--- title: "RL of building top from stations A and B" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:building-top-rl-two-stations-rod" source: https://www.bctengineer.com/q/semester_4/surveying_ii/building-top-rl-two-stations-rod site: BCT Engineering --- # RL of building top from stations A and B Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2076 Bhadra, 6 marks - Video: [Problem on determination of RL of top Of Object (Trigonometric levelling)](https://www.youtube.com/watch?v=DEIcbWQaAwc) — Tech Reinforce - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/building-top-rl-two-stations-rod - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question The top of the building was sighted from two instrument stations A and B at very different level. The stations A and B lie on the same line with the building. The angle of elevation from A and B to the top of building was 35°21' and 19°28' respectively. The angle of elevation from B to a rod of 2 m height held at A was 14°11' respectively. The height of instrument at A and B were 1.540 m and 1.410 m respectively. The horizontal distance between A and B was 130.570 m and RL of B was 1234.250 m. Find the RL of top of the building. ## Short answer From RL of B and instrument heights, compute RL of axes: B 1235.660 m, A 1268.199 m. Using elevation angles 35°21' and 19°28', solve horizontal distance x = 168.821 m, then RL of top = 1295.332 m. ## Answer A and B are on a line with the building. B sees the rod at A at an angle of elevation, so A is higher and nearer the building. All angles are angles of elevation; AB = 130.570 m is horizontal. #### RL of instrument axes - Axis of B = RL of B + HI = 1234.250 + 1.41 = 1235.660 m - Sight B to A: elevation 14°11'. The top of the rod (2 m above its foot at A) is at 1235.660 + 130.57 tan(14°11') = 1268.659 m, so the ground at A = 1266.659 m and axis of A = 1266.659 + 1.54 = 1268.199 m. Let T be the top of the building. Elevation angles to T: from B = 19°28'00", from A = 35°21'00". Take D = 130.57 m as the horizontal distance BA. #### Distance and RL of T Both stations lie in the same vertical plane as T, with A between B and T. Let $x$ = horizontal distance from B to T, so the distance from A to T is $x - D$ with $D = 130.57$ m. ```text T _.--'| _.--' | A _.--' aA | B _.--' aB - - - - - - - - | |<--- D --->|<--- (x - D) --->| |<------------ x ------------>| ``` Height of T above axis of B = $x\tan\alpha_B$ and above axis of A = $(x-D)\tan\alpha_A$. The RL of T is the same from both stations: $$ \begin{aligned} RL_{axis\,B} + x\tan\alpha_B &= RL_{axis\,A} + (x-D)\tan\alpha_A \\ x &= \frac{(RL_{axis\,A} - RL_{axis\,B}) - D\tan\alpha_A}{\tan\alpha_B - \tan\alpha_A} \end{aligned} $$ With $\alpha_B = 19°28'00"$ ($\tan = 0.35346$), $\alpha_A = 35°21'00"$ ($\tan = 0.70935$), axis RL of B = 1235.660 m and of A = 1268.199 m: $$ x = \frac{(1268.199 - 1235.660) - 130.57\times 0.70935}{0.35346 - 0.70935} = 168.821\ \text{m} $$ $$ RL_{T} = 1235.660 + 168.821\times 0.35346 = 1295.332\ \text{m} $$ Check from A: $1268.199 + (168.821 - 130.57)\times 0.70935 = 1295.332$ m (same). **Answer: RL of the top of the building (T) = 1295.332 m.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. B sees the rod at A at an elevation of 14°11'. What can be said about A? A. A is lower than B B. A is higher than B and nearer the building C. A is on the far side of the building D. A is at the same level as B Answer: B. An upward sight from B means A is higher; as the stations are in line with the building, A is nearer it. (Key point: Position of A) 2. What is the axis RL of B? A. 1234.250 m B. 1235.660 m C. 1232.840 m D. 1268.199 m Answer: B. Axis of B = RL of B + HI = 1234.250 + 1.41 = 1235.660 m. (Key point: Axis of B) 3. The vane at A (rod 2 m) is at 1268.659 m; HI of A is 1.540 m. What is the axis of A? A. 1268.199 m B. 1266.659 m C. 1270.199 m D. 1268.659 m Answer: A. Ground at A = 1268.659 - 2 = 1266.659; axis = 1266.659 + 1.54 = 1268.199 m. (Key point: Axis of A) 4. Given x = 168.821 m from B to the building, what is the RL of the top? A. 1295.332 m, from RL axis B + x tan(19°28') B. 1268.199 m, the axis of A C. 1304.200 m, from RL axis B + x tan(35°21') D. 1235.660 m, the axis of B Answer: A. RL = 1235.660 + 168.821 × 0.35346 = 1295.332 m, which also agrees from A. 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