1447,"style":167},[166],[29,1449,1451],{"className":1450},[171,172,173,174],[29,1452,113],{"className":1453},[136,140,174],[29,1455,182],{"className":1456},[181],[29,1458,1460],{"className":1459},[154],[29,1461,1463],{"className":1462,"style":189},[158],[29,1464],{},[29,1466],{"className":1467,"style":141},[290],[29,1469,1406],{"className":1470},[614],[29,1472],{"className":1473,"style":141},[290],[29,1475,1477,1480],{"className":1476},[127],[29,1478],{"className":1479,"style":702},[131],[29,1481,1483,1486],{"className":1482},[136],[29,1484,110],{"className":1485,"style":141},[136,140],[29,1487,1489],{"className":1488},[145],[29,1490,1492,1512],{"className":1491},[149,150],[29,1493,1495,1509],{"className":1494},[154],[29,1496,1498],{"className":1497,"style":159},[158],[29,1499,1500,1503],{"style":162},[29,1501],{"className":1502,"style":167},[166],[29,1504,1506],{"className":1505},[171,172,173,174],[29,1507,16],{"className":1508},[136,140,174],[29,1510,182],{"className":1511},[181],[29,1513,1515],{"className":1514},[154],[29,1516,1518],{"className":1517,"style":741},[158],[29,1519],{},") for a solid with center heights ",[29,1522,1524,1543],{"className":1523},[89],[29,1525,1527],{"className":1526},[93],[95,1528,1529],{"xmlns":97},[99,1530,1531,1540],{},[102,1532,1533],{},[105,1534,1535,1538],{},[108,1536,1537],{},"H",[214,1539,216],{},[115,1541,1542],{"encoding":117},"H_1",[29,1544,1546],{"className":1545,"ariaHidden":123},[122],[29,1547,1549,1552],{"className":1548},[127],[29,1550],{"className":1551,"style":132},[131],[29,1553,1555,1559],{"className":1554},[136],[29,1556,1537],{"className":1557,"style":1558},[136,140],"margin-right:0.0813em;",[29,1560,1562],{"className":1561},[145],[29,1563,1565,1586],{"className":1564},[149,150],[29,1566,1568,1583],{"className":1567},[154],[29,1569,1571],{"className":1570,"style":259},[158],[29,1572,1574,1577],{"style":1573},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[29,1575],{"className":1576,"style":167},[166],[29,1578,1580],{"className":1579},[171,172,173,174],[29,1581,216],{"className":1582},[136,174],[29,1584,182],{"className":1585},[181],[29,1587,1589],{"className":1588},[154],[29,1590,1592],{"className":1591,"style":189},[158],[29,1593],{}," and ",[29,1596,1598,1616],{"className":1597},[89],[29,1599,1601],{"className":1600},[93],[95,1602,1603],{"xmlns":97},[99,1604,1605,1613],{},[102,1606,1607],{},[105,1608,1609,1611],{},[108,1610,1537],{},[214,1612,227],{},[115,1614,1615],{"encoding":117},"H_2",[29,1617,1619],{"className":1618,"ariaHidden":123},[122],[29,1620,1622,1625],{"className":1621},[127],[29,1623],{"className":1624,"style":132},[131],[29,1626,1628,1631],{"className":1627},[136],[29,1629,1537],{"className":1630,"style":1558},[136,140],[29,1632,1634],{"className":1633},[145],[29,1635,1637,1657],{"className":1636},[149,150],[29,1638,1640,1654],{"className":1639},[154],[29,1641,1643],{"className":1642,"style":259},[158],[29,1644,1645,1648],{"style":1573},[29,1646],{"className":1647,"style":167},[166],[29,1649,1651],{"className":1650},[171,172,173,174],[29,1652,227],{"className":1653},[136,174],[29,1655,182],{"className":1656},[181],[29,1658,1660],{"className":1659},[154],[29,1661,1663],{"className":1662,"style":189},[158],[29,1664],{}," and side slopes ",[29,1667,1669,1683],{"className":1668},[89],[29,1670,1672],{"className":1671},[93],[95,1673,1674],{"xmlns":97},[99,1675,1676,1681],{},[102,1677,1678],{},[108,1679,1680],{},"S",[115,1682,1680],{"encoding":117},[29,1684,1686],{"className":1685,"ariaHidden":123},[122],[29,1687,1689,1692],{"className":1688},[127],[29,1690],{"className":1691,"style":359},[131],[29,1693,1680],{"className":1694,"style":1695},[136,140],"margin-right:0.0576em;"," is ",[29,1698,46],{},[29,1700,1702,1751],{"className":1701},[89],[29,1703,1705],{"className":1704},[93],[95,1706,1707],{"xmlns":97},[99,1708,1709,1748],{},[102,1710,1711,1715,1717,1723,1725,1727,1733,1735,1741],{},[1712,1713,1714],"mtext",{},"Correction",[218,1716,387],{},[389,1718,1719,1721],{},[108,1720,347],{},[214,1722,46],{},[108,1724,1680],{},[218,1726,398],{"stretchy":397},[105,1728,1729,1731],{},[108,1730,1537],{},[214,1732,216],{},[218,1734,1406],{},[105,1736,1737,1739],{},[108,1738,1537],{},[214,1740,227],{},[1742,1743,1744,1746],"msup",{},[218,1745,192],{"stretchy":397},[214,1747,227],{},[115,1749,1750],{"encoding":117},"\\text{Correction} = \\frac{L}{6} S (H_1 - H_2)^2",[29,1752,1754,1776,1905],{"className":1753,"ariaHidden":123},[122],[29,1755,1757,1760,1767,1770,1773],{"className":1756},[127],[29,1758],{"className":1759,"style":359},[131],[29,1761,1764],{"className":1762},[136,1763],"text",[29,1765,1714],{"className":1766},[136],[29,1768],{"className":1769,"style":471},[290],[29,1771,387],{"className":1772},[475],[29,1774],{"className":1775,"style":471},[290],[29,1777,1779,1782,1850,1853,1856,1896,1899,1902],{"className":1778},[127],[29,1780],{"className":1781,"style":485},[131],[29,1783,1785,1788,1847],{"className":1784},[136],[29,1786],{"className":1787},[492,493],[29,1789,1791],{"className":1790},[389],[29,1792,1794,1839],{"className":1793},[149,150],[29,1795,1797,1836],{"className":1796},[154],[29,1798,1800,1814,1822],{"className":1799,"style":506},[158],[29,1801,1802,1805],{"style":509},[29,1803],{"className":1804,"style":513},[166],[29,1806,1808],{"className":1807},[171,172,173,174],[29,1809,1811],{"className":1810},[136,174],[29,1812,46],{"className":1813},[136,174],[29,1815,1816,1819],{"style":525},[29,1817],{"className":1818,"style":513},[166],[29,1820],{"className":1821,"style":533},[532],[29,1823,1824,1827],{"style":536},[29,1825],{"className":1826,"style":513},[166],[29,1828,1830],{"className":1829},[171,172,173,174],[29,1831,1833],{"className":1832},[136,174],[29,1834,347],{"className":1835},[136,140,174],[29,1837,182],{"className":1838},[181],[29,1840,1842],{"className":1841},[154],[29,1843,1845],{"className":1844,"style":558},[158],[29,1846],{},[29,1848],{"className":1849},[564,493],[29,1851,1680],{"className":1852,"style":1695},[136,140],[29,1854,398],{"className":1855},[492],[29,1857,1859,1862],{"className":1858},[136],[29,1860,1537],{"className":1861,"style":1558},[136,140],[29,1863,1865],{"className":1864},[145],[29,1866,1868,1888],{"className":1867},[149,150],[29,1869,1871,1885],{"className":1870},[154],[29,1872,1874],{"className":1873,"style":259},[158],[29,1875,1876,1879],{"style":1573},[29,1877],{"className":1878,"style":167},[166],[29,1880,1882],{"className":1881},[171,172,173,174],[29,1883,216],{"className":1884},[136,174],[29,1886,182],{"className":1887},[181],[29,1889,1891],{"className":1890},[154],[29,1892,1894],{"className":1893,"style":189},[158],[29,1895],{},[29,1897],{"className":1898,"style":141},[290],[29,1900,1406],{"className":1901},[614],[29,1903],{"className":1904,"style":141},[290],[29,1906,1908,1912,1952],{"className":1907},[127],[29,1909],{"className":1910,"style":1911},[131],"height:1.0641em;vertical-align:-0.25em;",[29,1913,1915,1918],{"className":1914},[136],[29,1916,1537],{"className":1917,"style":1558},[136,140],[29,1919,1921],{"className":1920},[145],[29,1922,1924,1944],{"className":1923},[149,150],[29,1925,1927,1941],{"className":1926},[154],[29,1928,1930],{"className":1929,"style":259},[158],[29,1931,1932,1935],{"style":1573},[29,1933],{"className":1934,"style":167},[166],[29,1936,1938],{"className":1937},[171,172,173,174],[29,1939,227],{"className":1940},[136,174],[29,1942,182],{"className":1943},[181],[29,1945,1947],{"className":1946},[154],[29,1948,1950],{"className":1949,"style":189},[158],[29,1951],{},[29,1953,1955,1958],{"className":1954},[564],[29,1956,192],{"className":1957},[564],[29,1959,1961],{"className":1960},[145],[29,1962,1964],{"className":1963},[149],[29,1965,1967],{"className":1966},[154],[29,1968,1971],{"className":1969,"style":1970},[158],"height:0.8141em;",[29,1972,1974,1977],{"style":1973},"top:-3.063em;margin-right:0.05em;",[29,1975],{"className":1976,"style":167},[166],[29,1978,1980],{"className":1979},[171,172,173,174],[29,1981,227],{"className":1982},[136,174],[11,1984,1986],{"id":1985},"field-workflow","Field Workflow",[1988,1989,1990,1994,2077,2081,2116,2120,2125,2129,2208,2212],"steps",{},[55,1991,1993],{"id":1992},"cross-sectioning","Cross-Sectioning",[16,1995,1996,1997,2037,2038,2076],{},"Take