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Levelling in Surveying — Complete Guide with 3 Examples & 10 GATE MCQs

⏱ 18 min read📅 June 2026✅ Survey Practice🎓 GATE relevant
Levelling is the surveying operation that determines the difference in elevation between points on the earth's surface. Whether you're setting floor levels for a building, designing road gradients, or planning drainage, levelling is the fundamental tool. This guide covers the complete theory — backsight/foresight/intermediate sight, the HI method and rise-fall method, arithmetic checks, curvature and refraction corrections — with three worked examples and 10 GATE MCQs.

📋 Table of Contents

  1. Introduction
  2. Concept and Theory
  3. Terminology and Definitions
  4. Key Formulas
  5. Important Tables
  6. Procedure
  7. Worked Examples (3)
  8. GATE MCQs (10)
  9. Common Mistakes
  10. Revision Summary
  11. Related Articles

1. Introduction

Levelling is the backbone of civil engineering site work. Before any construction begins, you must establish the reduced levels (RLs) of the existing ground, set benchmarks for construction reference, check that the foundation excavation is at the correct depth, and verify that floor slabs are poured at the right level. All of this requires levelling.

The instrument used is the dumpy level (or auto level / digital level), which provides a perfectly horizontal line of sight. By reading a graduated staff held vertically at different points, the surveyor determines how high or low each point is relative to a known benchmark.

2. Concept and Theory

The principle of levelling

A level instrument creates a horizontal line of sight. When you read a levelling staff at point A and get a reading of 1.500m, it means the horizontal line is 1.500m above point A. If you then read the staff at point B and get 2.300m, point B is 0.800m lower than point A (because the staff had to be extended more to reach the horizontal line). This difference in staff readings directly gives the difference in elevation.

Backsight, foresight, and intermediate sight

The backsight (BS) is the first reading taken from any instrument setup — it is always on a point of known RL (the benchmark or a previously determined point). The foresight (FS) is the last reading taken before shifting the instrument. Intermediate sights (IS) are readings taken on points between the BS and FS without moving the instrument. Understanding this terminology is critical — GATE questions frequently test it.

Two methods of booking

The Height of Instrument (HI) method is faster for field work: you calculate the height of the line of sight (HI = RL of BS point + BS reading), then subtract each staff reading from HI to get the RL of that point. The Rise and Fall method compares consecutive staff readings: if the next reading is smaller (staff shorter → ground is higher), it is a rise; if larger, it is a fall. Both methods give the same final RLs and have the same arithmetic check.

3. Terminology and Definitions

TermDefinition
Benchmark (BM)A point of known elevation (RL). GTS benchmarks are established by the Survey of India. Temporary BMs (TBMs) are set up for local work.
Reduced Level (RL)The elevation of a point above the reference datum (usually mean sea level).
Backsight (BS)First staff reading from an instrument position, taken on a point of known RL.
Foresight (FS)Last staff reading before shifting the instrument. Determines the RL of the next change point.
Intermediate Sight (IS)Any staff reading between BS and FS from the same instrument position.
Change Point (CP)A point where both a FS (from old position) and BS (from new position) are taken. It transfers the level.
Height of Instrument (HI)RL of the line of sight = RL of BS point + BS reading.

4. Key Formulas

HI Method
HI = RL of known point + BS reading
RL of any point = HI − staff reading (IS or FS)

Arithmetic check: ΣBS − ΣFS = Last RL − First RL
Rise and Fall Method
If next reading < previous reading → Rise = previous − next
If next reading > previous reading → Fall = next − previous
RL of next point = RL of previous point + Rise (or − Fall)

Arithmetic check: ΣBS − ΣFS = ΣRise − ΣFall = Last RL − First RL
Curvature and Refraction Correction
Combined correction = 0.0673 × d² (metres), where d is in km
Curvature makes distant objects appear lower → correction is subtracted
Refraction raises the line of sight → partially offsets curvature
Net: true RL is HIGHER than observed RL by 0.0673d²

5. Important Tables

Permissible Closing Error in Levelling

OrderPermissible ErrorApplication
Precise (1st order)±4√K mm (K in km)Geodetic surveys, major projects
Ordinary (2nd order)±12√K mmEngineering surveys, construction
Rough±25√K mmReconnaissance, preliminary

6. Step-by-Step Procedure

  1. Set up the level on a firm tripod. Level the bubble.
  2. Take BS reading on the staff held at the benchmark (known RL).
  3. Calculate HI = RL of BM + BS.
  4. Take IS readings on intermediate points. RL = HI − IS.
  5. Take FS reading on the change point before shifting.
  6. Shift instrument to new position. Take BS on the same change point.
  7. Repeat until all points are surveyed.
  8. Close on a known BM if possible (closing check).
  9. Perform arithmetic check: ΣBS − ΣFS = Last RL − First RL.

