FS Exam Formula Sheet

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Fundamentals of Surveying — 63 formulas

Surveying Math / COGO

Distance Between Two Points

d = distance; (x₁,y₁) and (x₂,y₂) are point coordinates

Azimuth from Coordinates

Alpha = azimuth (apply quadrant adjustment); Delta E = easting difference, Delta N = northing difference

Latitude (N/S Component)

d = distance; alpha = azimuth or bearing angle

Departure (E/W Component)

d = distance; alpha = azimuth or bearing angle

Area by Coordinates (Shoelace)

Coordinate method; vertices listed sequentially, with (x_{n+1},y_{n+1}) = (x₁,y₁)

Area of Triangle (SAS)

a, b = two sides; C = included angle

Law of Sines

a, b, c = sides; A, B, C = opposite angles

Law of Cosines

c = side opposite angle C; a, b = other two sides

Traverse

Linear Error of Closure

Sum of adjusted latitudes and departures after balancing

Relative Error of Closure

E_L = linear error; P = total perimeter; express as 1:n

Compass Rule Correction (Latitude)

d_i = length of course i; P = perimeter; Sigma Lat = total latitude misclosure

Compass Rule Correction (Departure)

d_i = length of course i; P = perimeter; Sigma Dep = total departure misclosure

Curves (Horizontal)

Degree of Curve (Arc Definition)

D = degree of curve; R = radius in feet

Tangent Length

T = tangent distance; R = radius; Delta = deflection angle

Curve Length

L = arc length; Delta in degrees; R = radius

Long Chord

C = chord from PC to PT; R = radius; Delta = central angle

External Distance

E = external distance; sec = 1/cos; Delta = central angle

Middle Ordinate

M = middle ordinate distance from midpoint of chord to midpoint of arc

Curves (Vertical)

Vertical Curve Elevation

y_BVC = elevation at BVC; g₁, g₂ = grades (decimal); x = distance from BVC; L = curve length

High/Low Point Station

x = distance from BVC to high/low point; g₁, g₂ = grades (decimal); L = curve length

Rate of Grade Change

r = rate of change per station; g₁, g₂ = grades; L = curve length

Stopping Sight Distance (crest, S < L)

A = |g₂ − g₁| (in %); S = sight distance; h₁ = driver eye height; h₂ = object height. Crest curve where sight distance < curve length.

Stopping Sight Distance (sag, S < L)

A = |g₂ − g₁|; H = headlight height (≈ 2.0 ft); β = headlight upward angle (≈ 1°). AASHTO sag-curve formula.

Geodesy & Datums

Orthometric vs Ellipsoid Height

H = orthometric height (NAVD88); h = ellipsoid height (GPS); N = geoid undulation from a geoid model (e.g., GEOID18)

Lambert Conformal Conic — point scale factor

k₀ = scale on the central parallel; φ₀ = origin latitude; φ = point latitude; N = radius of curvature in the prime vertical; R = mean radius. Used for east–west zones.

Transverse Mercator — point scale factor

k₀ = scale on the central meridian (typically 0.9996 for UTM, 0.9999 for many SPCS); x = grid easting from central meridian; R = mean radius.

Combined Factor (CF)

Multiplies a ground distance to obtain grid distance; k = grid scale factor; h = orthometric height of the project area; R ≈ 20,902,000 ft = 6,371,000 m.

Convergence (Lambert)

γ = grid convergence (azimuth − bearing-from-grid-north); λ₀ = central meridian; λ = point longitude; φ₀ = origin latitude.