cross-sections at regular intervals (e.g., every ",[29,1998,2000,2018],{"className":1999},[89],[29,2001,2003],{"className":2002},[93],[95,2004,2005],{"xmlns":97},[99,2006,2007,2015],{},[102,2008,2009,2012],{},[214,2010,2011],{},"20",[1712,2013,2014],{}," m",[115,2016,2017],{"encoding":117},"20\\text{ m}",[29,2019,2021],{"className":2020,"ariaHidden":123},[122],[29,2022,2024,2028,2031],{"className":2023},[127],[29,2025],{"className":2026,"style":2027},[131],"height:0.6444em;",[29,2029,2011],{"className":2030},[136],[29,2032,2034],{"className":2033},[136,1763],[29,2035,2014],{"className":2036},[136]," or ",[29,2039,2041,2058],{"className":2040},[89],[29,2042,2044],{"className":2043},[93],[95,2045,2046],{"xmlns":97},[99,2047,2048,2055],{},[102,2049,2050,2053],{},[214,2051,2052],{},"50",[1712,2054,2014],{},[115,2056,2057],{"encoding":117},"50\\text{ m}",[29,2059,2061],{"className":2060,"ariaHidden":123},[122],[29,2062,2064,2067,2070],{"className":2063},[127],[29,2065],{"className":2066,"style":2027},[131],[29,2068,2052],{"className":2069},[136],[29,2071,2073],{"className":2072},[136,1763],[29,2074,2014],{"className":2075},[136],").",[55,2078,2080],{"id":2079},"area-calculation","Area Calculation",[16,2082,2083,2084,2112,2113,32],{},"Calculate the area (",[29,2085,2087,2100],{"className":2086},[89],[29,2088,2090],{"className":2089},[93],[95,2091,2092],{"xmlns":97},[99,2093,2094,2098],{},[102,2095,2096],{},[108,2097,212],{},[115,2099,212],{"encoding":117},[29,2101,2103],{"className":2102,"ariaHidden":123},[122],[29,2104,2106,2109],{"className":2105},[127],[29,2107],{"className":2108,"style":359},[131],[29,2110,212],{"className":2111},[136,140],") of each cross-section using coordinate geometry or a planimeter ",[29,2114,2115],{},"9, 10",[55,2117,2119],{"id":2118},"selection-of-rule","Selection of Rule",[16,2121,2122,2123,32],{},"If the areas between sections change uniformly, End-Area is sufficient. If the ground is highly irregular or precision is required for payment, use Prismoidal ",[29,2124,31],{},[55,2126,2128],{"id":2127},"mid-section-interpolation","Mid-Section Interpolation",[16,2130,2131,2132,2201,2202,2205,2206,32],{},"Note: The \"Mid-Area\" (",[29,2133,2135,2152],{"className":2134},[89],[29,2136,2138],{"className":2137},[93],[95,2139,2140],{"xmlns":97},[99,2141,2142,2150],{},[102,2143,2144],{},[105,2145,2146,2148],{},[108,2147,212],{},[108,2149,773],{},[115,2151,976],{"encoding":117},[29,2153,2155],{"className":2154,"ariaHidden":123},[122],[29,2156,2158,2161],{"className":2157},[127],[29,2159],{"className":2160,"style":132},[131],[29,2162,2164,2167],{"className":2163},[136],[29,2165,212],{"className":2166},[136,140],[29,2168,2170],{"className":2169},[145],[29,2171,2173,2193],{"className":2172},[149,150],[29,2174,2176,2190],{"className":2175},[154],[29,2177,2179],{"className":2178,"style":159},[158],[29,2180,2181,2184],{"style":262},[29,2182],{"className":2183,"style":167},[166],[29,2185,2187],{"className":2186},[171,172,173,174],[29,2188,773],{"className":2189},[136,140,174],[29,2191,182],{"className":2192},[181],[29,2194,2196],{"className":2195},[154],[29,2197,2199],{"className":2198,"style":189},[158],[29,2200],{},") in the Prismoidal rule is ",[20,2203,2204],{},"not"," the average of the two end areas. It must be calculated from the mid-dimensions (mid-height, mid-width) ",[29,2207,77],{},[55,2209,2211],{"id":2210},"summation","Summation",[16,2213,2214],{},"Sum the volumes of all segments to find the project total.",[11,2216,2218],{"id":2217},"step-by-step-example","Step-by-Step Example",[16,2220,2221,2224,2225,2263],{},[20,2222,2223],{},"Problem:"," Calculate the volume between two sections ",[29,2226,2228,2245],{"className":2227},[89],[29,2229,2231],{"className":2230},[93],[95,2232,2233],{"xmlns":97},[99,2234,2235,2242],{},[102,2236,2237,2240],{},[214,2238,2239],{},"100",[1712,2241,2014],{},[115,2243,2244],{"encoding":117},"100\\text{ m}",[29,2246,2248],{"className":2247,"ariaHidden":123},[122],[29,2249,2251,2254,2257],{"className":2250},[127],[29,2252],{"className":2253,"style":2027},[131],[29,2255,2239],{"className":2256},[136],[29,2258,2260],{"className":2259},[136,1763],[29,2261,2014],{"className":2262},[136]," apart.",[2265,2266,2267,2339,2410],"ul",{},[2268,2269,2270,2271],"li",{},"Section 1: Area = ",[29,2272,2274,2295],{"className":2273},[89],[29,2275,2277],{"className":2276},[93],[95,2278,2279],{"xmlns":97},[99,2280,2281,2292],{},[102,2282,2283,2286],{},[214,2284,2285],{},"208",[1742,2287,2288,2290],{},[1712,2289,2014],{},[214,2291,227],{},[115,2293,2294],{"encoding":117},"208\\text{ m}^2",[29,2296,2298],{"className":2297,"ariaHidden":123},[122],[29,2299,2301,2304,2307],{"className":2300},[127],[29,2302],{"className":2303,"style":1970},[131],[29,2305,2285],{"className":2306},[136],[29,2308,2310,2316],{"className":2309},[136],[29,2311,2313],{"className":2312},[136,1763],[29,2314,2014],{"className":2315},[136],[29,2317,2319],{"className":2318},[145],[29,2320,2322],{"className":2321},[149],[29,2323,2325],{"className":2324},[154],[29,2326,2328],{"className":2327,"style":1970},[158],[29,2329,2330,2333],{"style":1973},[29,2331],{"className":2332,"style":167},[166],[29,2334,2336],{"className":2335},[171,172,173,174],[29,2337,227],{"className":2338},[136,174],[2268,2340,2341,2342],{},"Mid-Section: Area = ",[29,2343,2345,2366],{"className":2344},[89],[29,2346,2348],{"className":2347},[93],[95,2349,2350],{"xmlns":97},[99,2351,2352,2363],{},[102,2353,2354,2357],{},[214,2355,2356],{},"292.6",[1742,2358,2359,2361],{},[1712,2360,2014],{},[214,2362,227],{},[115,2364,2365],{"encoding":117},"292.6\\text{ m}^2",[29,2367,2369],{"className":2368,"ariaHidden":123},[122],[29,2370,2372,2375,2378],{"className":2371},[127],[29,2373],{"className":2374,"style":1970},[131],[29,2376,2356],{"className":2377},[136],[29,2379,2381,2387],{"className":2380},[136],[29,2382,2384],{"className":2383},[136,1763],[29,2385,2014],{"className":2386},[136],[29,2388,2390],{"className":2389},[145],[29,2391,2393],{"className":2392},[149],[29,2394,2396],{"className":2395},[154],[29,2397,2399],{"className":2398,"style":1970},[158],[29,2400,2401,2404],{"style":1973},[29,2402],{"className":2403,"style":167},[166],[29,2405,2407],{"className":2406},[171,172,173,174],[29,2408,227],{"className":2409},[136,174],[2268,2411,2412,2413,363,2481],{},"Section 2: Area = ",[29,2414,2416,2437],{"className":2415},[89],[29,2417,2419],{"className":2418},[93],[95,2420,2421],{"xmlns":97},[99,2422,2423,2434],{},[102,2424,2425,2428],{},[214,2426,2427],{},"378",[1742,2429,2430,2432],{},[1712,2431,2014],{},[214,2433,227],{},[115,2435,2436],{"encoding":117},"378\\text{ m}^2",[29,2438,2440],{"className":2439,"ariaHidden":123},[122],[29,2441,2443,2446,2449],{"className":2442},[127],[29,2444],{"className":2445,"style":1970},[131],[29,2447,2427],{"className":2448},[136],[29,2450,2452,2458],{"className":2451},[136],[29,2453,2455],{"className":2454},[136,1763],[29,2456,2014],{"className":2457},[136],[29,2459,2461],{"className":2460},[145],[29,2462,2464],{"className":2463},[149],[29,2465,2467],{"className":2466},[154],[29,2468,2470],{"className":2469,"style":1970},[158],[29,2471,2472,2475],{"style":1973},[29,2473],{"className":2474,"style":167},[166],[29,2476,2478],{"className":2477},[171,172,173,174],[29,2479,227],{"className":2480},[136,174],[29,2482,2483],{},"8",[2485,2486,2487,2742,2923],"ol",{},[2268,2488,2489,2544],{},[20,2490,2491,2492,2543],{},"Using Prismoidal Rule (",[29,2493,2495,2513],{"className":2494},[89],[29,2496,2498],{"className":2497},[93],[95,2499,2500],{"xmlns":97},[99,2501,2502,2510],{},[102,2503,2504,2506,2508],{},[108,2505,347],{},[218,2507,387],{},[214,2509,2239],{},[115,2511,2512],{"encoding":117},"L=100",[29,2514,2516,2534],{"className":2515,"ariaHidden":123},[122],[29,2517,2519,2522,2525,2528,2531],{"className":2518},[127],[29,2520],{"className":2521,"style":359},[131],[29,2523,347],{"className":2524},[136,140],[29,2526],{"className":2527,"style":471},[290],[29,2529,387],{"className":2530},[475],[29,2532],{"className":2533,"style":471},[290],[29,2535,2537,2540],{"className":2536},[127],[29,2538],{"className":