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

# Subtense bar principle, fixed hair derivation

Surveying II · Chapter 2: Tacheometry

- Asked in: 2066 Magh (old course), 7 marks
- Video: [Tacheometry survey: Easiest way to derive Fixed hair method when angle of elevation in hindi](https://www.youtube.com/watch?v=Jl9jDLZMt5E) — Diksha Point
- 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-fixed-hair-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

Describe the working principle of subtense bar. Derive an expression to find the horizontal distance and height difference between the instrument station and staff point in the case of fixed hair method, line of sight is inclined and staff held vertical.

## Short answer

A subtense bar is a 2 m horizontal bar; the horizontal angle β it subtends gives D = (l/2) cot(β/2). In fixed-hair stadia with inclined sight, staff intercept s, normal intercept s cosθ, inclined distance L = K s cosθ + C, yielding D = Ks cos²θ + C cosθ and V = ½ Ks sin2θ + C sinθ.

## Answer

#### Working principle of the subtense bar
A horizontal bar of fixed length $l$ (usually 2 m) with targets at its ends is set perpendicular to the line of sight. The small horizontal angle $\beta$ it subtends is measured with a theodolite. From the isosceles triangle:

$$
D=\frac{l}{2}\cot\frac{\beta}{2}
$$

Since $\beta$ is measured in the horizontal plane, $D$ is the horizontal distance even when the sight is inclined. The bar is used to find distances over rough ground and to extend control.

#### Fixed hair (stadia) method, line of sight inclined, staff vertical
In the fixed hair method, the stadia hairs are fixed a constant distance $i$ apart in the diaphragm, and the intercept $s$ on the staff is read. The fixed hairs fix the parallactic angle, so $K=f/i$ is constant.

```text
        A (top hair)
        |\
        | \  s
 O ----------\-------- L = K s' + C
        | /
        B (bottom hair)
```
Let the line of sight make an angle $\theta$ with the horizontal, and let the vertical staff intercept be $AB=s$. The intercept normal to the line of sight is $s'=s\cos\theta$, so the inclined distance from the instrument axis to the staff is

$$
L=Ks'+C=Ks\cos\theta+C
$$

Horizontal distance and height difference:

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

The reduced level of the staff station is

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

where $h$ is the axial hair reading (use $-V$ for an angle of depression).

## Check yourself

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

1. How does the subtense bar give distance?
   A. Measure the small angle β subtended by a bar of length l: D = (l/2)cot(β/2)
   B. Measure the intercept on a staff
   C. Measure the vertical angle only
   D. Time the travel of light
   Answer: A. The isosceles triangle with base l and apex angle β gives D. (Key point: Subtense bar principle)

2. With an inclined sight and vertical staff, intercept s relates to normal intercept s' by?
   A. s' = s sinθ
   B. s' = s cosθ
   C. s' = s/cosθ
   D. s' = s
   Answer: B. The vertical intercept is foreshortened by cosθ when viewed normal to the line of sight. (Key point: Intercept normal to sight)

3. What is D for fixed hair method?
   A. D = Ks cosθ + C
   B. D = Ks sin^2θ + C sinθ
   C. D = Ks cos^2θ + C cosθ
   D. D = Ks + C cosθ
   Answer: C. L = Ks cosθ + C, and D = L cosθ. (Key point: Horizontal distance)

4. What is V?
   A. V = Ks cosθ
   B. V = (1/2)Ks cos2θ + C cosθ
   C. V = Ks tanθ
   D. V = (1/2)Ks sin2θ + C sinθ
   Answer: D. V = L sinθ = Ks cosθ sinθ + C sinθ. (Key point: Vertical component)

## Related in Surveying II

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- [Subtense bar, elevation difference and gradient P-Q](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-staff-elevation-difference-gradient-pq) — Surveying II, chapter 2
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- [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

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