---
title: "Chain stations and lines; well-conditioned triangle"
subject: "Surveying I"
chapter: 3
key: "semester_3/surveying_i:factors-stations-well-conditioned-triangle"
source: https://www.bctengineer.com/q/semester_3/surveying_i/factors-stations-well-conditioned-triangle
site: BCT Engineering
---

# Chain stations and lines; well-conditioned triangle

Surveying I · Chapter 3: Chain Survey

- Asked in: 2081 Baisakh, 2+2 marks
- Video: [what is chaining | chain surveying | Triangulation | well conditioned | Ill conditioned triangle](https://www.youtube.com/watch?v=j4BAO6B9wZw) — EASY CIVIL
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/factors-stations-well-conditioned-triangle
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-3

## Question

What factors should be considered in selecting stations as well as chain lines of a chain survey? Why is it necessary to use a well-conditioned triangle?

## Short answer

Stations must be intervisible, form well-conditioned triangles, and keep lines near details for short offsets; a long base line on level ground with check lines is essential. A well-conditioned triangle (angles 30–120 degrees) minimizes plotting error because a small measurement error causes large point displacement if intersection angle is small (displacement is about error/sinθ).

## Answer

#### Factors in selecting stations and chain lines
1. **Intervisibility.** Stations should be visible from each other so that lines can be ranged and chained easily.
2. **Well-conditioned triangles.** The lines should make triangles with all angles between $30^\circ$ and $120^\circ$.
3. **Base line.** Choose the longest line on level ground through the middle of the area; it carries the main framework.
4. **Short offsets.** Lines should pass close to the details so that offsets are not longer than 15 m.
5. **Minimum number of stations** and lines, to reduce work and errors.
6. **Ground.** Avoid steep slopes, ponds, buildings and thick bushes on lines.
7. **Check and tie lines** must be provided.
8. Stations should be on firm, permanent ground and tied to objects.

#### Why a well-conditioned triangle is necessary
A triangle is plotted from the lengths of its three sides by intersecting arcs. If an angle is very small or very large (an **ill-conditioned** triangle), the arcs intersect at a flat angle, and a small error in a measured length moves the plotted point by a large amount:

$$
\text{displacement} \approx \frac{\delta}{\sin\theta}
$$

where $\delta$ is the error in the side and $\theta$ the intersection angle. For $\theta = 15^\circ$, an error of 1 mm shifts the point 3.9 mm; for $\theta = 60^\circ$ only 1.15 mm. An equilateral triangle is the best shape; angles from $30^\circ$ to $120^\circ$ are acceptable. This keeps the plotting error small and the framework accurate.

## Check yourself

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

1. Why must stations be intervisible?
   A. So the stations can be at the same level
   B. To avoid check lines
   C. So the lines can be ranged and chained easily
   D. To shorten the offsets
   Answer: C. Ranging needs a clear view along each line. (Key point: Intervisibility)

2. Why is a well-conditioned triangle necessary?
   A. It needs fewer stations
   B. It removes the need for offsets
   C. It makes the lines shorter
   D. A small error in a side shifts the plotted point only slightly
   Answer: D. Arcs cut sharply so error is not magnified. (Key point: Why well-conditioned)

3. In an ill-conditioned triangle the arcs:
   A. Intersect at a flat angle, magnifying the position error
   B. Never intersect
   C. Intersect at exactly 90 degrees
   D. Are drawn with the same radius
   Answer: A. Displacement is about d / sin(theta), large for small theta. (Key point: Ill-conditioned effect)

4. Which should be avoided for chain lines?
   A. Level firm ground
   B. Steep slopes, ponds, buildings and thick bushes
   C. Fence lines
   D. Roads
   Answer: B. They create obstacles to chaining and ranging. (Key point: Ground for lines)

## Related in Surveying I

- [Principle of chain surveying; well-conditioned triangle](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Achain-principle-well-conditioned-vs-error) — Surveying I, chapter 3
- [Well-conditioned triangle; importance of tie line](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Awell-conditioned-triangle-tie-line) — Surveying I, chapter 3
- [Factors for choosing stations; offset types](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Afactors-stations-and-offset-types) — Surveying I, chapter 3
- [Selecting main stations; number of offsets](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Afactors-main-station-number-of-offsets) — Surveying I, chapter 3
- [Obstacles in chain survey and remedies](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aobstacles-chain-survey) — Surveying I, chapter 3, asked 4×
- [Field book and obstacles in chain survey](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Afield-book-and-obstacles) — Surveying I, chapter 3

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