AT5

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Centro Federal de Educação de Minas Gerais 
Unidade Varginha 
Departamento de Computação e Engenharia Civil 
 
Topografia – Atividade 5 - Lorena Leocádio Gomes 
 
Karen de Fátima Oliveira 
 
RESULTADOS 
Estação 
Altura 
Instr. 
Ponto 
visado 
Leitura Ângulo 
Zenital 
Ângulo 
Horizontal FS FM FI 
B 1,61 
B1 1,484 1,242 1,000 86°00'00'' 32°12'00'' 
B2 1,623 1,111 0,600 97°12"00'' 46°53'00'' 
B3 2,070 1,635 1,200 91°14'00'' 115°14'00'' 
B4 1,722 1,500 1,278 79°38'00'' 86°30'00'' 
B5 2,285 2,000 1,715 82°56'00'' 145°24'00'' 
B6 1,284 1,142 1,000 93°53'00'' 120°08'00'' 
B7 2,000 1,630 1,260 98°21'00'' 208°33'00'' 
B8 2,200 2,002 1,805 105°14'00'' 275°10'00'' 
B9 1,665 1,333 1,000 81°10'00'' 304°58'00'' 
B10 1,280 1,040 0,800 86°44'00'' 320°45'00'' 
 
Ponto 
visado 
Azimute 
Distância 
Horizontal 
Desnível Delta X Delta Y 
Coordenadas (m) 
X Y Z 
B1 358°37'32'' 48,164 m 3,736 m -1,552 48,15 198,845 148,15 5,346 
B2 225°30'32'' 100,693 m -12,221 m -71,83 -70,565 128,17 29,434 -10,611 
B3 160°44'32'' 86,960 m -1,897 m 28,681 -82,094 228, 681 17,906 -0,287 
B4 67°14'32'' 42,962 m 7,969 m 39,617 16,619 239,617 116,619 9,579 
B5 32°38'32'' 56,137 m 6,569 m 30,28 47,323 230,28 147,27 8,179 
B6 332°46'32'' 28,270 m -1,451 m -12,933 25,138 187,067 125,138 0,159 
B7 1°19'32'' 72,439 m -10,652 m 1,676 72,42 201,676 172,42 -9,042 
B8 96°29'32'' 36,773 m -10,406 m 36,537 -4,158 236,537 95,842 -8,796 
B9 221°27'32'' 64,932 m 10,368 m -42,99 -48,662 157, 010 51,338 11,978 
B10 2°12'32'' 47,844 m 3,301 m 1,844 47,808 201,844 148,15 4,911 
 
 Ponto anterior a B 
 Az = 146°25’32’’ 
 X=200m 
 Y=100m 
 Z = 36m 
 
MEMORIAL DE CÁLCULOS 
1. Azimute 
Ponto visado Azimute 
B1 (146°25'32'' + 32°12'00'') + 180° = 358°37'32'' 
B2 (358°37'32'' + 46°53'00'') - 180° = 225°30'32'' 
B3 (225°30'32'' + 115°14'00'') - 180° = 160°44'32'' 
B4 (160°44'32'' + 86°30'00'') - 180° = 67°14'32'' 
B5 (67°14'32'' + 145°24'00'') - 180° = 32°38'32'' 
B6 (32°38'32'' + 120°08'00'') + 180° = 332°46'32'' 
B7 (332°46'32'' + 208°33'00'') - 540° = 1°19'32'' 
B8 (1°19'32'' + 275°10'00'') - 180° = 96°29'32'' 
B9 (96°29'32'' + 304°58'00'') - 180° = 221°27'32'' 
B10 (221°27'32'' + 320°45'00'') - 540° = 2°12'32'' 
 
2. Distância Horizontal 
G Dh = G*k*sen²(Z) 
0,484 0,484 * 100 * sen² (86°00'00'') = 48,164 m 
1,023 1,023 * 100 * sen² (97°12"00'') = 100,693 m 
0,870 0,870 * 100 *sen² (91°14'00'') = 86,960 m 
0,444 0,444 * 100 * sen² (79°38'00'') = 42,962 m 
0,570 0,570 * 100 * sen² (82°56'00'') = 56,137 m 
0,284 0,284 * 100 * sen² (93°53'00'') = 28,270 m 
0,740 0,740 * 100 * sen² (98°21'00'') = 72,439 m 
0,395 0,395 * 100 * sen² (105°14'00'') = 36,773 m 
0,665 0,665 * 100 * sen² (81°10'00'') = 64,932 m 
0,480 0,480 * 100 * sen² (86°44'00'') = 47,844 m 
 
