---
title: "Plain, diagonal and vernier scales"
subject: "Surveying I"
chapter: 1
key: "semester_3/surveying_i:plain-diagonal-vernier-scales"
source: https://www.bctengineer.com/q/semester_3/surveying_i/plain-diagonal-vernier-scales
site: BCT Engineering
---

# Plain, diagonal and vernier scales

Surveying I · Chapter 1: Introduction

- Asked in: 2065 Shrawan, 8 marks; 2063 Baisakh, 3 marks (asked 2 times)
- Video: [Surveying: Different types of Scale, Plain, Vernier, Shrinkage factor, Representative fraction](https://www.youtube.com/watch?v=bKLWrtdMKJ8) — ShreeDevi Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/plain-diagonal-vernier-scales
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-1

## Question

What are plain, diagonal and vernier scales? Make appropriate sketches and show some measurements.

## Short answer

A plain scale reads two units like metres and decimetres. A diagonal scale reads three units using diagonals and parallel lines to subdivide further, e.g., metres, decimetres, centimetres. A vernier scale is a sliding scale; if n vernier divisions equal n-1 main divisions, its least count is main division value divided by n.

## Answer

A **scale** is a ruler-like graduated line used to measure ground distances on a map. Plain, diagonal and vernier scales are the three common types; each one reads to a smaller unit than the one before.

#### Plain scale
A plain scale has main divisions and one set of sub-divisions. It reads two units, e.g. metres and decimetres, but cannot read a third unit exactly. Example: RF = 1/200, so 1 cm = 2 m; a scale 10 cm long represents 20 m. Each main division (1 cm) is 2 m, and the left division is divided into 10 parts of 0.2 m each.

```text
 m 2    1   0      2    4    6    8   (m)
   |....|...|------|----|----|----|----|
   <-sub-> <-- main divisions (2 m) -->
```
Reading: 7.4 m = 3 main divisions (6 m) + 7 sub-divisions on the left (1.4 m).

#### Diagonal scale
A diagonal scale reads three units (e.g. 100 m, 10 m, 1 m). Each sub-division is divided further by a set of equally spaced horizontal lines and diagonal lines drawn across them; the principle is that of similar triangles.

Example: RF = 1/4000 (1 cm = 40 m), to read 1 m up to 400 m. Length of scale = 400/4000 m = 10 cm. Divide it into 4 main divisions of 2.5 cm (100 m). Divide the left main division into 10 sub-divisions of 0.25 cm (10 m). Draw 10 equally spaced horizontal lines above the base. Join the top end of the first sub-division to the zero point on the base line, and draw diagonals parallel to it through the other sub-divisions. Each horizontal line moves the diagonal along by $\frac{1}{10}$ of 10 m = 1 m.

```text
   10m
 10|  /./././././././/|
  8| / /  .. 
  6|/ / 
  4|  
  2|__|_____|_____|___|
   100 0   100  200 300
```
Measurement of 234 m: 200 m (2 main divisions) + 30 m (3 sub-divisions) + 4 m (4th horizontal line).

#### Vernier scale
A vernier scale is a small sliding scale used with a main scale. It reads a fraction of the smallest main division. If $n$ divisions of the vernier equal $(n-1)$ divisions of the main scale, then

$$
\text{least count} = \frac{\text{value of 1 main division}}{n}
$$

Example: 1 main division = 1 m and $n$ = 10, so the least count = 0.1 m. If the zero of the vernier is just past 4 m and the 3rd vernier line coincides with a main-scale line, the reading is $4 + 3 \times 0.1 = 4.3$ m.

The vernier is mostly used on instruments such as theodolites, target levelling staves and compasses, while plain and diagonal scales are drawn on maps.

## Check yourself

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

1. A plain scale can read
   A. three units
   B. two units, such as metres and decimetres
   C. only one unit
   D. any unit using diagonal lines
   Answer: B. A plain scale has main divisions and one set of sub-divisions so reads only two units. (Key point: Plain scale)

2. A diagonal scale reads the third unit by using
   A. similar triangles with parallel horizontal lines and diagonals
   B. a sliding scale
   C. a divider and the plain scale only
   D. a magnifying glass
   Answer: A. Diagonals crossing equally spaced horizontal lines divide a sub-division into tenths by similar triangles. (Key point: Diagonal scale principle)

3. For RF 1/4000, how long is a diagonal scale to read up to 400 m?
   A. 16 cm
   B. 1 cm
   C. 10 cm
   D. 40 cm
   Answer: C. 400 m = 40000 cm; divided by 4000 gives 10 cm. (Key point: Diagonal example)

4. If 1 main division = 1 m and n = 10 vernier divisions, the least count is
   A. 1 m
   B. 0.01 m
   C. 10 m
   D. 0.1 m
   Answer: D. Least count = value of 1 main division / n = 1/10 = 0.1 m. (Key point: Vernier least count)

## Related in Surveying I

- [Obstacles; field sketch for detail survey](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aobstacles-and-field-sketch) — Surveying I, chapter 3

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