---
title: "Tacheometry principle, stadia distance and RL"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:tacheometry-principle-stadia-formula"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometry-principle-stadia-formula
site: BCT Engineering
---

# Tacheometry principle, stadia distance and RL

Surveying II · Chapter 2: Tacheometry

- Asked in: 2075 Baisakh, 8 marks
- Video: [Tacheometric Surveying: Introduction, Field Procedure, Principle and Distance Measurement](https://www.youtube.com/watch?v=HuJMSS3T070) — CIVIL Solved
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometry-principle-stadia-formula
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

Explain the principle of Tacheometric survey and also derive the formula to determine the horizontal distance and RL of the object with respect to instrument station when the staff is held in vertical.

## Short answer

Two stadia hairs subtend a fixed angle, so the staff intercept s is proportional to distance. For horizontal sight D=Ks+C; for inclined sight D=Ks cos^2 θ + C cosθ and V=1/2 Ks sin2θ + C sinθ, where K=f/i and C=f+d. RL of staff = RL of axis ± V − h.

## Answer

**Principle of stadia tacheometry:** the telescope has two additional 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 $C=f+d$ the **additive constant** (about 0.3 m for an external focusing telescope, nearly zero for an internal focusing telescope and exactly zero for an anallatic lens).

#### Inclined sight, staff vertical
Let the line of sight be inclined at $\theta$ and the vertical staff intercept be $AB=s$. The intercept normal to the line of sight is $s'=s\cos\theta$ (the hair angle is small). The inclined distance from the axis is $L=Ks\cos\theta+C$. Then

$$
\begin{aligned}
D&=L\cos\theta=Ks\cos^2\theta+C\cos\theta\\
V&=L\sin\theta=\tfrac12Ks\sin2\theta+C\sin\theta
\end{aligned}
$$

#### Reduced level
For a staff station with axial hair reading $h$:

$$
\text{RL of staff station}=\text{RL of instrument axis}\pm V-h
$$

(+ for an angle of elevation, − for depression), where RL of axis $=$ RL of the instrument station $+$ height of instrument.

The principle is used for contouring, detail surveys and traversing in hilly ground.

## Check yourself

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

1. Why is the staff intercept proportional to distance?
   A. Rays through the stadia hairs make a fixed small angle at the focus
   B. The staff length is fixed
   C. The vertical angle is fixed
   D. The hair interval varies with distance
   Answer: A. The fixed angle between rays means the intercept grows linearly with distance. (Key point: Stadia principle)

2. For a horizontal sight, D equals
   A. (f/i) s + (f + d)
   B. (i/f) s
   C. f s / i - d
   D. s/(f i)
   Answer: A. From similar triangles D = (f/i)s + (f + d) = Ks + C. (Key point: Horizontal sight formula)

3. What are K and C?
   A. K = f/i (about 100), C = f + d (about 0.3 m external focusing, zero for anallatic)
   B. K = i/f, C = f - d
   C. K = 0.3, C = 100
   D. K = f x i, C = 0 always
   Answer: A. K = f/i is the multiplying constant; C = f + d is the additive constant. (Key point: Constants)

4. With vertical staff and sight inclined at theta, D is
   A. Ks cos^2(theta) + C cos(theta)
   B. Ks cos(theta) + C
   C. Ks sin^2(theta) + C
   D. Ks / cos(theta)
   Answer: A. Intercept normal to sight is s cos(theta), so L = Ks cos(theta) + C and D = L cos(theta). (Key point: Inclined sight formulas)

5. What is the RL of the staff station?
   A. RL of axis ± V - h (+ for elevation, - for depression)
   B. RL of axis ± D + h
   C. RL of station - HI
   D. RL of axis x V
   Answer: A. RL = RL of axis ± V - h, where RL of axis = RL of station + HI. (Key point: RL)

## Related in Surveying II

- [RL, distance, gradient P to Q, instrument RL 1325.750](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-rl-distance-gradient-pq-1325) — Surveying II, chapter 2
- [Tacheometric RL of B, C, gradient BC](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-rl-b-c-gradient-bc-2) — Surveying II, chapter 2, asked 2×
- [Tacheometric RL of B, C, gradient BC](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-rl-b-c-gradient-bc) — Surveying II, chapter 2, asked 2×
- [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
- [Stadia principle, tangential method depression](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Astadia-principle-tangential-depression) — Surveying II, chapter 2
- [Tacheometry without staff, procedure](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Astaffless-tacheometry-procedure) — Surveying II, chapter 2

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