---
title: "Measuring horizontal and vertical angles"
subject: "Surveying I"
chapter: 7
key: "semester_3/surveying_i:horizontal-and-vertical-angle-methods"
source: https://www.bctengineer.com/q/semester_3/surveying_i/horizontal-and-vertical-angle-methods
site: BCT Engineering
---

# Measuring horizontal and vertical angles

Surveying I · Chapter 7: Transit and Theodolite

- Asked in: 2062 Poush, 9 marks
- Video: [Measuring horizontal and vertical angles (Theodolite Surveying Part-I)](https://www.youtube.com/watch?v=Mb2jbdqMJHA) — The Engineering Surveyor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/horizontal-and-vertical-angle-methods
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-7

## Question

Describe with sketches the methods of measuring horizontal and vertical angles by theodolite.

## Short answer

Horizontal angles are measured by ordinary (single), repetition (adding angle several times), or reiteration (direction) methods. Vertical angles are obtained by levelling the altitude bubble, bisecting the target on both faces, and computing vertical angle = 90° − zenith distance after removing index error.

## Answer

#### Horizontal angles
**1. Ordinary method (single angle).** Set up and level the theodolite at O. Set zero (or any reading) on the circle, bisect A using the lower clamp, then release the upper clamp, bisect B and read both verniers. Change face, repeat, and take the mean. The angle AOB = reading on B - reading on A.

**2. Repetition method** - a single angle is added several times on the circle by using the upper and lower clamps alternately; the angle = total / number of repetitions.

**3. Reiteration (direction) method** - several directions are observed from a reference object, the angles are found from differences of directions, and sets are taken with different zero settings.

```text
        B
        /
   A --- O          horizontal angle AOB at O
```

#### Vertical angles
A vertical angle is the angle between the line of sight and the horizontal plane (an angle of elevation or depression).

1. Set up the theodolite and level it with reference to the altitude bubble. Set the vertical-circle verniers to zero with the vertical clamp and tangent screw, and bring the altitude bubble central with the clip screw; the line of sight is then horizontal.
2. On face left, bisect the target using the vertical clamp and tangent screw. Check that the altitude bubble is central and read both verniers of the vertical circle.
3. Change face (transit the telescope) and repeat. The mean of the two results is the vertical angle; the index error is removed.

$$
\text{Index error} = \frac{(FL + FR) - 360^\circ}{2}, \quad \text{Vertical angle} = 90^\circ - Z
$$

where $Z = (FL - FR + 360^\circ)/2$ is the mean zenith distance, for a zenith-type circle.

```text
                    target P
                  /
      horizontal /  alpha (angle of elevation)
   O ------------+
```

## Check yourself

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

1. In the ordinary method, how is angle AOB obtained?
   A. Reading on A + reading on B
   B. Reading on B - reading on A
   C. Total / number of repetitions
   D. Mean of the vertical readings
   Answer: B. The angle is the difference of the readings on the two targets, averaged over both faces. (Key point: Ordinary method)

2. What is a vertical angle?
   A. The angle between the line of sight and the horizontal plane
   B. The angle between two ground lines
   C. The angle between the trunnion axis and vertical axis
   D. The angle of the plate bubble
   Answer: A. It is the angle of elevation or depression of the line of sight from the horizontal. (Key point: Vertical angle definition)

3. Given FL and FR zenith readings, what is the index error?
   A. (FL - FR)/2
   B. (FL + FR - 360°)/2
   C. 90° - Z
   D. FL + FR
   Answer: B. For a correct instrument FL + FR = 360°, so index error = ((FL + FR) - 360°)/2. (Key point: Index error)

4. Why is the vertical angle observed on both faces?
   A. To close the horizon
   B. To remove the index error by taking the mean
   C. To reduce centring error
   D. To avoid levelling
   Answer: B. The index error acts oppositely on the two faces and cancels in the mean. (Key point: Why both faces for vertical angle)

## Related in Surveying I

- [Methods of measuring horizontal angles](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Amethods-of-measuring-horizontal-angles) — Surveying I, chapter 7, asked 2×
- [Horizontal angles from field notes ABCDA (2065)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atheodolite-field-notes-abcda-65-shrawan) — Surveying I, chapter 7
- [Horizontal angles from field notes, triangle ABC](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atheodolite-field-notes-abc-64-jestha) — Surveying I, chapter 7
- [Mean horizontal angles, station O (P, Q, R)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Amean-horizontal-angles-o-pqrp) — Surveying I, chapter 7
- [Theodolite traverse ABCDA with missing VCR (68°30′10″)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atheodolite-traverse-abcda-hcr-vcr-68-30-10) — Surveying I, chapter 7
- [Theodolite traverse ABCDA, booking format with VCR](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atraverse-abcda-booking-format-vcr-120-14-20) — Surveying I, chapter 7

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