---
title: "Tangential tacheometry, both angles depression"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:tangential-system-depression-expressions"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/tangential-system-depression-expressions
site: BCT Engineering
---

# Tangential tacheometry, both angles depression

Surveying II · Chapter 2: Tacheometry

- Asked in: 2072 Magh, 4 marks
- Video: [Tangential Method: Both angle are angle of depression](https://www.youtube.com/watch?v=6XLfyiDH4oo) — Online Education
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/tangential-system-depression-expressions
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

Develop expression for H, V, and R.L. for the tangential system of tachometry when the both sightings are angles of depression.

## Short answer

For both depression angles α1 to lower target and α2 to upper target, H=s/(tanα1−tanα2), V=H tanα2, and RL of staff station = RL of axis − H tanα1 − h, where h is lower target height above the station.

## Answer

In the tangential system the vertical angles to two targets (vanes) on a vertical staff, a known distance $s$ apart, are measured. The staff is below the line of collimation, so both angles are angles of depression.

```text
 O ------------------------------ horizontal
  \ \        alpha2 < alpha1
   \  \_______   upper target B   |
    \  alpha1\__ lower target A   | s
     \                           _|
      <--------- H ------------>  staff station C
```
Let $\alpha_1$ be the angle of depression to the lower target A and $\alpha_2$ to the upper target B ($\alpha_1>\alpha_2$). Let $H$ be the horizontal distance, $V$ the vertical depth of the upper target B below the horizontal line of sight through the instrument axis, and $h$ the height of the lower target above the staff station C.

#### Horizontal distance H
$$
\begin{aligned}
\text{Depth of A}&=V+s=H\tan\alpha_1\\
\text{Depth of B}&=V=H\tan\alpha_2\\
s&=H(\tan\alpha_1-\tan\alpha_2)\\
H&=\frac{s}{\tan\alpha_1-\tan\alpha_2}
\end{aligned}
$$

#### Vertical distance V
$$
V=H\tan\alpha_2=\frac{s\tan\alpha_2}{\tan\alpha_1-\tan\alpha_2}
$$

#### Reduced level of the staff station
The lower target A is at depth $V+s=H\tan\alpha_1$ below the axis, and the staff station C is $h$ below A:

$$
\text{R.L. of C}=\text{R.L. of axis}-H\tan\alpha_1-h
$$

or, using the upper target, $\text{R.L. of C}=\text{R.L. of axis}-H\tan\alpha_2-(s+h)$, where R.L. of axis $=$ R.L. of the instrument station $+$ height of instrument.

## Check yourself

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

1. With both angles of depression, which target has the larger angle?
   A. The lower target, α1 > α2
   B. The upper target
   C. Both are equal
   D. Cannot be said
   Answer: A. The lower target is deeper below the horizontal, so its depression angle is larger. (Key point: Which angle is larger)

2. What is H for both depression angles?
   A. H = s/(tanα1 + tanα2)
   B. H = s/(tanα1 - tanα2)
   C. H = s(tanα1 - tanα2)
   D. H = s/(tanα2 - tanα1)
   Answer: B. Depth of A is V + s = H tanα1 and depth of B is V = H tanα2; subtract to get s = H(tanα1 - tanα2). (Key point: Horizontal distance)

3. What is V (depth of upper target below the horizontal line of sight)?
   A. V = H tanα1
   B. V = s tanα1/(tanα1 - tanα2)
   C. V = H tanα2 = s tanα2/(tanα1 - tanα2)
   D. V = H/ tanα2
   Answer: C. The upper target is at depth V = H tanα2; substituting H gives the expression. (Key point: Vertical distance V)

4. What is the RL of staff station C?
   A. RL of axis + H tanα1 + h
   B. RL of axis + H tanα1 - h
   C. RL of axis - H tanα2 + h
   D. RL of axis - H tanα1 - h
   Answer: D. A is H tanα1 below the axis and C is a further h below A, so subtract both. (Key point: RL of staff station)

## Related in Surveying II

- [Stadia principle, tangential method depression](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Astadia-principle-tangential-depression) — Surveying II, chapter 2
- [Tangential system expressions, angles of elevation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-system-elevation-expressions) — Surveying II, chapter 2
- [Tacheometry, methods, tangential formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atacheometry-methods-tangential-derivation) — Surveying II, chapter 2
- [Tangential tacheometry, total station procedure](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-tacheometry-total-station-field-procedure) — Surveying II, chapter 2
- [Tacheometer depression angles, distance A to B and RL](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atachometer-depression-bm-distance-ab-rl) — Surveying II, chapter 2
- [Methods of tacheometry, tangential horizontal distance](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atacheometry-methods-tangential-horizontal-distance) — Surveying II, chapter 2

## Related in other subjects

- [Internal angles in quadrantal bearing system](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aqb-system-internal-angles) — Surveying I, chapter 4