Chapter 3 of 16

Chain & Compass Surveying

In the PSSSB JE Civil syllabus under Surveying · 2 parts

Chain & Tape Surveying · Compass Surveying & Bearings

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📑 Contents (22 sections)

Part 1 of 2

Chain & Tape Surveying

Last reviewed 16 Sept 2026 · 10 min read

Principle of chain surveying

Chain surveying is the simplest method of surveying in which only linear measurements are made in the field. The area is divided into a network of triangles whose sides are measured; details are located by offsets from the survey lines.

The principle is triangulation in its simplest form — a triangle is the only figure that can be plotted from its sides alone.
Triangles should be well-conditioned — no angle less than about 30° or greater than about 120° (ideally equilateral) — so that plotting errors are small.

Suitability: small, fairly level, open areas with simple details. Unsuitable for large areas, crowded or wooded areas, or undulating ground.

Instruments

Chains

Chain	Details
Metric chain	20 m (100 links, each 0.2 m) and 30 m (150 links); brass handles at ends (length measured over handles); tallies at regular intervals (e.g. every 2 m on a 20 m chain) and small rings at intermediate metre marks for easy reading
Gunter's chain	66 ft long with 100 links (each 0.66 ft) — convenient for acres (10 square chains = 1 acre) and miles (80 chains = 1 mile)
Engineer's chain	100 ft with 100 links of 1 ft
Revenue chain	33 ft with 16 links — used in cadastral (revenue) surveys

Chains are robust and easily repaired but change length (stretching, bending of links, wear) and are heavy.

Tapes

Tape	Features
Cloth or linen tape	Light, flexible, but stretches and shrinks easily — rough work, offsets
Metallic tape	Linen interwoven with fine brass/copper wires — common for offsets and ordinary work
Steel tape	Accurate; for precise linear measurement; corrodes and breaks if kinked
Invar tape	Alloy of nickel (about 36%) and steel — very low coefficient of thermal expansion; for base line measurement and precise work; expensive and delicate
Synthetic (fibreglass) tapes	Non-conducting, water resistant

Accessories

Arrows (chaining pins) — mark ends of each chain length.
Ranging rods — 2 m or 3 m long, painted in alternate red and white (or black and white) bands of 20 cm, with a pointed shoe — mark stations and range lines.
Offset rods — with a hook to pull the chain through obstructions and notches for setting out right angles.
Pegs — mark survey stations.
Plumb bob — transfer points and chaining on slopes.
Cross staff (open, French, adjustable), optical square and prism square — to set out perpendicular offsets (optical and prism squares use reflection to give 90°).
Line ranger — to locate intermediate points on a line between two stations.
Clinometer / Abney level — to measure slopes.

Survey stations and lines

Main stations — at the ends of main survey lines; subsidiary (tie) stations — on main lines, to run tie lines.
Base line — the longest line running through the middle of the area, on which the framework is built; measured most accurately.
Main survey lines — sides of the main triangles.
Check (proof) lines — run from a station to a point on the opposite side (or between points on two sides) to check the accuracy of the framework.
Tie lines — join tie stations to locate interior details and provide checks.

Selection of stations: mutually visible adjacent stations; well-conditioned triangles; minimum number of lines; lines through level ground and close to details (short offsets); avoid obstacles to chaining and ranging; lines not crossing busy roads frequently.

Ranging

Ranging is the process of establishing intermediate points on a straight line between two end points.

Direct ranging — when the ends are intervisible: intermediate ranging rods are aligned by eye (or line ranger).
Indirect (reciprocal) ranging — when the ends are not intervisible (e.g. due to a rise in ground): two persons with rods on intermediate points alternately align each other with the far end until both are in line.

Chaining on sloping ground

Direct method (stepping): the tape is held horizontal in short steps and the end is transferred to the ground by a plumb bob.
Indirect methods: measure the slope distance $L$ and:
- Angle measured: horizontal distance $= L cos θ$ ; correction $= L (1 - cos θ)$ .
- Difference in level $h$ measured: horizontal distance $= L^{2} - h^{2}$ ; slope correction $\approx \frac{h ^{2}}{2 L}$ .
- Hypotenusal allowance: at each chain length on a slope $θ$ , the arrow is placed ahead by $100 (sec θ - 1)$ links per chain, so that the horizontal distance recorded is directly correct.

