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Home β€Ί Formulas β€Ί Surveying

Surveying & Levelling
Formula Reference

Essential equations for chain surveying, leveling, theodolite traversing, tacheometry, curve setting out, and area/volume calculations.

πŸ“ Chain Surveying & Corrections
βˆ’
Chain Correction (Incorrect Length)
Ltrue = L' Γ— (ltrue / l')
Correct measured distance if the chain/tape length is wrong.

L' = Measured distance

l' = Actual length of chain/tape used

ltrue = Correct length of chain/tape

Area Correction
Atrue = A' Γ— (ltrue / l')Β²
Corrected area when the measuring chain is incorrect.

A' = Measured area

Slope Correction
Cs = l βˆ’ d = l (1 βˆ’ cosΞΈ)
Difference between slope distance and horizontal distance. Always negative.

l = Slope distance

d = Horizontal distance

ΞΈ = Angle of slope

Approximation for small angles: Cs β‰ˆ hΒ² / (2l)
Temperature Correction
Ct = Ξ± Γ— (Tm βˆ’ To) Γ— l
Correction due to thermal expansion/contraction of tape.

Ξ± = Coefficient of thermal expansion

Tm = Mean temperature during measurement

To = Standard temperature at calibration

Pull Correction
Cp = (Pm βˆ’ Po) Γ— l / (A Γ— E)
Correction for difference in applied pull vs. standard pull.

Pm = Applied pull

Po = Standard pull

A = Cross-sectional area of tape

E = Modulus of elasticity

Refraction & Curvature Correction
C = 0.0673 Γ— DΒ²
Combined correction for earth curvature and atmospheric refraction in leveling.

D = Distance in km

C = Correction in meters (subtract from staff reading)

Curvature alone: Cc = 0.0785 DΒ² | Refraction alone: Cr = 0.0112 DΒ²
βš–οΈ Leveling Calculations
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Rise and Fall Method
RLn = RLn-1 Β± Rise/Fall
Calculate Reduced Level using difference between consecutive staff readings.

Rise = Backsight (BS) βˆ’ Foresight (FS) (if BS > FS)

Fall = FS βˆ’ BS (if FS > BS)

Height of Instrument (HI) Method
RL = HI βˆ’ Staff Reading
Calculate RL using the plane of collimation.

HI = RLbm + Backsight

Arithmetic Check
Ξ£BS βˆ’ Ξ£FS = Ξ£Rise βˆ’ Ξ£Fall = Last RL βˆ’ First RL
Verification formula for leveling field book. Must hold true for both methods.

All sums must be consistent.

Reciprocal Leveling Error
e = (hA + hB) / 2
Error due to earth curvature/refraction eliminated by reciprocal leveling.

hA = Difference in level from A

hB = Difference in level from B

πŸ”­ Theodolite & Traverse
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Magnetic Declination
WCB = MB Β± Declination
Convert Magnetic Bearing (MB) to Whole Circle Bearing (WCB).

Add declination if East, subtract if West

Latitude & Departure
Lat = L Γ— cos ΞΈ | Dep = L Γ— sin ΞΈ
Components of a traverse line in North-South and East-West directions.

L = Length of line

ΞΈ = Bearing (Azimuth)

Closing Error
e = √(ΣLat² + ΣDep²)
Magnitude of the closing error in a closed traverse.

Ξ£Lat = Algebraic sum of latitudes

Ξ£Dep = Algebraic sum of departures

Accuracy Ratio
AR = e / Perimeter
Measure of precision of the traverse.

Acceptable AR depends on order of survey (e.g., 1:5000 for detailed, 1:3000 for preliminary)

Bowditch's Rule (Compass)
Ξ΄L = (e / P) Γ— L
Distribute linear error of closure proportionally to length of each leg.

Ξ΄L = Correction to latitude/departure of line

e = Total closing error

P = Perimeter of traverse

L = Length of the specific line

πŸ“ Tacheometry
+
Horizontal Distance (Horizontal Sight)
D = k Γ— s + c
Distance calculated from staff intercept s when telescope is horizontal.

k = Multiplying constant (usually 100)

c = Additive constant (usually 0 for anallatic lens)

s = Staff intercept (Top βˆ’ Bottom hair)

Vertical Component (Inclined Sight)
V = Β½ k Γ— s Γ— sin(2Ξ±)
Vertical distance from instrument axis to staff.

Ξ± = Vertical angle

Reduced Level with Inclined Sight
RL = HI + V βˆ’ m
RL of staff point when telescope is inclined upwards.

HI = Height of Instrument

V = Vertical component

m = Middle hair reading

πŸ›£οΈ Curve Setting Out
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Deflection Angle
Ξ” = 180Β° βˆ’ I
Angle between tangent lines. I = Intersection angle.

I = Angle between back tangent and forward tangent

Tangent Length
T = R Γ— tan(Ξ” / 2)
Distance from Point of Intersection (PI) to Tangent Point (PC or PT).

R = Radius of curve

Length of Curve
L = (Ο€ Γ— R Γ— Ξ”) / 180Β°
Arc length of the simple circular curve.

Ξ” = Deflection angle in degrees

External Distance
E = R Γ— [sec(Ξ”/2) βˆ’ 1]
Distance from PI to the midpoint of the curve.
Mid-Ordinate
M = R Γ— [1 βˆ’ cos(Ξ”/2)]
Maximum offset from the long chord to the curve midpoint.
Rankine's Method (Tangential Angles)
Ξ΄ = 1718.9 Γ— (c / R) seconds
Deflection angle for a chord of length c.

c = Chord length

R = Radius

1718.9 = Conversion factor (180Γ—60 / Ο€)

πŸ—ΊοΈ Area & Volume Calculation
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Trapezoidal Rule
Area = d/2 Γ— [(O₁ + Oβ‚™) + 2(Oβ‚‚ + ... + Oₙ₋₁)]
Area under a curve divided into trapezoids. Assumes straight lines between ordinates.

d = Common distance between ordinates

O = Ordinate values

Simpson's 1/3rd Rule
Area = d/3 Γ— [(O₁ + Oβ‚™) + 4(Oβ‚‚ + Oβ‚„...) + 2(O₃ + Oβ‚…...)]
More accurate than trapezoidal. Requires odd number of ordinates (even number of divisions).

d = Common distance

Average End Area (Volume)
V = L Γ— (A₁ + Aβ‚‚) / 2
Volume between two cross-sections.

A₁, Aβ‚‚ = Areas of cross sections

L = Distance between sections

Prismoidal Formula (Volume)
V = L/6 Γ— (A₁ + 4Aβ‚˜ + Aβ‚‚)
More accurate volume calculation. Requires mid-area Aβ‚˜.

Aβ‚˜ = Area of middle section (not average of A₁, Aβ‚‚)

Coordinate Method (Gauss)
2A = Ξ£(xα΅’yα΅’β‚Šβ‚) βˆ’ Ξ£(yα΅’xα΅’β‚Šβ‚)
Area of a polygon given coordinates of vertices in order.

Vertices must be listed clockwise or counter-clockwise