---
title: "RL of temple top from stations A and B"
subject: "Surveying II"
chapter: 3
key: "semester_4/surveying_ii:temple-top-rl-two-stations"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/temple-top-rl-two-stations
site: BCT Engineering
---

# RL of temple top from stations A and B

Surveying II · Chapter 3: Trigonometric Leveling

- Asked in: 2078 Baisakh, 6 marks
- Video: [Trigonometric Levelling | RL of top of Temple Numerical  |Board exam solution](https://www.youtube.com/watch?v=TzYNf7c8CF4) — OPengineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/temple-top-rl-two-stations
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3

## Question

The top of a Temple (T) is sighted from two stations (A and B) at very different level, maintaining same vertical plane with the temple. Find the R.L. of the top of temple (T) from the following observed data:

| Inst. Stn. | Target | Vertical Angle | Staff Reading | RL |
|---|---|---|---|---|
| A | Bench Mark (BM) | -6°34'40" | 2.955, 2.843, 2.732 | 1200.00 m (BM) |
| | T (Temple) | 22°26'40" | | |
| | B | 10°21'50" | 1.614, 1.518, 1.423 | |
| B | T (Temple) | 27°32'40" | | |

## Short answer

Using stadia reduction, instrument axis RL at A is 1205.381 m and at B is 1207.243 m. Solving for horizontal distance from A to temple with elevation angles gives 71.696 m, so RL of temple top = 1234.997 m.

## Answer

Assumptions: stadia constants $k = 100$, $c = 0$, staff vertical. The height of instrument at B is not given, so the axis of B is taken at the level of the peg B (the RL of T would change by the actual height of instrument at B). Distances are horizontal.

#### Step 1: RL of the axis of A (from the BM)

- $s = 2.955 - 2.732 = 0.223$ m, $\theta = -6^\circ34'40''$
- $V = \frac{ks}{2}\sin 2\theta = 50\times 0.223\times\sin(-13^\circ09'20'') = -2.538$ m
- Axis of A $= RL_{BM} + r - V = 1200.000 + 2.843 - (-2.538) = 1205.381$ m

#### Step 2: distance AB and RL of axis at B

- $s = 1.614 - 1.423 = 0.191$ m, $\theta = +10^\circ21'50''$
- $D_{AB} = k s\cos^2\theta = 18.482$ m
- $V = 50 s\sin2\theta = 3.380$ m
- Ground at B $= 1205.381 + 3.380 - 1.518 = 1207.243$ m, so the axis of B is 1207.243 m (peg level).

#### Step 3: RL of the top T (A, B and T in one vertical plane, B between A and T)

Elevation angles of T: $\alpha_A = 22^\circ26'40''$, $\alpha_B = 27^\circ32'40''$. Let $x$ = horizontal distance A to T.

$$
x = \frac{(axis_B - axis_A) - D\tan\alpha_B}{\tan\alpha_A - \tan\alpha_B} = \frac{(1207.243 - 1205.381) - 18.482\times 0.52155}{0.41308 - 0.52155} = 71.696\ \text{m}
$$

$$
RL_T = 1205.381 + 71.696\times 0.41308 = 1234.997\ \text{m}
$$

Check from B: $1207.243 + (71.696 - 18.482)\times 0.52155 = 1234.997$ m.

**Answer: RL of the top of the temple = 1234.997 m.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The sight from A to the BM is a depression (-6°34'40''), giving V = -2.538 m. How is the axis RL of A found?
   A. 1200.000 + 2.843 + 2.538 = 1205.381 m (axis = RL BM + r - V)
   B. 1200.000 + 2.843 - 2.538 = 1200.305 m
   C. 1200.000 - 2.843 - 2.538 = 1194.619 m
   D. 1200.000 + 2.538 = 1202.538 m
   Answer: A. Axis A = RL BM + r - V. As V is negative for a depression, the axis lies above the BM by the central reading plus 2.538 m. (Key point: Axis of A from the BM)

2. For a staff intercept s = 0.191 m at a vertical angle of 10°21'50'', which formula gives the horizontal distance AB (k = 100, c = 0)?
   A. D = k s sin^2(theta)
   B. D = (k s / 2) sin 2(theta)
   C. D = k s cos^2(theta)
   D. D = k s cos(theta) + c
   Answer: C. Horizontal distance by stadia is D = k s cos^2(theta); the half sin 2(theta) expression is for the vertical component. (Key point: Stadia horizontal distance)

3. The height of instrument at B is not given. What does the answer assume?
   A. HI of B is taken equal to that of A, so RL of T is unaffected
   B. The axis of B is taken at the peg level; the true RL of T would change by the actual HI of B
   C. The axis of B is taken 1.518 m above the peg
   D. RL of T does not depend on the axis of B
   Answer: B. With HI unknown, axis B = peg RL (1207.243 m). The real axis would be higher by HI, changing RL of T. (Key point: Assumption for the axis of B)

4. The elevation of T is 22°26'40'' from A and 27°32'40'' from B. Which statement is correct?
   A. B is farther from T, so x is measured from B
   B. The two stations are equally distant from T
   C. T is below the axis of A
   D. A is farther (smaller elevation); B lies between A and T, so x is A to T and B to T is x - D
   Answer: D. The smaller angle means the greater distance. So A is farther and B to T is x - 18.482 m. (Key point: Farther station)

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