Errors in chaining

Sources: incorrect length of chain or tape (standardisation), bad ranging (always makes measured length too long), bad straightening (too long), non-horizontality on slopes (too long), sag (too long), temperature variations (either sign), variation in pull (either sign), displacement of arrows, and personal mistakes (miscounting chains, wrong reading of tallies, wrong booking).

Errors such as bad ranging, bad straightening, sag and slope are positive (cumulative) errors — the measured length is too long.

Tape corrections

FormulaCorrections to measured length (L)

Correction	Formula	Sign
Standardisation (absolute)	$C_{a} = \frac{L c}{l}$ ( $c$ = correction per tape length $l$ )	+ if tape too long; − if too short
Temperature	$C_{t} = α (T_{m} - T_{0}) L$	+ if $T_{m} > T_{0}$
Pull (tension)	$C_{p} = \frac{( P - P _{0} ) L}{A E}$	+ if $P > P_{0}$
Sag	$C_{s} = \frac{L W ^{2}}{24 P ^{2}} = \frac{w ^{2} l ^{3}}{24 P ^{2}}$ per span	Always −
Slope	$C_{s l} = L (1 - cos θ) \approx \frac{h ^{2}}{2 L}$	Always −
Reduction to mean sea level	$C_{m c} = \frac{L h}{R}$ ( $h$ = mean elevation, $R$ = earth's radius ≈ 6370 km)	− for places above MSL
Alignment	$\frac{d ^{2}}{2 L}$ ( $d$ = deviation)	Always −

$α$ = coefficient of thermal expansion; $T_{0}$ = standard temperature; $P_{0}$ = standard pull; $A$ = cross-sectional area; $E$ = modulus of elasticity; $W$ = total weight of tape between supports; $w$ = weight per unit length; $l$ = span length.

FormulaNormal tension

The normal tension $P_{n}$ is the pull at which the elongation due to pull equals the shortening due to sag (the two corrections cancel):

P_{n} = \frac{0.204 W A E}{P _{n} - P _{0}}

(solved by trial).

Erroneous length of chain

If the true (designated) length of a chain is $L$ and its actual length is $L^{'}$ :

FormulaIncorrect chain length

True length = $\frac{L ^{'}}{L} \times$ measured length
True area = $(\frac{L ^{'}}{L})^{2} \times$ measured area
True volume = $(\frac{L ^{'}}{L})^{3} \times$ measured volume

If the chain length changed gradually during the work (tested at the start and end), use the mean of the initial and final errors.

Offsets

Offsets are lateral measurements from survey lines to locate details.

Perpendicular offsets — at right angles to the chain line (set by eye, cross staff, optical/prism square).
Oblique offsets — at any angle; two oblique offsets from two points on the chain line fix a detail by intersection.
Long offsets need accurate right angles; short offsets can be set by eye. The limiting length of offsets depends on the scale and the accuracy of setting the right angle.

Obstacles in chaining

Case	Examples	Methods
Chaining free, vision obstructed	Rising ground, hillock, small wood	Reciprocal ranging, or measuring around the obstacle using perpendiculars
Chaining obstructed, vision free	Pond, lake, river	For ponds: go around using perpendicular offsets, equal perpendiculars or similar triangles; for rivers: similar triangles, perpendicular and angle methods (e.g. setting perpendiculars and using geometric relations)
Both chaining and vision obstructed	Building	Establish parallel lines using perpendiculars or equilateral triangles to carry the line past the obstacle, then return to the original line

Field book

The field book records measurements. In single-line booking a single central line represents the chain line; in double-line booking the chainages are written between two parallel lines. Records include station names, chainages, offsets with sketches of details, and remarks. Entries start from the bottom of the page and go upwards.

Worked examples

Worked ExampleExample 1 — incorrect chain length

A 20 m chain was found to be 20.05 m long after measuring a distance of 750 m and an area of 2.5 ha. Find the true distance and area.

