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TEC-GIS45-04 · Lesson 4 · 12 clock hours · lab

Measuring Horizontal and Vertical Angles

Objective. Students measure horizontal and vertical angles to three targets using direct and reverse face readings and compute the mean angle, identifying and correcting for index error.

⚠ Pending professional review

This lesson, its worked examples and its practice questions has not yet been verified by an industry-certified professional or by a certification or approval body. It is a study draft and may contain errors. Do not use it as instructions for work on a job site or on live equipment. Always follow your instructor, your supervising professional, and current official references and regulations.

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Why Direct and Reverse Face Readings Matter

A total station measures horizontal and vertical angles electronically, but no instrument is perfectly manufactured. Small mechanical imperfections—collimation error, trunnion-axis tilt, and vertical index error—introduce systematic bias into every reading. The standard field procedure for eliminating most of these errors is to observe each target in two instrument orientations: face left (the vertical circle is on the observer's left, also called direct) and face right (the vertical circle is on the observer's right, also called reverse). When you average the two readings correctly, the systematic errors cancel mathematically. This is one of the most important habits a survey technician develops, and it is tested on the NSPS CST Level 1–2 exam.

Setting Up the Total Station

Before taking any angle measurement, the instrument must be centered over the control point and leveled. Use the optical or laser plummet to center the instrument, then level the plate bubble and electronic level display to within the manufacturer's tolerance—typically one division of the plate bubble or a fraction of the displayed tilt value.

Safety first: When working on or near a roadway, parking lot, or any area with vehicle or foot traffic, set up safety cones or barricades before touching the instrument. Wear a high-visibility vest whenever you are within a highway right-of-way. Never leave the tripod unattended where it could be struck; a falling total station causes serious injury and costly damage. When carrying the instrument between setups, always use both hands and keep the instrument secured in its case for any move longer than a few steps.

Recording Face-Left and Face-Right Horizontal Angles

For horizontal angles, the procedure is:

1. With the instrument in face left, set the horizontal circle to zero (or a convenient initial value) while sighting the reference target (Target 1). Record the reading.

2. Rotate clockwise and sight Target 2; record the horizontal circle reading. Then sight Target 3; record.

3. Plunge (flip) the telescope to put the instrument in face right. Re-sight Target 3 first and record. Then rotate counterclockwise to Target 2 and Target 1, recording each reading.

4. For each target, the face-right reading should differ from the face-left reading by approximately 180°. The mean horizontal angle between two targets is:

Mean = ( Face-Left angle + (Face-Right angle ± 180°) ) ÷ 2

Record every reading in your field book the moment you take it—never reconstruct readings from memory. Use the standard columnar booking format shown in class, with columns for Point, Face, Hz Circle Reading, Zenith Angle, and Remarks.

Vertical Angles and Index Error

The total station's vertical circle should read exactly 90° (zenith angle) when the telescope is horizontal, and 0° when pointing straight up. If the instrument has a small index error (also called vertical circle index error or zero error), both face-left and face-right zenith readings are shifted by the same amount in opposite directions, making it detectable.

For a single target:

For a correctly adjusted instrument, Z_L + Z_R = 360°. Any departure from 360° is twice the index error:

Index Error (i) = ( Z_L + Z_R − 360° ) ÷ 2

The corrected mean zenith angle is:

Z_mean = Z_L − i OR equivalently Z_mean = (Z_L + (360° − Z_R)) ÷ 2

A small index error does not make the instrument unusable—the mean of face-left and face-right always removes it. However, if the index error exceeds the manufacturer's published limit (often a few seconds of arc for a construction-grade instrument), the instrument should be recalibrated or returned for service.

Comparing to Check Values and Identifying Systematic Errors

After computing your mean horizontal and vertical angles, compare them to the instructor's check values. The allowable discrepancy is two times the instrument's stated angular accuracy. For example, if the total station is rated at 5" (five arc-seconds), your mean angle should agree with the check value within 10". If you consistently exceed this tolerance on all three targets, suspect a systematic error such as:

Random errors that vary from target to target are normal and expected within the tolerance. A bias that shifts all your angles in the same direction suggests a setup or procedural problem.

Field Book Discipline

Your field book is a legal record. Use pencil, make corrections with a single line (never erase), initial any correction, and record the date, weather, instrument serial number, and your name on every page. A well-kept field book is required for the pass criterion of this lab and reflects professional practice expected by every licensed surveyor.

Key terms8

Face Left (Direct)Instrument orientation in which the vertical circle is on the observer's left side when sighting through the eyepiece; the starting position for a two-face observation set.
Face Right (Reverse)Instrument orientation in which the vertical circle is on the observer's right; achieved by plunging the telescope 180° after a face-left observation.
Zenith AngleThe vertical angle measured from the zenith (directly overhead, 0°) downward to the line of sight; a horizontal line of sight reads 90° zenith angle.
Vertical Index ErrorA systematic offset in the vertical circle reading caused by the instrument's index not being exactly at the true zero position; detected when the sum of face-left and face-right zenith angles departs from 360°.
Mean AngleThe arithmetic average of the face-left and face-right angle values, which cancels most systematic instrumental errors.
Collimation ErrorA systematic error caused by the line of sight (collimation axis) not being perpendicular to the trunnion axis; it is largely eliminated by averaging face-left and face-right readings.
Plunging the TelescopeRotating the telescope 180° about the trunnion axis to change from face left to face right (or vice versa) without moving the tripod.
Angular AccuracyThe manufacturer's published standard deviation or tolerance for a single direction measurement, typically expressed in arc-seconds (e.g., 5"); used to set pass/fail tolerances for field observations.