2539,"style":2027},[131],[29,2541,2239],{"className":2542},[136],"):",[29,2545,2547,2591],{"className":2546},[89],[29,2548,2550],{"className":2549},[93],[95,2551,2552],{"xmlns":97},[99,2553,2554,2588],{},[102,2555,2556,2558,2560,2566,2569,2571,2573,2575,2577,2579,2581,2583,2585],{},[108,2557,110],{},[218,2559,387],{},[389,2561,2562,2564],{},[214,2563,2239],{},[214,2565,46],{},[218,2567,2568],{"stretchy":397},"[",[214,2570,2285],{},[218,2572,407],{},[214,2574,1068],{},[218,2576,398],{"stretchy":397},[214,2578,2356],{},[218,2580,192],{"stretchy":397},[218,2582,407],{},[214,2584,2427],{},[218,2586,2587],{"stretchy":397},"]",[115,2589,2590],{"encoding":117},"V = \\frac{100}{6} [208 + 4(292.6) + 378]",[29,2592,2594,2612,2703,2730],{"className":2593,"ariaHidden":123},[122],[29,2595,2597,2600,2603,2606,2609],{"className":2596},[127],[29,2598],{"className":2599,"style":359},[131],[29,2601,110],{"className":2602,"style":141},[136,140],[29,2604],{"className":2605,"style":471},[290],[29,2607,387],{"className":2608},[475],[29,2610],{"className":2611,"style":471},[290],[29,2613,2615,2619,2688,2691,2694,2697,2700],{"className":2614},[127],[29,2616],{"className":2617,"style":2618},[131],"height:1.1901em;vertical-align:-0.345em;",[29,2620,2622,2625,2685],{"className":2621},[136],[29,2623],{"className":2624},[492,493],[29,2626,2628],{"className":2627},[389],[29,2629,2631,2677],{"className":2630},[149,150],[29,2632,2634,2674],{"className":2633},[154],[29,2635,2638,2652,2660],{"className":2636,"style":2637},[158],"height:0.8451em;",[29,2639,2640,2643],{"style":509},[29,2641],{"className":2642,"style":513},[166],[29,2644,2646],{"className":2645},[171,172,173,174],[29,2647,2649],{"className":2648},[136,174],[29,2650,46],{"className":2651},[136,174],[29,2653,2654,2657],{"style":525},[29,2655],{"className":2656,"style":513},[166],[29,2658],{"className":2659,"style":533},[532],[29,2661,2662,2665],{"style":536},[29,2663],{"className":2664,"style":513},[166],[29,2666,2668],{"className":2667},[171,172,173,174],[29,2669,2671],{"className":2670},[136,174],[29,2672,2239],{"className":2673},[136,174],[29,2675,182],{"className":2676},[181],[29,2678,2680],{"className":2679},[154],[29,2681,2683],{"className":2682,"style":558},[158],[29,2684],{},[29,2686],{"className":2687},[564,493],[29,2689,2568],{"className":2690},[492],[29,2692,2285],{"className":2693},[136],[29,2695],{"className":2696,"style":141},[290],[29,2698,407],{"className":2699},[614],[29,2701],{"className":2702,"style":141},[290],[29,2704,2706,2709,2712,2715,2718,2721,2724,2727],{"className":2705},[127],[29,2707],{"className":2708,"style":624},[131],[29,2710,1068],{"className":2711},[136],[29,2713,398],{"className":2714},[492],[29,2716,2356],{"className":2717},[136],[29,2719,192],{"className":2720},[564],[29,2722],{"className":2723,"style":141},[290],[29,2725,407],{"className":2726},[614],[29,2728],{"className":2729,"style":141},[290],[29,2731,2733,2736,2739],{"className":2732},[127],[29,2734],{"className":2735,"style":624},[131],[29,2737,2427],{"className":2738},[136],[29,2740,2587],{"className":2741},[564],[2268,2743,2744,2747],{},[20,2745,2746],{},"Calculation:",[29,2748,2750,2796],{"className":2749},[89],[29,2751,2753],{"className":2752},[93],[95,2754,2755],{"xmlns":97},[99,2756,2757,2793],{},[102,2758,2759,2761,2763,2766,2769,2771,2773,2775,2778,2780,2782,2784,2786,2788,2790],{},[108,2760,110],{},[218,2762,387],{},[214,2764,2765],{},"16.667",[218,2767,2768],{},"×",[218,2770,2568],{"stretchy":397},[214,2772,2285],{},[218,2774,407],{},[214,2776,2777],{},"1170.4",[218,2779,407],{},[214,2781,2427],{},[218,2783,2587],{"stretchy":397},[218,2785,387],{},[214,2787,2765],{},[218,2789,2768],{},[214,2791,2792],{},"1756.4",[115,2794,2795],{"encoding":117},"V = 16.667 \\times [208 + 1170.4 + 378] = 16.667 \\times 