Leveling

Curvature & Refraction (US)

F = distance in thousands of feet; h = correction in feet

Curvature & Refraction (Metric)

D = distance in km; h = correction in meters

Differential Leveling

BS = backsight reading; FS = foresight reading

Allowable Leveling Closure

k = constant per order (e.g. 0.012 ft for 2nd order); M = distance in miles

EDM / Distance

Slope to Horizontal Distance

H = horizontal distance; S = slope distance; v = vertical angle from horizontal

Sea Level Correction

R = earth radius (~20,902,000 ft, or 6,371,000 m); h = elevation above sea level; H = measured distance

Combined Scale Factor

Grid factor from state plane projection; elevation factor = R/(R+h)

Grid Distance

Convert ground distance to grid (state plane) distance using combined scale factor

Slope to Horizontal

S = slope distance; θ = vertical angle from horizontal

Tape Temperature Correction

α ≈ 6.45×10⁻⁶ /°F (steel); T = field temp, T₀ = standardization temp (typ. 68°F); L = measured length

Tape Pull (Tension) Correction

P = field pull; P₀ = standard pull; A = cross-section area; E = modulus of elasticity (29×10⁶ psi for steel)

Tape Sag Correction

w = weight per unit length; L = unsupported length; P = applied pull. Always negative (taut tape always reads long).

Stadia Horizontal Distance

K = stadia interval factor (typ. 100); s = rod intercept; α = vertical angle; C = stadia constant (typ. 0 for internal-focusing instruments)

Stadia Vertical Distance

V = vertical component; K = stadia factor; s = rod intercept; α = vertical angle

Error Theory / Statistics

Standard Deviation

v_i = residual (observation minus mean); n = number of observations

Standard Error of the Mean

sigma = standard deviation; n = number of observations

Error Propagation (Sum)

Combined error when adding independent measurements

Most Probable Value

Simple mean of n equally weighted observations

Weighted Mean

w_i = weight for observation i (often proportional to 1/sigma^2)

90% Confidence Interval

Multiply standard deviation by 1.6449 for 90% confidence

95% Confidence Interval

Multiply standard deviation by 1.96 for 95% confidence

Error Propagation (general / partial derivatives)

For y = f(x_1, ..., x_n) with independent x_i. Reduces to the sum-of-squares rule for additive functions.

Error Propagation (product)

For y = a · b with independent a, b. Relative errors combine in quadrature.

Least-Squares (Parametric Form)

V = residuals; A = design matrix; X = unknowns; L = observed minus computed; W = weight matrix (typically diagonal of 1/σ²).

Reference Variance (post-adjustment)

σ₀² = a-posteriori variance of unit weight; r = degrees of freedom (n_obs − n_unknowns).

Photogrammetry

Photo Scale

f = focal length; H = flying height (above datum); h = ground elevation

Relief Displacement

r = radial distance from photo center; h = object height; H = flying height above base

Ground Distance from Photo

S = photo scale (as a fraction); D_photo = measured distance on photo

Air Base (forward overlap)

B = air base (distance between photo centers); G = ground side length covered by one photo; p_e = end (forward) overlap percent (typ. 60%)

Stereoscopic Parallax — height

Difference in parallax between two points yields height. B = air base, f = focal length, H = flying height. Hands-on photogrammetry classic.

Number of Photos to Cover an Area

A = total area; a = ground area covered by one photo; p_e = end overlap, p_s = side overlap (percent)

GNSS / GPS

Position DOP (PDOP)

PDOP combines horizontal and vertical dilution of precision; lower is better. PDOP < 6 typically required for survey-grade work.

GNSS Position Standard Error

UERE = User Equivalent Range Error (combined satellite, atmospheric, receiver, and multipath error)

Carrier Wavelength (L1)

c = speed of light, f_L1 = 1575.42 MHz. L2 ≈ 24.4 cm; L5 ≈ 25.5 cm.

Differential Pseudorange Correction

DGPS/RTK reduces error by applying base-station-derived corrections to rover pseudoranges or carrier phases.

Unit Conversions

From	To	Factor
1 chain	feet	66
1 rod / pole / perch	feet	16.5
1 link	feet	0.66
1 vara (TX)	inches	33.333 in = 2.7778 ft
1 acre	sq ft	43,560
1 acre	sq chains	10
1 hectare	acres	2.471
1 sq mile	acres	640
1 meter	feet	3.28084
1 ft (US Survey)	meters	0.30480061
1 ft (International)	meters	0.3048