Worked examples4

1. A total station rated at 5" angular accuracy sights Target 2 from a reference direction set to 0°00'00" on face left. The face-left horizontal circle reads 47°18'24" for Target 2. The face-right reading for the same target (measured back from the same reference after plunging) is 227°18'32". Compute the mean horizontal angle to Target 2.
  1. Step 1 — Adjust the face-right reading: subtract 180° from the face-right reading to bring it onto the same arc as the face-left reading. 227°18'32" − 180°00'00" = 47°18'32".
  2. Step 2 — Average the two values: Mean = (47°18'24" + 47°18'32") ÷ 2 = 94°36'56" ÷ 2 = 47°18'28".
  3. Step 3 — Check the difference between the two: 47°18'32" − 47°18'24" = 0°00'08" = 8". This 8" spread is within 2 × 5" = 10", so the observation is acceptable.
Answer: Mean horizontal angle to Target 2 = 47°18'28"
2. For Target 1, the face-left zenith angle is 88°42'10" and the face-right zenith angle is 271°18'18". Calculate the vertical index error and the corrected mean zenith angle.
  1. Step 1 — Sum the two readings: 88°42'10" + 271°18'18" = 360°00'28".
  2. Step 2 — Find twice the index error: 360°00'28" − 360°00'00" = 0°00'28" = +28".
  3. Step 3 — Index error i = +28" ÷ 2 = +14".
  4. Step 4 — Corrected mean zenith angle = Z_L − i = 88°42'10" − 14" = 88°41'56".
  5. Step 5 — Verify with the face-right: (360° − 271°18'18") = 88°41'42"; average with face-left: (88°42'10" + 88°41'42") ÷ 2 = 177°23'52" ÷ 2 = 88°41'56". Confirms the answer.
Answer: Index error = +14"; corrected mean zenith angle = 88°41'56"
3. The instructor's check value for the mean horizontal angle to Target 3 is 112°05'14". A student computes a mean of 112°05'36". The total station is rated at 5" angular accuracy. Does the student's result pass or fail the lab tolerance?
  1. Step 1 — Find the discrepancy: 112°05'36" − 112°05'14" = 22".
  2. Step 2 — Compute the allowable tolerance: 2 × 5" = 10".
  3. Step 3 — Compare: 22" > 10", so the discrepancy exceeds the lab tolerance.
Answer: Fail — the 22" discrepancy exceeds the 10" tolerance; the student must identify and correct the source of error before re-observing.
4. A student's face-left and face-right horizontal readings for Target 1 (the reference) are 0°00'00" and 180°00'10" respectively. What is the difference from the expected 180°, and what does this indicate?
  1. Step 1 — Expected face-right reference reading: 0° + 180° = 180°00'00".
  2. Step 2 — Actual face-right reading: 180°00'10".
  3. Step 3 — Difference: 180°00'10" − 180°00'00" = 10".
  4. Step 4 — Interpretation: a 10" inconsistency on the reference target suggests a pointing error or a collimation error. For a 5" instrument the allowable spread on a single pointing is typically twice the rated accuracy (10"), so this is at the margin. The student should re-bisect the target carefully in both faces and re-record before proceeding.
Answer: Difference = 10"; at the margin of acceptance for a 5" instrument — re-observe before continuing.

The math in this lesson2

Learners reduce face-right readings by 180 degrees and average direct and reverse angles in degrees-minutes-seconds.
N-Q.1N-Q.3
Brush up in the math halls at newSchoolWorks.com:
In class: Sight three prism poles set at measured distances; record face-left and face-right readings in a standard field book format and compute mean horizontal angles · Calculate the vertical index error from the face-left and face-right zenith angle readings and correct the mean vertical angle · Compare student-derived angles to instructor-provided check values and identify any systematic errors exceeding the instrument's stated angular accuracy
Materials: Total station, three prism poles with reflectors, field book, scientific calculator, angle booking form, instructor check-angle sheet
Check: Mean horizontal angles agree with check values within two times the instrument's stated angular accuracy; field book formatted per instruction—both required for pass

Practice check8

Answer each question, then check it. Nothing is recorded; this is your own practice.

NSPS CST L1-2 — Field Procedures and Angle Measurement
1. What is the primary reason a surveyor observes each target in both face left and face right?
NSPS CST L1-2 — Vertical Angles and Index Error
2. A total station's face-left zenith angle to a target is 94°06'20" and the face-right zenith angle is 265°54'16". What is the vertical index error?
NSPS CST L1-2 — Vertical Angles and Index Error
3. Using the data from the previous question (Z_L = 94°06'20", index error = +18"), what is the corrected mean zenith angle?
NSPS CST L1-2 — Quality Control and Error Tolerance
4. A total station rated at 3" angular accuracy produces a mean horizontal angle that differs from the instructor's check value by 8". What is the correct conclusion?
NSPS CST L1-2 — Field Procedures and Angle Measurement
5. When computing the mean horizontal angle, the face-right reading must be adjusted before averaging. Which adjustment is correct?
NSPS CST L1-2 — Professional Practice and Documentation
6. Which of the following field book practices is required for a legally acceptable survey record?
NSPS CST L1-2 — Instrument Setup and Error Sources
7. A student sets up the total station but forgets to level the instrument fully before observing. What type of error is most likely introduced?
NSPS CST L1-2 — Workplace Safety
8. When observing near a roadway with a total station, which personal protective equipment and site precaution are required?
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