1756.4",[29,2797,2799,2817,2836,2857,2875,2896,2914],{"className":2798,"ariaHidden":123},[122],[29,2800,2802,2805,2808,2811,2814],{"className":2801},[127],[29,2803],{"className":2804,"style":359},[131],[29,2806,110],{"className":2807,"style":141},[136,140],[29,2809],{"className":2810,"style":471},[290],[29,2812,387],{"className":2813},[475],[29,2815],{"className":2816,"style":471},[290],[29,2818,2820,2824,2827,2830,2833],{"className":2819},[127],[29,2821],{"className":2822,"style":2823},[131],"height:0.7278em;vertical-align:-0.0833em;",[29,2825,2765],{"className":2826},[136],[29,2828],{"className":2829,"style":141},[290],[29,2831,2768],{"className":2832},[614],[29,2834],{"className":2835,"style":141},[290],[29,2837,2839,2842,2845,2848,2851,2854],{"className":2838},[127],[29,2840],{"className":2841,"style":624},[131],[29,2843,2568],{"className":2844},[492],[29,2846,2285],{"className":2847},[136],[29,2849],{"className":2850,"style":141},[290],[29,2852,407],{"className":2853},[614],[29,2855],{"className":2856,"style":141},[290],[29,2858,2860,2863,2866,2869,2872],{"className":2859},[127],[29,2861],{"className":2862,"style":2823},[131],[29,2864,2777],{"className":2865},[136],[29,2867],{"className":2868,"style":141},[290],[29,2870,407],{"className":2871},[614],[29,2873],{"className":2874,"style":141},[290],[29,2876,2878,2881,2884,2887,2890,2893],{"className":2877},[127],[29,2879],{"className":2880,"style":624},[131],[29,2882,2427],{"className":2883},[136],[29,2885,2587],{"className":2886},[564],[29,2888],{"className":2889,"style":471},[290],[29,2891,387],{"className":2892},[475],[29,2894],{"className":2895,"style":471},[290],[29,2897,2899,2902,2905,2908,2911],{"className":2898},[127],[29,2900],{"className":2901,"style":2823},[131],[29,2903,2765],{"className":2904},[136],[29,2906],{"className":2907,"style":141},[290],[29,2909,2768],{"className":2910},[614],[29,2912],{"className":2913,"style":141},[290],[29,2915,2917,2920],{"className":2916},[127],[29,2918],{"className":2919,"style":2027},[131],[29,2921,2792],{"className":2922},[136],[2268,2924,2925,363,2928,363,3012,32],{},[20,2926,2927],{},"Result:",[29,2929,2931,2958],{"className":2930},[89],[29,2932,2934],{"className":2933},[93],[95,2935,2936],{"xmlns":97},[99,2937,2938,2955],{},[102,2939,2940,2943,2945,2948],{},[214,2941,2942],{},"29",[218,2944,220],{"separator":123},[214,2946,2947],{},"273",[1742,2949,2950,2952],{},[1712,2951,2014],{},[214,2953,2954],{},"3",[115,2956,2957],{"encoding":117},"29,273\\text{ m}^3",[29,2959,2961],{"className":2960,"ariaHidden":123},[122],[29,2962,2964,2968,2971,2974,2977,2980],{"className":2963},[127],[29,2965],{"className":2966,"style":2967},[131],"height:1.0085em;vertical-align:-0.1944em;",[29,2969,2942],{"className":2970},[136],[29,2972,220],{"className":2973},[286],[29,2975],{"className":2976,"style":291},[290],[29,2978,2947],{"className":2979},[136],[29,2981,2983,2989],{"className":2982},[136],[29,2984,2986],{"className":2985},[136,1763],[29,2987,2014],{"className":2988},[136],[29,2990,2992],{"className":2991},[145],[29,2993,2995],{"className":2994},[149],[29,2996,2998],{"className":2997},[154],[29,2999,3001],{"className":3000,"style":1970},[158],[29,3002,3003,3006],{"style":1973},[29,3004],{"className":3005,"style":167},[166],[29,3007,3009],{"className":3008},[171,172,173,174],[29,3010,2954],{"className":3011},[136,174],[29,3013,2483],{},[11,3015,3017],{"id":3016},"practical-tips","Practical Tips",[2265,3019,3020,3172],{},[2268,3021,3022,3025,3026,3169,3170,32],{},[20,3023,3024],{},"The Mid-Area Trap:"," For the Prismoidal rule, you must calculate the area of a cross-section using the average linear dimensions of the end sections. Simply averaging the areas (",[29,3027,3029,3061],{"className":3028},[89],[29,3030,3032],{"className":3031},[93],[95,3033,3034],{"xmlns":97},[99,3035,3036,3058],{},[102,3037,3038,3044,3046,3052,3056],{},[105,3039,3040,3042],{},[108,3041,212],{},[214,3043,216],{},[218,3045,407],{},[105,3047,3048,3050],{},[108,3049,212],{},[214,3051,227],{},[108,3053,3055],{"mathvariant":3054},"normal","\u002F",[214,3057,227],{},[115,3059,3060],{"encoding":117},"A_1+A_2\u002F2",[29,3062,3064,3119],{"className":3063,"ariaHidden":123},[122],[29,3065,3067,3070,3110,3113,3116],{"className":3066},[127],[29,3068],{"className":3069,"style":132},[131],[29,3071,3073,3076],{"className":3072},[136],[29,3074,212],{"className":3075},[136,140],