Solution. True distance $= \frac{20.05}{20} \times 750 = 751.9 m$ True area $= (\frac{20.05}{20})^{2} \times 2.5 = 1.00501 \times 2.5 = 2.5125 ha$

Worked ExampleExample 2 — temperature, pull and sag corrections

A 30 m steel tape was standardised at 20 °C under a pull of 50 N. It is used at 35 °C under a pull of 100 N, suspended between two supports. Take $α = 11.6 \times 1 0^{- 6}$ /°C, $A$ = 6 mm², $E = 2.1 \times 1 0^{5}$ N/mm² and tape weight 6 N for the span. Find the corrections for one tape length.

Solution. Temperature: $11.6 \times 1 0^{- 6} \times 15 \times 30 = + 0.00522$ m Pull: $\frac{( 100 - 50 ) \times 30 000}{6 \times 2.1 \times 1 0 ^{5}} = + 1.19$ mm $= + 0.00119$ m Sag: $\frac{30 \times 6 ^{2}}{24 \times 10 0 ^{2}} = - 0.0045$ m Net correction $= 0.00522 + 0.00119 - 0.0045 = + 0.0019 m$ per tape length

Worked ExampleExample 3 — slope correction

A slope length of 30 m has a difference in level of 1.5 m between its ends. Find the slope correction and horizontal distance.

Solution. $C_{s l} = \frac{1. 5 ^{2}}{2 \times 30} = 0.0375 m$ → horizontal distance $= 30 - 0.0375 = 29.9625 m$ (Exact: $3 0^{2} - 1. 5^{2} = 29.9625$ m ✓)

Worked ExampleExample 4 — hypotenusal allowance

Find the hypotenusal allowance per chain length (100 links) on a slope of 10°.

Solution. $100 (sec 10° - 1) = 100 \times 0.0154 = 1.54 links$

Frequently tested points

Chain surveying: only linear measurements; well-conditioned triangles (30°–120°); small open level areas.
Metric chains 20 m (100 links) and 30 m (150 links); Gunter's chain 66 ft; engineer's chain 100 ft; revenue chain 33 ft.
Invar tape — very low thermal expansion — base line measurement.
Ranging rods 2–3 m with 20 cm bands; optical/prism squares for right angles.
Base line, check lines (for checking), tie lines (details and checks).
Reciprocal ranging when ends are not intervisible.
Hypotenusal allowance $100 (sec θ - 1)$ links per chain.
Bad ranging, bad straightening, sag, slope → measured length too long.
Corrections: standardisation ±, temperature ±, pull ±, sag −, slope −, MSL −; normal tension balances pull and sag.
True length $(L^{'} / L) \times$ measured; area $(L^{'} / L)^{2}$ ; volume $(L^{'} / L)^{3}$ .

Common MistakeCommon mistakes

Adding the sag correction (it is always subtracted).
Using $(L^{'} / L)$ instead of $(L^{'} / L)^{2}$ for areas measured with a wrong chain.
Chaining along the slope and recording it as the horizontal distance without correction.

Revision SummaryChapter summary

Chain surveying measures only distances and fixes points using a framework of well-conditioned triangles.
Chains, tapes (especially steel and invar), ranging rods, arrows and right-angle instruments are the main equipment.
Main, check and tie lines form the framework; ranging and chaining on slopes need special methods.
Systematic errors are corrected by standardisation, temperature, pull, sag, slope and MSL corrections; incorrect chain length affects lengths, areas and volumes.
Offsets locate details and special methods overcome obstacles to chaining and ranging.

Part 2 of 2

Compass Surveying & Bearings

Last reviewed 16 Sept 2026 · 10 min read

Compass surveying

Compass surveying is a method of traversing in which directions (bearings) of survey lines are measured with a magnetic compass and lengths with a chain or tape.

Suitability: large areas where details are few, rough or preliminary surveys, wooded or crowded areas where triangulation by chain is impossible, and where speed is more important than high accuracy. Unsuitable near magnetic disturbances (iron structures, power lines, ore bodies).

Traverse — a series of connected survey lines whose lengths and directions are measured:

Closed traverse — starts and ends at the same point, or at points of known position (provides a check).
Open traverse — starts and ends at different unknown points (roads, rivers, pipelines); no inherent check.

Meridians and bearings

Bearing of a line is its horizontal angle with respect to a reference direction called the meridian.