3. Coordenadas X e Y 
Xvis = Xest + Dh * sen (Az) yvis = yest + Dh * cos (Az) 
200 + 48,164 * sen (358°37'32'') = 198,845 m 100 + 48,164 * cos (358°37'32'') = 148,150 m 
200 + 100,693 * sen (225°30'32'') = 128,170 m 100 + 100,693 * cos (225°30'32'') = 29,434 m 
200 + 86,960 * sen (160°44'32'') = 228, 681 m 100 + 86,960 * cos (160°44'32'') = 17,906 m 
200 + 42,962 * sen (67°14'32'') = 239,617 m 100 + 42,962 * cos (67°14'32'') = 116,619 m 
200 + 56,137 * sen (32°38'32'') = 230,280 m 100 + 56,137 * cos (32°38'32'') = 147,270 m 
200 + 28,270 * sen (332°46'32'') = 187,067 m 100 + 28,270 * cos (332°46'32'') = 125,138 m 
200 + 72,439 * sen (1°19'32'') = 201,676 m 100 + 72,439 * cos (1°19'32'') = 172,420 m 
200 + 36,773 * sen (96°29'32'') = 236,537 m 100 + 36,773 * cos (96°29'32'') = 95,842 m 
200 + 64,932 * sen (221°27'32'') = 157, 010 m 100 + 64,932 * cos (221°27'32'') = 51,338 m 
200 + 47,844 * sen (2°12'32'') = 201,844 m 100 + 47,844 * cos (2°12'32'') = 148,150 m 
4. Δh e coordenada Z 
Δh = hi – hs + Dh * cotg (Z) Hvis = Hest + Δh 
1,61 - 1,242 + 48,164 * cotg (86°00'00'') = 3,736 5,346 
1,61 - 1,111 + 100,693 * cotg (97°12"00'') = -12,221 -10,611 
1,61 -1,635 + 86,960 * cotg (91°14'00'') = - 1,897 -0,287 
1,61 - 1,500 + 42,962 * cotg (79°38'00'') = 7,969 9,579 
1,61 - 2,000 + 56,137 *cotg (82°56'00'') = 6,569 8,179 
1,61 - 1,142 + 28,270 * cotg (93°53'00'') = -1,451 0,159 
1,61 - 1,630 + 72,439 * cotg (98°21'00'') = -10,652 -9,042 
1,61 - 2,002 + 36,773 * cotg (105°14'00'') = -10,406 -8,796 
1,61 - 1,333 + 64,932 * cotg (81°10'00'') = 10,368 11,978 
1,61 - 1,040 + 47,844 * cotg (86°44'00'') = 3,301 4,911 
 
5. ΔX e ΔY 
Δx = Dh * sen (Az) Δy = Dh * cos (Az) 
48,164 * sen (358°37'32'') = -1,552 48,164 * cos (358°37'32'') = 48,150 
100,693 * sen (225°30'32'') = -71,830 100,693 * cos (225°30'32'') = -70,565 
86,960 * sen (160°44'32'') = 28,681 86,960 * cos (160°44'32'') = -82,094 
42,962 * sen (67°14'32'') = 39,617 42,962 * cos (67°14'32'') = 16,619 
56,137 * sen (32°38'32'') = 30,280 56,137 * cos (32°38'32'') = 47,323 
28,270 * sen (332°46'32'') = -12,933 28,270 * cos (332°46'32'') = 25,138 
72,439 * sen (1°19'32'') = 1,676 72,439 * cos (1°19'32'') = 72,420 
36,773 * sen (96°29'32'') = 36,537 36,773 * cos (96°29'32'') = -4,158 
64,932 * sen (221°27'32'') = -42,990 64,932 * cos (221°27'32'') = -48,662 
47,844 * sen (2°12'32'') = 1,844 47,844 * cos (2°12'32'') = 47,808 
 
*Observação: algumas contas mais simples, como G e Hvisado foram realizadas 
diretamente no excel, por este motivo não estão expressas neste documento.