--- title: "Importance of trig levelling, inaccessible RL" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:trig-leveling-importance-inaccessible-object" source: https://www.bctengineer.com/q/semester_4/surveying_ii/trig-leveling-importance-inaccessible-object site: BCT Engineering --- # Importance of trig levelling, inaccessible RL Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2065 Kartik (old course), 10 marks - Video: [Trigonometry leveling derivation inaccessible of different level.fifth semester surveying unit 1](https://www.youtube.com/watch?v=9qZIwhivKXw) — CTEVT PROBLEM - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/trig-leveling-importance-inaccessible-object - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question Discuss importance of trigonometrical levelling. Derive an expression to find the R.L. of an inaccessible object when the instrument stations are in the different vertical plane. ## Short answer Trigonometric levelling is vital in rough terrain and for inaccessible points. When stations are in different vertical planes, horizontal distances are found by sine rule from base and horizontal angles; then RL of object = axis RL + distance × tan(vertical angle). ## Answer #### Importance of trigonometric levelling Trigonometric levelling finds the difference in elevation between two points from the measured vertical angle and the horizontal (or slope) distance. It is important because: - it is the only practical method in **rough, hilly or mountainous country** (as in Nepal) where spirit levelling is slow and costly; - it gives the elevation of **inaccessible points** such as the top of a tower, chimney, hill, mountain peak, transmission tower, or the other bank of a river; - it is quick, needs few staff, and gives the elevations of **triangulation stations** and control points over a large area; - it is used to fix the heights of the stations of a traverse and to check the levels run by other methods; - the accuracy is sufficient for contouring, topographic survey and the height of structures. Spirit levelling is more accurate and is preferred when high accuracy is required on gentle ground. #### RL of an inaccessible object when the stations are in different vertical planes If the instrument stations X and Y are not in the same vertical plane as the object P, a base line XY is measured and horizontal angles are observed at X and Y. The horizontal distances XP and YP are found by the sine rule in the plan triangle XYP, and the height of P is found from the vertical angles. ```text P (top) / \ XP / \ YP / \ / \ X -------- Y alpha b beta ``` #### Derivation Let the base line be $b = XY$, the horizontal angles $\alpha = \angle YXP$ and $\beta = \angle XYP$, so $\angle XPY = 180^\circ - (\alpha + \beta)$. $$ \begin{aligned} XP &= \frac{b\sin\beta}{\sin(\alpha+\beta)} \\ YP &= \frac{b\sin\alpha}{\sin(\alpha+\beta)} \end{aligned} $$ Let $\theta_X$ and $\theta_Y$ be the vertical angles (elevation) of P from X and Y, and $h_X$, $h_Y$ the heights of the instrument axes above their stations. If a BM of known RL is sighted from X and Y with back-sights $BS_X$, $BS_Y$: $$ \begin{aligned} RL_{\text{axis X}} &= RL_{BM} + BS_X, \qquad RL_{\text{axis Y}} = RL_{BM} + BS_Y \\ RL_P &= RL_{\text{axis X}} + XP\tan\theta_X \quad (\text{from X}) \\ RL_P &= RL_{\text{axis Y}} + YP\tan\theta_Y \quad (\text{from Y}) \end{aligned} $$ For zenith angle $z$, $\theta = 90^\circ - z$. The mean of the two values is the RL of P. If the stations are at different levels, the difference of the axis RLs is already included through the BM sights. When the sights are long, the correction for curvature and refraction $0.0673\,d^2$ (d in km) is added to each height. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is trigonometric levelling especially important in hilly or mountainous country such as Nepal? A. It is more accurate than spirit levelling on gentle ground B. Spirit levelling is slow and costly there, and this method works with vertical angles and distances C. It needs no angle measurement at all D. It can only be used on flat ground Answer: B. The method gives height differences from vertical angle and distance, so it is practical where spirit levelling is slow and costly. (Key point: Why trigonometric levelling) 2. When stations X and Y are not in the same vertical plane as P, how are the horizontal distances XP and YP found? A. By the sine rule in the plan triangle XYP, using base XY and horizontal angles at X and Y B. By measuring them directly with a chain over the hill C. By using the vertical angles only D. By taking the average of the two vertical angles Answer: A. A base XY is measured and horizontal angles alpha and beta are observed, so the sine rule in triangle XYP gives XP and YP. (Key point: Plan triangle for distances) 3. With base b = XY, horizontal angles alpha at X and beta at Y, what is XP? A. b sin(alpha) / sin(alpha+beta) B. b sin(beta) / sin(alpha+beta) C. b sin(alpha+beta) / sin(beta) D. b cos(beta) / cos(alpha+beta) Answer: B. XP is opposite angle beta at Y, and the angle at P is 180 - (alpha+beta), whose sine equals sin(alpha+beta); so XP = b sin(beta)/sin(alpha+beta). (Key point: Sine rule result) 4. After finding XP, how is the RL of P obtained from station X? A. RL of axis of X + XP tan(theta_X) B. RL of BM + XP sin(theta_X) C. RL of axis of X - XP cos(theta_X) D. RL of X + XY tan(theta_X) Answer: A. The height of P above the instrument axis is XP tan(theta); this is added to the RL of the axis (RL of BM + back-sight). (Key point: RL of the object) ## Related in Surveying II - [Inaccessible object, stations in different vertical plane](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ainaccessible-object-different-vertical-plane) — Surveying II, chapter 3, asked 3× - [Inaccessible object, stations at different levels](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ainaccessible-object-different-levels-same-plane) — Surveying II, chapter 3 - [RL of tower top, base inaccessible](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atower-inaccessible-base-rl-two-stations) — Surveying II, chapter 3 - [Reciprocal trigonometrical levelling formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Areciprocal-trig-leveling-formula) — Surveying II, chapter 3 - [Derive RL on parabolic vertical curve](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aparabolic-vertical-curve-rl-expression) — Surveying II, chapter 6 - [RL of hill top from two stations (A-B 118 m)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-two-stations-2079j) — Surveying II, chapter 3, asked 4× ## Related in other subjects - [Reciprocal levelling; RL of closed traverse stations](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-closed-traverse-bcdefgb) — Surveying I, chapter 5