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

# Subtense bar principle, increasing precision

Surveying II · Chapter 2: Tacheometry

- Asked in: 2065 Chaitra (old course), 7 marks
- 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-precision
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

Write working principle of a subtense bar. How precision can be increased by using subtense bar for computed distance.

## Short answer

A horizontal subtense bar of length l subtends a small angle β, giving D=(l/2) cot(β/2). Precision improves by using a 1″ theodolite with repeated measurements, a longer invar bar, serial measurement of short sections, an auxiliary base, and keeping lines under 150 m with exact perpendicularity.

## Answer

#### Working principle
A horizontal subtense bar of fixed length $l$ (usually 2 m) is placed at the far end of the line, perpendicular to it, and the small horizontal angle $\beta$ it subtends at the theodolite is measured. The horizontal distance is $D=\frac{l}{2}\cot\frac{\beta}{2}\approx l/\beta$.

#### Increasing the precision
The error in distance for a small error $d\beta$ in the angle is

$$
dD=-\frac{D^2}{l}\,d\beta
$$

so the error increases with the square of the distance, and the relative error $dD/D=D\,d\beta/l$ grows as $D$ increases. The precision is improved by:
1. **Measuring $\beta$ more accurately:** use a 1" theodolite, repeat the angle several times (the error reduces by $\sqrt{n}$ for $n$ repetitions) and observe on both faces.
2. **Using a longer bar** (for example 4 m, or two bars end to end), which gives a larger angle.
3. **Serial method:** divide a long line into short sections and measure each section separately with the bar, so each angle is large.
4. **Auxiliary base method:** at one end of a long line, set out a short auxiliary base perpendicular to it and measure that base with the subtense bar; then measure the angle the auxiliary base subtends at the other end and compute the long line from it.
5. **Keeping lines short** (below about 150 m), with the bar exactly horizontal and perpendicular to the line and the targets steady.
6. **Using an invar bar** of accurately known length, so that its length hardly changes with temperature.

Example: for $D=100$ m, $l=2$ m, $d\beta=1''=4.85\times10^{-6}$ rad, $dD=100^2\times4.85\times10^{-6}/2=0.024$ m (about 1 in 4000).

## Check yourself

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

1. A subtense bar of length l subtends a small angle β at the theodolite. The distance is:
   A. D = (l/2) cot(β/2)
   B. D = l tan β
   C. D = 2l cot β
   D. D = (l/2) tan(β/2)
   Answer: A. From the isosceles triangle, D = (l/2) cot(β/2), approximately l/β for small β. (Key point: Subtense bar principle)

2. How is the subtense bar placed at the far end of the line?
   A. Horizontal and perpendicular to the line
   B. Vertical, along the line
   C. Horizontal, along the line
   D. Inclined parallel to the ground slope
   Answer: A. The bar must be horizontal and at right angles to the line so that the measured horizontal angle gives the distance. (Key point: Bar set-up)

3. Since dD = -(D^2/l) dβ, how does the error in distance vary with distance for a given angular error?
   A. It is independent of D
   B. It increases in proportion to D
   C. It decreases as D increases
   D. It increases with the square of D
   Answer: D. The error is proportional to D^2, so the precision dD/D falls as D increases. (Key point: Error with distance)

4. Which of these does NOT improve precision of subtense bar distances?
   A. Repeating the angle several times on both faces
   B. Using a longer bar or two bars end to end
   C. Using the auxiliary base method for long lines
   D. Observing lines as long as possible in one setting
   Answer: D. Long lines increase error as D^2; lines should be kept short (below about 150 m) or split by an auxiliary base. (Key point: Improving precision)

## Related in Surveying II

- [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, fixed hair derivation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-fixed-hair-derivation) — Surveying II, chapter 2
- [Subtense bar principle and formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-principle-formula) — Surveying II, chapter 2, asked 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
- [Subtense bar, level difference and gradient A-B](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-tacheometer-level-diff-gradient) — Surveying II, chapter 2

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