Meridian	Definition
True (geographic) meridian	Line through the true north and south poles; fixed; found by astronomical observations
Magnetic meridian	Direction indicated by a freely suspended, balanced magnetic needle; changes with time and place
Grid meridian	Parallel to the central meridian of a map projection (used for national grid systems)
Arbitrary meridian	Any convenient direction (e.g. to a permanent object) — for small local surveys

Bearings accordingly are true, magnetic, grid or arbitrary bearings.

Systems of bearings

Whole circle bearing (WCB)	Quadrantal bearing (QB) / reduced bearing (RB)
Measured clockwise from north from 0° to 360°	Measured from north or south (whichever is nearer), towards east or west, from 0° to 90°
E.g. 45°, 135°, 225°, 315°	E.g. N 45° E, S 45° E, S 45° W, N 45° W
Used in the prismatic compass	Used in the surveyor's compass

FormulaConversion of WCB to QB

WCB	Quadrant	QB (RB)
0°–90°	NE	N θ E
90°–180°	SE	S (180° − θ) E
180°–270°	SW	S (θ − 180°) W
270°–360°	NW	N (360° − θ) W

Fore bearing and back bearing

Fore bearing (FB) of a line — bearing measured in the direction of progress of the survey (e.g. AB measured at A towards B).
Back bearing (BB) — bearing in the opposite direction (BA measured at B towards A).

B B = F B \pm 180°

(+ if FB < 180°, − if FB > 180°.) In the QB system, the back bearing has the same numerical value with opposite letters (N ↔ S, E ↔ W): e.g. FB = N 30° E → BB = S 30° W.

Included angles from bearings

The angle between two lines meeting at a point is the difference of their bearings measured from that point:

Included angle at B = bearing of BC - bearing of BA

(add 360° if negative). For a closed traverse of $n$ sides, the sum of interior angles $= (2 n - 4) \times 90°$ and of exterior angles $= (2 n + 4) \times 90°$ .

Computing bearings from included angles: bearing of the next line = bearing of the previous line + included angle ± 180° (for clockwise angles), adjusting by 360° as necessary.

Compasses

Prismatic compass

A graduated aluminium ring attached to a broad magnetic needle rotates with it; the ring is graduated in WCB with 0° at the south end (so that the reading through the prism gives the bearing of the line sighted), figures written inverted.
Sighting is done through the prism slit and the object vane (with a horse hair); the reading is taken simultaneously through the prism.
Can be used on a tripod or held in hand.

Surveyor's compass

The graduated ring is attached to the box (does not rotate); the needle moves freely over it.
Graduated in quadrantal bearings with E and W interchanged (so that the reading at the north end of the needle gives the correct quadrant).
Sighting and reading are not simultaneous; reading taken directly by looking at the needle from the top; must be used on a tripod.

Feature	Prismatic compass	Surveyor's compass
Needle	Broad (edge bar) type, rotates with the graduated ring	Thin needle, rotates over fixed ring
Graduated ring	Attached to the needle; rotates	Attached to the box; fixed
Graduations	WCB, 0° at south end, figures inverted	QB, 0° at N and S, E and W interchanged
Sighting	Prism slit and object vane with hair	Eye vane with fine slit and object vane
Reading	Through the prism — simultaneous with sighting	Directly from the top — after sighting
Tripod	Optional (hand-held possible)	Essential

Magnetic declination and dip

Declination

Magnetic declination is the horizontal angle between the true meridian and the magnetic meridian at a place.

East declination — magnetic north is east of true north.
West declination — magnetic north is west of true north.

FormulaTrue bearing from magnetic bearing

True bearing = Magnetic bearing + Declination (East)

True bearing = Magnetic bearing - Declination (West)

Isogonic lines — lines joining places of equal declination; agonic line — zero declination.

Variations in declination:

Secular variation — slow, long-period change (over centuries) — most important in surveying (old surveys must be corrected).
Annual variation — small yearly periodic change.
Diurnal (daily) variation — daily swing, larger in summer and near the poles.
Irregular variation — due to magnetic storms, earthquakes, solar flares.

Dip

The magnetic needle, if freely suspended, dips from the horizontal towards the nearer magnetic pole. Angle of dip is 0° at the magnetic equator and 90° at the magnetic poles. To keep the needle horizontal, a small sliding rider (weight) is placed on the higher end. Isoclinic lines join points of equal dip; the aclinic line is the magnetic equator.

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