7. Worked Examples

Example 1 — Simple Two-Setup Levelling (HI Method)
BM RL = 100.000m. Readings: BS₁ = 1.500, IS = 2.100, IS = 1.800, FS₁ = 2.700 (CP). BS₂ = 0.900, IS = 1.400, FS₂ = 2.200.
Setup 1
HI₁ = 100.000 + 1.500 = 101.500
RL(IS₁) = 101.500 − 2.100 = 99.400
RL(IS₂) = 101.500 − 1.800 = 99.700
RL(CP) = 101.500 − 2.700 = 98.800
Setup 2
HI₂ = 98.800 + 0.900 = 99.700
RL(IS₃) = 99.700 − 1.400 = 98.300
RL(last) = 99.700 − 2.200 = 97.500
Arithmetic Check
ΣBS = 1.500 + 0.900 = 2.400
ΣFS = 2.700 + 2.200 = 4.900
ΣBS − ΣFS = 2.400 − 4.900 = −2.500
Last RL − First RL = 97.500 − 100.000 = −2.500 ✅ Check passed
Example 2 — Rise and Fall Method (Intermediate)
Same data as Example 1, processed using rise-fall method.
Consecutive Comparisons
BS₁=1.500 → IS=2.100: 2.100 > 1.500 → Fall = 0.600 → RL = 100 − 0.6 = 99.400
IS=2.100 → IS=1.800: 1.800 < 2.100 → Rise = 0.300 → RL = 99.4 + 0.3 = 99.700
IS=1.800 → FS=2.700: 2.700 > 1.800 → Fall = 0.900 → RL = 99.7 − 0.9 = 98.800
BS₂=0.900 → IS=1.400: 1.400 > 0.900 → Fall = 0.500 → RL = 98.8 − 0.5 = 98.300
IS=1.400 → FS=2.200: 2.200 > 1.400 → Fall = 0.800 → RL = 98.3 − 0.8 = 97.500
Check
ΣRise = 0.300, ΣFall = 0.600+0.900+0.500+0.800 = 2.800
ΣRise − ΣFall = 0.300 − 2.800 = −2.500 = Last RL − First RL ✅
Example 3 — Curvature and Refraction Correction
A level reading is taken at a distance of 2 km. What is the combined curvature and refraction correction?
Correction
C = 0.0673 × d² = 0.0673 × 2² = 0.0673 × 4 = 0.269m
The true RL of the distant point is 0.269m higher than the observed RL. At 2 km, this is significant — nearly 27 cm!

8. GATE MCQs

Q1. A backsight in levelling is:
  1. (a) The last reading from an instrument setup
  2. (b) The first reading on a point of known RL
  3. (c) A reading taken looking backward
  4. (d) Always the smallest reading
Answer: (b)
BS = first reading from any setup, always on a known RL point. The name "backsight" doesn't mean looking backward — it refers to the point you came from.
Q2. In the HI method, the RL of a point is calculated as:
  1. (a) HI + staff reading
  2. (b) HI − staff reading
  3. (c) Staff reading − HI
  4. (d) BS − FS
Answer: (b)
RL = HI − staff reading. HI is the height of the line of sight above datum; staff reading is how far below the line of sight the point is.
Q3. The arithmetic check in levelling is:
  1. (a) ΣBS + ΣFS = Last RL
  2. (b) ΣBS − ΣFS = Last RL − First RL
  3. (c) ΣBS = ΣFS always
  4. (d) ΣRise = ΣFall always
Answer: (b)
ΣBS − ΣFS = ΣRise − ΣFall = Last RL − First RL. This is the universal check for both methods.
Q4. Combined curvature and refraction correction for a distance of 1 km is:
  1. (a) 0.0673m
  2. (b) 0.0785m
  3. (c) 0.112m
  4. (d) 0.0341m
Answer: (a)
C = 0.0673 × d² = 0.0673 × 1² = 0.0673m = 67.3mm.
Q5. A change point in levelling is:
  1. (a) Where the instrument is set up
  2. (b) Where both FS and BS are taken
  3. (c) The benchmark
  4. (d) Where only IS is taken
Answer: (b)
A change point (CP) is where the staff remains while the instrument shifts. FS from old position and BS from new position are both taken here.
Q6. If BS = 2.500 and FS = 1.800 at the same HI, the ground at FS is:
  1. (a) 0.700m higher than BS point
  2. (b) 0.700m lower than BS point
  3. (c) At the same level
  4. (d) Cannot be determined
Answer: (a)
Smaller staff reading = ground is higher (staff didn't need to extend as much). Rise = 2.500 − 1.800 = 0.700m.
Q7. The advantage of the Rise and Fall method over the HI method is:
  1. (a) Faster computation
  2. (b) Provides a check on intermediate sights
  3. (c) Requires fewer readings
  4. (d) No arithmetic check needed
Answer: (b)
The R&F method compares every consecutive pair, so errors in IS readings are caught. In the HI method, IS errors only affect the individual point — the arithmetic check doesn't catch them.
Q8. Curvature of the earth makes distant objects appear:
  1. (a) Higher than actual
  2. (b) Lower than actual
  3. (c) At the same level
  4. (d) Shifted sideways
Answer: (b)
Earth curves away from the horizontal line of sight, so distant points drop below the line → they appear lower. The correction adds to the observed RL.
Q9. Permissible closing error in ordinary levelling over 4 km is:
  1. (a) ±24mm
  2. (b) ±48mm
  3. (c) ±12mm
  4. (d) ±100mm
Answer: (a)
±12√K = 12√4 = 12 × 2 = ±24mm.
Q10. If ΣBS = 8.400 and ΣFS = 10.200, and the starting RL is 100.000, the last RL is:
  1. (a) 101.800
  2. (b) 98.200
  3. (c) 108.400
  4. (d) 100.000
Answer: (b)
Last RL = First RL + (ΣBS − ΣFS) = 100.000 + (8.4 − 10.2) = 100 − 1.8 = 98.200.

9. Common Mistakes

Mistake 1: Confusing BS and FS. BS = first reading on known point. FS = last reading before shifting. It has nothing to do with which direction you're looking.
Mistake 2: Forgetting that smaller staff reading = higher ground. The staff reading is measured from the ground UP to the line of sight. If the ground is higher, the staff reading is smaller.
Mistake 3: Not performing the arithmetic check. ΣBS − ΣFS = Last RL − First RL must be verified. If it doesn't match, there's a booking or calculation error.
Mistake 4: Ignoring curvature correction for long sights. At 1 km, the correction is 67mm — significant for precise work. Always apply for distances > 300m.

10. Quick Revision Summary

Memorise:

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