---
title: "Tacheometry, methods, tangential formula"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:tacheometry-methods-tangential-derivation"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometry-methods-tangential-derivation
site: BCT Engineering
---

# Tacheometry, methods, tangential formula

Surveying II · Chapter 2: Tacheometry

- Asked in: 2079 Jestha, 1+1+4 marks
- Video: [Tangential Method of Tacheometric Surveying](https://www.youtube.com/watch?v=Vt0gh_NqghA) — 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-methods-tangential-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

What is tacheometric surveying? State the methods of tachometry. Derive an expression for determining distance and elevation using tangential method for both vertical angle being angle of depression and angle of elevation.

## Short answer

Tacheometric surveying uses optical instruments to measure distance and elevation without taping. Methods: stadia, tangential, subtense bar. In tangential method, D = s/(tanθ1−tanθ2) for elevation; for depression D = s/(tanα1−tanα2); V = D tanθ2 or D tanα2 accordingly.

## Answer

**Tacheometric surveying** is a rapid method of surveying in which the horizontal distance and the difference in elevation of a point are found from optical observations on a staff with a tacheometer, without taping or levelling.

#### Methods of tacheometry
1. **Stadia method:** fixed hair (hairs fixed, staff intercept varies) and movable hair (hairs moved to fixed marks on the staff).
2. **Tangential method:** vertical angles to two targets on the staff.
3. **Subtense bar method:** the horizontal angle subtended by a bar of fixed length.
4. **Special methods:** use of the Beaman stadia arc, or an EDM/total station.

#### Tangential method
In the **tangential method** the stadia hairs are not used. The telescope is pointed at two targets (vanes) on a staff, a known distance $s$ apart, and the two vertical angles are measured. The distance is calculated from the tangents of these angles. It is used when the telescope has no stadia hairs, or when the distance is too large for the stadia intercept to be read accurately.

```text
                   _ B  (upper target)
               _ -  |
        theta1 -    | s
    O  ----------   A  (lower target)  
        theta2 \_   |
 axis ----------\__ |       horizontal
                  D
```
Let $D$ be the horizontal distance of the staff from the instrument axis, $V$ the vertical height of the lower target A above (or below) the axis, and $h$ the height of the lower target above the foot of the staff (the staff station Q).

#### Case 1: both angles are angles of elevation
The angle to the upper target is $\theta_1$ and to the lower target $\theta_2$ ($\theta_1>\theta_2$):

$$
\begin{aligned}
V+s&=D\tan\theta_1,\qquad V=D\tan\theta_2\\
s&=D(\tan\theta_1-\tan\theta_2)\\
D&=\frac{s}{\tan\theta_1-\tan\theta_2},\qquad V=\frac{s\tan\theta_2}{\tan\theta_1-\tan\theta_2}
\end{aligned}
$$

Reduced level of the staff station Q:

$$
\text{RL}_Q=\text{RL of axis}+V-h
$$

#### Case 2: both angles are angles of depression
The angle to the lower target is $\alpha_1$ and to the upper target $\alpha_2$ ($\alpha_1>\alpha_2$). The lower target is at depth $D\tan\alpha_1$ below the axis and the upper one at $D\tan\alpha_2$:

$$
\begin{aligned}
s&=D\tan\alpha_1-D\tan\alpha_2=D(\tan\alpha_1-\tan\alpha_2)\\
D&=\frac{s}{\tan\alpha_1-\tan\alpha_2},\qquad V=D\tan\alpha_2\ (\text{depth of the upper target})\\
\text{RL}_Q&=\text{RL of axis}-V-s-h
\end{aligned}
$$

since the lower target is $s$ below the upper one, so the staff station is $V+s+h$ below the axis (equivalently, $\text{RL}_Q=\text{RL of axis}-D\tan\alpha_1-h$).

#### Case 3: one angle of elevation $\theta$ (upper target) and one of depression $\alpha$ (lower target)
The upper target is $D\tan\theta$ above the axis and the lower target $D\tan\alpha$ below it:

$$
\begin{aligned}
s&=D\tan\theta+D\tan\alpha=D(\tan\theta+\tan\alpha)\\
D&=\frac{s}{\tan\theta+\tan\alpha},\qquad V=D\tan\alpha\ (\text{depth of the lower target})\\
\text{RL}_Q&=\text{RL of axis}-V-h
\end{aligned}
$$

## Check yourself

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

1. What is tacheometric surveying?
   A. Rapid surveying where horizontal distance and elevation difference come from optical measurements on a staff, without taping or levelling
   B. Surveying using only a chain and tape
   C. Surveying by magnetic bearings alone
   D. Surveying by aerial photographs
   Answer: A. Tacheometry finds distance and elevation difference rapidly by optical readings. (Key point: Definition)

2. What is measured in the tangential method?
   A. The staff intercept between two stadia hairs
   B. Vertical angles to two targets a known distance s apart on a staff
   C. The horizontal angle subtended by a 2 m bar
   D. The magnetic dip
   Answer: B. Stadia hairs are not used; two vertical angles to targets s apart give the distance. (Key point: Tangential method principle)

3. For both angles of elevation theta1 (upper) and theta2 (lower), what is D?
   A. s/(tan theta1 + tan theta2)
   B. s/(tan theta1 - tan theta2)
   C. s (tan theta1 - tan theta2)
   D. s/(sin theta1 - sin theta2)
   Answer: B. V + s = D tan theta1 and V = D tan theta2, so s = D(tan theta1 - tan theta2). (Key point: Both angles of elevation)

4. For both angles of depression (alpha1 to the lower target, alpha2 to the upper), what is RL of the staff station?
   A. RL axis + V + s + h
   B. RL axis - V - s - h, with V = D tan alpha2
   C. RL axis - V + s - h
   D. RL axis + V - s - h
   Answer: B. The station lies V + s + h below the axis, where V = D tan alpha2 is the depth of the upper target. (Key point: Angles of depression)

5. With one angle of elevation theta (upper) and one of depression alpha (lower), what is D?
   A. s/(tan theta - tan alpha)
   B. s/(tan theta + tan alpha)
   C. s (tan theta + tan alpha)
   D. s/(tan theta x tan alpha)
   Answer: B. The target separation is the sum D tan theta + D tan alpha, so D = s/(tan theta + tan alpha). (Key point: Elevation and depression mix)

## 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
- [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
- [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 method distance and elevation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-method-distance-measurement) — Surveying II, chapter 2
- [Degree of curve, tangential angle formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Adegree-of-curve-tangential-angle-derivation) — Surveying II, chapter 6
- [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

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