---
title: "Subtense bar principle and formula"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:subtense-bar-principle-formula"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/subtense-bar-principle-formula
site: BCT Engineering
---

# Subtense bar principle and formula

Surveying II · Chapter 2: Tacheometry

- Asked in: 2078 Chaitra, 2+4 marks; 2072 Asoj, 2+4 marks (asked 2 times)
- Video: [Subtense Bar Method | Tachymetry Methods of Horizontal Distance Measurement | Surveying](https://www.youtube.com/watch?v=gbuOsFrpkI4) — Jonas' Class Notes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/subtense-bar-principle-formula
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

Describe the working principle of subtense bar and derive subtense bar formula for computing horizontal and vertical distances when line of sight is inclined upward.

## Short answer

A subtense bar is a horizontal 2 m bar with targets at ends. The horizontal angle β is measured; horizontal distance D = (l/2) cot(β/2). For an upward inclined sight, D is unchanged and vertical distance V = D tan α, where α is the vertical angle.

## Answer

A **subtense bar** is a horizontal bar of fixed length $l$ (usually 2 m) with targets at its ends. It is mounted on a tripod over the station, levelled, and set perpendicular to the line joining it to the instrument by sighting through a small alignment telescope on the bar. The small **horizontal** angle $\beta$ subtended by the two targets at the theodolite is measured (to seconds, by repetition), and the distance is computed from it. It is used to measure distances over difficult ground (hilly terrain, water, thick vegetation) without taping.

```text
 Theodolite                       Subtense bar
     O  <------------ D ---------->  A |
       \  beta                       | l (2 m)
         \_____________________________ B
```
#### Working principle (plan view)
From the isosceles triangle $OAB$, with the bar normal to the line OM (M the middle of the bar):

$$
\frac{l}{2}=D\tan\frac{\beta}{2}\ \Rightarrow\ D=\frac{l}{2}\cot\frac{\beta}{2}\ \approx\ \frac{l}{\beta}\ (\beta\ \text{in radians})
$$

#### Line of sight inclined upward by angle $\alpha$ (derivation)
The bar is always kept **horizontal**, and the theodolite measures the angle $\beta$ on its horizontal circle, i.e. the angle projected onto the horizontal plane. Hence the triangle OAB above is a horizontal triangle whatever the inclination of the sight, and the distance found from it is directly the **horizontal distance**:

$$
D=\frac{l}{2}\cot\frac{\beta}{2}\qquad(\text{independent of }\alpha)
$$

The vertical angle $\alpha$ is then measured to the centre of the bar (the sighting target). From the vertical triangle formed by the instrument axis, the horizontal line and the inclined line of sight:

$$
\begin{aligned}
V&=D\tan\alpha=\frac{l}{2}\cot\frac{\beta}{2}\tan\alpha\\
S&=D\sec\alpha\quad(\text{inclined distance, if required})
\end{aligned}
$$

If the bar is not exactly normal to the line of sight in plan but is out by an angle $\delta$, the effective length is $l\cos\delta$ and the distance becomes $D=\frac{l\cos\delta}{2}\cot\frac{\beta}{2}$.

The RL of the bar station is RL of instrument axis $+V-h$, where $h$ is the height of the bar's sighting target above the station.

## Check yourself

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

1. What is measured with the theodolite in the subtense bar method?
   A. The vertical angle to the bar centre
   B. The small horizontal angle subtended by the bar targets
   C. The staff intercept between the stadia hairs
   D. The time of travel of a signal
   Answer: B. The bar has a fixed length l, and the small horizontal angle beta it subtends is measured. (Key point: Subtense bar principle)

2. For a horizontal line of sight, what is the distance D?
   A. D = (l/2) tan(beta/2)
   B. D = l tan(beta)
   C. D = (l/2) cot(beta/2)
   D. D = l sin(beta/2)
   Answer: C. From the isosceles triangle, l/2 = D tan(beta/2), so D = (l/2) cot(beta/2). (Key point: Horizontal distance formula)

3. When the line of sight is inclined upward by alpha, how are D and V obtained?
   A. D = S sin(alpha), V = S cos(alpha)
   B. D = S cos(alpha), V = S sin(alpha), where S = (l/2) cot(beta/2)
   C. D = S, V = S tan(alpha)
   D. D = S/cos(alpha), V = S/sin(alpha)
   Answer: B. The bar is normal to the inclined sight, so the inclined distance S is resolved into horizontal S cos(alpha) and vertical S sin(alpha). (Key point: Inclined line of sight)

4. If the bar is out of normal by angle delta, what changes?
   A. The effective length becomes l cos(delta)
   B. The angle beta is multiplied by cos(delta)
   C. The distance is unchanged
   D. The effective length becomes l / cos(delta)
   Answer: A. The bar's projection perpendicular to the line of sight is l cos(delta). (Key point: Bar not normal)

## Related in Surveying II

- [Subtense bar principle, fixed hair derivation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-fixed-hair-derivation) — Surveying II, chapter 2
- [Use and working principle of subtense bar](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-use-working-principle) — Surveying II, chapter 2
- [Subtense bar principle, increasing precision](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-principle-precision) — Surveying II, chapter 2
- [Subtense bar, RL difference, gradient, bearing AB](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-c-difference-rl-gradient) — Surveying II, chapter 2
- [Subtense bar and tacheometer, A-B distance and level](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-2m-theodolite-c-ab) — Surveying II, chapter 2
- [Subtense bar and tacheometer gradient A-B](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-3m-level-difference-gradient) — Surveying II, chapter 2

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