[29,3077,3079],{"className":3078},[145],[29,3080,3082,3102],{"className":3081},[149,150],[29,3083,3085,3099],{"className":3084},[154],[29,3086,3088],{"className":3087,"style":259},[158],[29,3089,3090,3093],{"style":262},[29,3091],{"className":3092,"style":167},[166],[29,3094,3096],{"className":3095},[171,172,173,174],[29,3097,216],{"className":3098},[136,174],[29,3100,182],{"className":3101},[181],[29,3103,3105],{"className":3104},[154],[29,3106,3108],{"className":3107,"style":189},[158],[29,3109],{},[29,3111],{"className":3112,"style":141},[290],[29,3114,407],{"className":3115},[614],[29,3117],{"className":3118,"style":141},[290],[29,3120,3122,3125,3165],{"className":3121},[127],[29,3123],{"className":3124,"style":624},[131],[29,3126,3128,3131],{"className":3127},[136],[29,3129,212],{"className":3130},[136,140],[29,3132,3134],{"className":3133},[145],[29,3135,3137,3157],{"className":3136},[149,150],[29,3138,3140,3154],{"className":3139},[154],[29,3141,3143],{"className":3142,"style":259},[158],[29,3144,3145,3148],{"style":262},[29,3146],{"className":3147,"style":167},[166],[29,3149,3151],{"className":3150},[171,172,173,174],[29,3152,227],{"className":3153},[136,174],[29,3155,182],{"className":3156},[181],[29,3158,3160],{"className":3159},[154],[29,3161,3163],{"className":3162,"style":189},[158],[29,3164],{},[29,3166,3168],{"className":3167},[136],"\u002F2",") is incorrect and will just give you the End-Area result ",[29,3171,77],{},[2268,3173,3174,3177,3178,32],{},[20,3175,3176],{},"Curvature Correction:"," On curved road sections, the volume must be corrected for the eccentricity of the centroid of the cross-section relative to the radius of the curve ",[29,3179,77],{},[11,3181,3183],{"id":3182},"common-mistakes","Common Mistakes",[2265,3185,3186,3238],{},[2268,3187,3188,3191,3192,3235,3236,32],{},[20,3189,3190],{},"Ignoring Side Slopes:"," When calculating mid-areas, ensure the side slopes (",[29,3193,3195,3214],{"className":3194},[89],[29,3196,3198],{"className":3197},[93],[95,3199,3200],{"xmlns":97},[99,3201,3202,3211],{},[102,3203,3204,3206,3209],{},[214,3205,216],{},[1712,3207,3208],{}," in ",[108,3210,1680],{},[115,3212,3213],{"encoding":117},"1 \\text{ in } S",[29,3215,3217],{"className":3216,"ariaHidden":123},[122],[29,3218,3220,3223,3226,3232],{"className":3219},[127],[29,3221],{"className":3222,"style":359},[131],[29,3224,216],{"className":3225},[136],[29,3227,3229],{"className":3228},[136,1763],[29,3230,3208],{"className":3231},[136],[29,3233,1680],{"className":3234,"style":1695},[136,140],") remain constant; otherwise, the prismoidal assumption fails ",[29,3237,46],{},[2268,3239,3240,3243,3244,3247,3248,32],{},[20,3241,3242],{},"Using Prismoidal for Even Sections:"," The Prismoidal rule (Simpson's) only works for an ",[20,3245,3246],{},"odd"," number of sections. If you have an even number, use Prismoidal for the bulk and the End-Area rule for the final segment ",[29,3249,77],{},[11,3251,3253],{"id":3252},"faq","FAQ",[3255,3256,3257,3266,3354],"accordion",{},[3258,3259,3261],"accordion-item",{"label":3260},"Which rule is more accurate?",[16,3262,3263,3264,32],{},"The Prismoidal rule is generally more accurate as it accounts for the non-linear change in area between cross-sections ",[29,3265,77],{},[3258,3267,3269],{"label":3268},"When should I use the Pyramid formula?",[16,3270,3271,3272,3350,3351,32],{},"When a section goes from a finite area to zero (a \"Grade Point\"), the volume should be treated as a pyramid (",[29,3273,3275,3305],{"className":3274},[89],[29,3276,3278],{"className":3277},[93],[95,3279,3280],{"xmlns":97},[99,3281,3282,3302],{},[102,3283,3284,3286,3289,3291,3294,3296,3298,3300],{},[108,3285,212],{},[108,3287,3288],{},"r",[108,3290,113],{},[108,3292,3293],{},"a",[218,3295,2768],{},[108,3297,347],{},[108,3299,3055],{"mathvariant":3054},[214,3301,2954],{},[115,3303,3304],{"encoding":117},"Area \\times L \u002F 