---
title: "Stadia principle, tangential method depression"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:stadia-principle-tangential-depression"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/stadia-principle-tangential-depression
site: BCT Engineering
---

# Stadia principle, tangential method depression

Surveying II · Chapter 2: Tacheometry

- Asked in: 2080 Chaitra, 2+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/stadia-principle-tangential-depression
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

Write down the principle of stadia tacheometry. Discuss measurement of horizontal distance and elevation by tangential method for both angles in depression.

## Short answer

Stadia tacheometry uses fixed stadia hairs subtending a constant angle; staff intercept s gives D=Ks+C. Tangential method with both depression angles: D=s/(tanα1-tanα2), V=H tanα2, and RL of staff station = RL of axis - H tanα1 - h.

## Answer

**Principle of stadia tacheometry:** the telescope has two extra horizontal hairs (stadia hairs), one above and one below the central cross-hair. The rays through these hairs make a fixed small angle at the focus, so the staff intercept $s$ between them is proportional to the distance of the staff from the instrument.

```text
 Diaphragm     Objective              Staff
   a |           |   \                  A  top
     | i         |     \  ...           |
  ---+-----------+-------F--------------+---- axis
     |           |      /               |
   b |           |   /                  B  bottom
     |<-- f --->|<--- D1 ---->|
```
#### Horizontal sight, staff vertical
From the similar triangles formed at the external focus $F$ (stadia hair interval $i$, focal length $f$, intercept $s$):

$$
\frac{D_1}{f}=\frac{s}{i}\ \Rightarrow\ D_1=\frac{f}{i}\,s
$$

Adding the distance of the focus from the instrument axis, $(f+d)$, where $d$ is the distance from the objective to the vertical axis:

$$
D=\frac{f}{i}\,s+(f+d)=Ks+C
$$

$K=f/i$ is the **multiplying constant** (usually 100) and $C=f+d$ is the **additive constant** (about 0.3 m for an external focusing telescope, and zero for an internal focusing or anallatic telescope).

#### Tangential method, both angles of depression
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 $\alpha_2$ that 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. In stadia tacheometry, the staff intercept is proportional to distance because:
   A. The stadia hairs subtend a fixed small angle at the instrument
   B. The staff length is fixed
   C. The focal length changes with distance
   D. The vertical angle is fixed
   Answer: A. Rays through the stadia hairs make a fixed angle, so the intercept increases with distance. (Key point: Stadia principle)

2. For D = Ks + C, K and C are:
   A. K = f/i multiplying constant; C = f + d additive constant
   B. K = i/f; C = f - d
   C. K = f + d; C = f/i
   D. K = 100 and C = 100 always
   Answer: A. K = f/i (usually 100) and C = f + d, about 0.3 m for external focusing, zero for anallatic. (Key point: Constants)

3. For both angles of depression, α1 (lower target) > α2 (upper target), the horizontal distance is:
   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 minus depth of B equals s, so s = H(tanα1 - tanα2). (Key point: Depression formula)

4. For both depressions, the RL of staff station C is:
   A. RL axis - H tanα1 - h
   B. RL axis + H tanα1 - h
   C. RL axis - H tanα1 + h
   D. RL axis + H tanα2 + s
   Answer: A. The lower target is H tanα1 below the axis and C is h below it. (Key point: RL of staff station)

## 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
- [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
- [Optical distance principle, tangential horizontal distance](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aoptical-principle-tangential-horizontal-distance) — 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
- [Tangential method distance and elevation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-method-distance-measurement) — Surveying II, chapter 2