---
title: "Angular vs linear precision in traversing"
subject: "Surveying II"
chapter: 1
key: "semester_4/surveying_ii:traverse-angular-linear-precision-relation"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/traverse-angular-linear-precision-relation
site: BCT Engineering
---

# Angular vs linear precision in traversing

Surveying II · Chapter 1: Traversing

- Asked in: 2080 Chaitra, 2+2+2 marks; 2075 Bhadra, 6 marks (asked 2 times)
- Video: [How to Calculate Linear Accuracy in Traverse Computations](https://www.youtube.com/watch?v=W3pzQQ3DsJ4) — Surveying Solutions
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/traverse-angular-linear-precision-relation
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1

## Question

Why degree of accuracy in between angular and linear measurements are to be maintained during traversing? By which expression angular versus linear and vice-versa precision are to be computed. State one of each example.

## Short answer

A traverse is only as accurate as its weaker measurement; mismatched accuracy wastes effort. Equivalent precision: 1/N = α" / 206265, so N = 206265/α" and α" = 206265/N. Example: 20" angles require 1:10,000 distances; 1:5000 tape needs 41.3" angles.

## Answer

#### Why angular and linear accuracy must be matched
A traverse is only as accurate as its weaker measurement. If the angles are measured to $1''$ but the lengths are taped to 1 in 1000, the extra effort on angles is wasted, and the reverse is equally true. An angular error $\alpha$ moves a station sideways by $L\alpha$ at a distance $L$, which acts like a linear error. So both are made to give the same position error, giving a balanced and economical survey (same accuracy for both lets Bowditch's rule be used).

#### Expressions
Linear precision is the closing error divided by the perimeter, $1/N$. An angular error $\alpha$ (in radians) gives a sideways error $L\alpha$ over a length $L$, so the equivalent linear precision is $\alpha$ itself:

$$
\frac{1}{N}=\alpha\ (\text{rad})=\frac{\alpha''}{206265}
$$

Hence

$$
\text{angular to linear: } N=\frac{206265}{\alpha''},\qquad \text{linear to angular: } \alpha''=\frac{206265}{N}
$$

#### Examples
- **Angular to linear:** angles measured to $20''$ need distances to $1/N=20/206265$, i.e. $N=206265/20\approx10313$, about 1 in 10,000.
- **Linear to angular:** lengths taped to 1 in 5000 need angles to $\alpha''=206265/5000=41.3''$, about $40''$ (a $1'$ least-count theodolite is enough for 1 in 3438, and a $20''$ one for 1 in 10,000).
- A quick rule: $1'\leftrightarrow1/3438$, $1''\leftrightarrow1/206265$.

## Check yourself

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

1. Why should angular and linear accuracy be kept in balance in a traverse?
   A. Angles are always more accurate than lengths
   B. Balanced accuracy removes all errors
   C. The traverse is only as accurate as its weaker measurement, so extra effort on the other is wasted
   D. Linear measurement does not affect position
   Answer: C. An angular error shifts a station sideways by L x alpha, which acts like a linear error, so both should give the same position error. (Key point: Why match accuracies)

2. Which expression connects linear precision 1/N and angular precision alpha in seconds?
   A. 1/N = alpha'' / 206265
   B. 1/N = alpha'' x 206265
   C. 1/N = alpha'' / 3438
   D. 1/N = 206265 / alpha''^2
   Answer: A. An angle alpha radians gives a sideways error L alpha, so precision 1/N = alpha (rad) = alpha''/206265. (Key point: Relation formula)

3. Angles measured to 20'' need distances measured to about what precision?
   A. About 1 in 1000
   B. About 1 in 10,000
   C. About 1 in 100,000
   D. About 1 in 3438
   Answer: B. N = 206265/20 = about 10313, i.e. around 1 in 10,000. (Key point: Angular to linear example)

4. Lengths taped to 1 in 5000 need angles measured to about what?
   A. About 1 second
   B. About 5 minutes
   C. About 3''
   D. About 41'' (roughly 40'')
   Answer: D. alpha'' = 206265/5000 = 41.3''. (Key point: Linear to angular example)

## Related in Surveying II

- [Field measurements in traversing, closed/open traverse](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-field-measurements-closed-open) — Surveying II, chapter 1
- [Field measurements in traversing, Bowditch rule](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-field-measurements-bowditch-rule) — Surveying II, chapter 1
- [Significance of traversing, accuracy parameters](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraversing-significance-accuracy-parameters) — Surveying II, chapter 1
- [Subtense bar principle, increasing precision](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-principle-precision) — Surveying II, chapter 2

## Related in other subjects

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- [Accuracy and precision defined](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aaccuracy-and-precision) — Surveying I, chapter 2, asked 2×
- [Accuracy, precision, sensitivity, resolution](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Aaccuracy-precision-sensitivity-resolution) — Instrumentation and Measurement, chapter 3
- [Linear and angular momentum of particle system](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Asystem-of-particles-linear-angular-momentum) — Applied Mechanics, chapter 9