3",[29,3306,3308,3337],{"className":3307,"ariaHidden":123},[122],[29,3309,3311,3315,3318,3322,3325,3328,3331,3334],{"className":3310},[127],[29,3312],{"className":3313,"style":3314},[131],"height:0.7667em;vertical-align:-0.0833em;",[29,3316,212],{"className":3317},[136,140],[29,3319,3288],{"className":3320,"style":3321},[136,140],"margin-right:0.0278em;",[29,3323,113],{"className":3324},[136,140],[29,3326,3293],{"className":3327},[136,140],[29,3329],{"className":3330,"style":141},[290],[29,3332,2768],{"className":3333},[614],[29,3335],{"className":3336,"style":141},[290],[29,3338,3340,3343,3346],{"className":3339},[127],[29,3341],{"className":3342,"style":624},[131],[29,3344,347],{"className":3345},[136,140],[29,3347,3349],{"className":3348},[136],"\u002F3",") rather than using End-Area ",[29,3352,3353],{},"6, 11",[3258,3355,3357],{"label":3356},"Does rock 'bulking' affect these formulas?",[16,3358,3359,3360,3363,3364,32],{},"The formulas calculate the ",[20,3361,3362],{},"geometric volume",". You must apply a bulking factor (e.g., 1.2 for rock) separately after the calculation ",[29,3365,3366],{},"12",[11,3368,3370],{"id":3369},"conclusion","Conclusion",[16,3372,3373],{},"Mastering the distinction between End-Area and Prismoidal volumes is vital for any surveyor involved in earthwork quantification. By applying the correct rule and checking for prismoidal excess, you ensure financial accuracy and project integrity.",[11,3375,3377],{"id":3376},"references","References",[16,3379,3380,3381,3385],{},"Schofield, W. (2001). ",[3382,3383,3384],"em",{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann.",{"title":3387,"searchDepth":3388,"depth":3388,"links":3389},"",2,[3390,3391,3392,3397,3404,3411,3412,3413,3414,3415,3416],{"id":13,"depth":3388,"text":14},{"id":35,"depth":3388,"text":36},{"id":49,"depth":3388,"text":50,"children":3393},[3394,3396],{"id":57,"depth":3395,"text":58},3,{"id":70,"depth":3395,"text":71},{"id":80,"depth":3388,"text":81,"children":3398},[3399,3401,3403],{"id":84,"depth":3395,"text":3400},"The End-Area Formula (VeV_eVe​)",{"id":670,"depth":3395,"text":3402},"The Prismoidal Formula (VpV_pVp​)",{"id":1381,"depth":3395,"text":1382},{"id":1985,"depth":3388,"text":1986,"children":3405},[3406,3407,3408,3409,3410],{"id":1992,"depth":3395,"text":1993},{"id":2079,"depth":3395,"text":2080},{"id":2118,"depth":3395,"text":2119},{"id":2127,"depth":3395,"text":2128},{"id":2210,"depth":3395,"text":2211},{"id":2217,"depth":3388,"text":2218},{"id":3016,"depth":3388,"text":3017},{"id":3182,"depth":3388,"text":3183},{"id":3252,"depth":3388,"text":3253},{"id":3369,"depth":3388,"text":3370},{"id":3376,"depth":3388,"text":3377},"Earthworks","A technical comparison of the Trapezoidal (End-Area) and Prismoidal rules for calculating earthwork volumes in civil engineering.","md","\u002Fimages\u002Fvolume-formulas.jpg",{"authorCredentials":3422,"commentCount":3423,"tags":3424,"relatedTools":3428,"relatedArticles":3431,"sources":3434},"P.Eng",0,[3417,3425,26,3426,3427],"Volumes","End-Area Method","Math",[3429,3430],"Volume Calculator","Excel",[3432,3433],"mass-haul-diagrams-economics","site-grading-controls",[3435],{"title":3436,"type":3437,"note":3438},"Engineering Surveying (Fifth Edition)","Book","Primary source for the derivation of Simpson's and Trapezoidal rules.",true,"\u002Fblog\u002Fprismoidal-vs-end-area-volumes","2024-04-05",16,{"title":5,"description":3418},{"loc":3440},"blog\u002F17.prismoidal-vs-end-area-volumes","axPanPh4qj8oNqswKI5x7Vib2PLR0onbsVsgk6bsOA4",[3448,3453],{"title":3449,"path":3450,"stem":3451,"description":3452,"children":-1},"Weighted Observations: Using Statistical Reliability to Find the MPV","\u002Fblog\u002Fweighted-observations-mpv","blog\u002F16.weighted-observations-mpv","A technical guide to applying weights to surveying measurements to determine the Most Probable Value (MPV) and assess indices of precision.",{"title":3454,"path":3455,"stem":3456,"description":3457,"children":-1},"Autoplumb and Verticality: Ensuring Plumbness in High-Rise Structures","\u002Fblog\u002Fautoplumb-high-rise-verticality","blog\u002F18.autoplumb-high-rise-verticality","A technical guide to controlling the verticality of tall structures using optical plumbing and the Autoplumb instrument.",1783851950225]