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

# Tangential system expressions, angles of elevation

Surveying II · Chapter 2: Tacheometry

- Asked in: 2076 Bhadra, 4 marks
- 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-elevation-expressions
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

Derive an expression for horizontal distance and RL for the tangential system of tacheometry when both sighting are angle of elevation.

## Short answer

For tangential method with both elevation angles α1 (lower) and α2 (upper) to vanes distance s apart: horizontal distance D = s/(tanα2 - tanα1). RL of staff station = RL of instrument + height of instrument + D tanα1 - height of lower vane above staff foot.

## Answer

#### Tangential system

In the tangential method the staff carries two vanes (targets) at a known vertical distance $s$ apart, and no stadia hairs are used. The vertical angles to the two vanes are measured. The method is used when the diaphragm has no stadia hairs.

#### Both angles of elevation

Let the instrument be at A with axis at height $h_i$ above the station, and the staff be held at B.
- $\alpha_1$ = angle of elevation to the lower vane, $\alpha_2$ = angle of elevation to the upper vane ($\alpha_2 > \alpha_1$),
- $s$ = vertical distance between the vanes, $h$ = height of the lower vane above the staff foot (or the staff reading of the lower vane), $D$ = horizontal distance AB.

```text
                          * upper vane
                     _.-' |  s
                _.-'      * lower vane
        a2 _.-' a1  . '   |  h
   A _.-'-------------------- B
   |<--------- D -------->|
```

From the two triangles formed with the horizontal through the instrument axis:

$$
\begin{aligned}
V_1 &= D\tan\alpha_1 \quad (\text{lower vane}) \\
V_2 &= D\tan\alpha_2 \quad (\text{upper vane}) \\
s &= V_2 - V_1 = D(\tan\alpha_2 - \tan\alpha_1)
\end{aligned}
$$

#### Horizontal distance

$$
\boxed{D = \frac{s}{\tan\alpha_2 - \tan\alpha_1}}
$$

#### Vertical component and RL

$$
V_1 = D\tan\alpha_1 = \frac{s\tan\alpha_1}{\tan\alpha_2 - \tan\alpha_1}
$$

The lower vane is at height $h$ above the staff foot B. The level of the vane equals the axis level plus $V_1$:

$$
RL_B + h = RL_A + h_i + V_1
$$

$$
\boxed{RL_B = RL_A + h_i + D\tan\alpha_1 - h}
$$

If the staff is held on a BM of known RL and the RL of the instrument station is wanted, $RL_A = RL_{BM} + h - h_i - D\tan\alpha_1$.

Here $RL_A$ = RL of the instrument station, $h_i$ = height of the instrument, and $h$ is read from the staff (or the height of the lower vane).

## Check yourself

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

1. In the tangential system the staff carries:
   A. Stadia hairs
   B. Two vanes a known distance s apart, with vertical angles measured to each
   C. A bubble level
   D. One vane and a bearing
   Answer: B. No stadia hairs are used, two vanes and two vertical angles. (Key point: Tangential method data)

2. For lower vane α1 and upper vane α2 (α2 > α1):
   A. D = s/(tanα2 - tanα1)
   B. D = s/(tanα2 + tanα1)
   C. D = s(tanα2 - tanα1)
   D. D = s/(tanα1 - tanα2)
   Answer: A. s = D tanα2 - D tanα1. (Key point: Distance for two elevations)

3. RL_B (staff foot) is:
   A. RL_A + h_i + D tanα1 - h
   B. RL_A - h_i - D tanα1 + h
   C. RL_A + D tanα1 + h
   D. RL_A + h_i - D tanα1 - h
   Answer: A. Vane level = axis + V1, and the foot is h below the lower vane. (Key point: RL expression)

4. The vertical component to the lower vane is:
   A. V1 = s tanα1/(tanα2 - tanα1)
   B. V1 = s/(tanα1)
   C. V1 = D cosα1
   D. V1 = s(tanα2 - tanα1)
   Answer: A. V1 = D tanα1 with D substituted. (Key point: Vertical component)

## Related in Surveying II

- [Tangential tacheometry, both angles depression](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-system-depression-expressions) — 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
- [Derive tangential angle 1718.87 C/R](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-angle-171887-derivation) — Surveying II, chapter 6
- [Tacheometry, methods, tangential formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atacheometry-methods-tangential-derivation) — Surveying II, chapter 2
- [Tangential method distance and elevation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-method-distance-measurement) — Surveying II, chapter 2
- [Tacheometry principle, stadia distance and RL](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atacheometry-principle-stadia-formula) — Surveying II, chapter 2

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