Most of the difficulty candidates experience on the Fundamentals of Surveying exam is not conceptual. It is that the mathematics is not automatic, so every question costs more time than it should, and the deficit compounds until there is no time left for the questions that would have been easy.
The distinction that matters is between being able to do something and being fluent at it. You can probably convert a bearing to an azimuth. Can you do it in five seconds, in the right quadrant, without drawing a sketch and without second-guessing the sign? That is fluency, and it is what the exam actually rewards.
Angles: the foundation everything else sits on
Surveying runs on angles expressed in degrees, minutes and seconds, and computation runs on decimal degrees or radians. Moving between them has to be reflexive.
42°17′30″ = 42 + 17/60 + 30/3600 = 42.291667°0.3125° = 0°18′45″- Convert between degrees-minutes-seconds and decimal degrees in both directions, on paper and on your calculator's dedicated function.
- Add and subtract angles in degrees-minutes-seconds, including borrowing across the 60 boundary, which is where careless errors cluster.
- Convert between degrees and radians, and know without checking which mode your calculator is in.
- Convert between bearings and azimuths in all four quadrants. N 42°17′30″ E is azimuth 42°17′30″; S 42°17′30″ W is azimuth 222°17′30″; N 42°17′30″ W is azimuth 317°42′30″.
- Reduce an azimuth outside the 0° to 360° range, and compute a back azimuth, without hesitation.
- Handle grads or mils if they appear, at least well enough to convert.
Trigonometry
Right-triangle trigonometry and the two oblique-triangle laws cover the overwhelming majority of computational questions. They should be recallable instantly, not looked up.
c² = a² + b² − 2ab cos Ca / sin A = b / sin B = c / sin Carea = ½ ab sin C = ½ × 245.30 × 318.75 × sin 67°14′20″ = 36,050.23 sq ft = 0.8276 acresAlso be comfortable with the reciprocal functions, the identities you will actually use such as sin²θ + cos²θ = 1, and enough spherical trigonometry to reason about a geodetic problem rather than to solve one from scratch.
Coordinate geometry
This is the working language of surveying computation, and it deserves the largest share of your mathematics practice.
- Forward computation: from a known point, a direction and a distance, compute the coordinates of a new point. Northing increases by distance times cosine of azimuth; easting by distance times sine of azimuth.
- Inverse computation: from two coordinate pairs, compute the direction and distance between them, with the direction placed in the correct quadrant.
- Intersections: bearing-bearing, bearing-distance and distance-distance. Know which of these can have two solutions and how to choose between them.
- Area from coordinates by the shoelace method, and the ability to apply it without mixing up the order of the cross products.
- Line and circle geometry: perpendicular distance from a point to a line, the foot of a perpendicular, and the intersection of a line with a circle.
- Curve geometry: the relationships among radius, deflection angle, tangent, arc length, chord, middle ordinate and external distance.
2A = Σ (xᵢ · yᵢ₊₁ − xᵢ₊₁ · yᵢ)Algebra, vectors and a little calculus
Algebra should be invisible: rearranging a formula, solving a quadratic, handling simultaneous equations and manipulating ratios and proportions ought to cost you no thought at all. Any hesitation here shows up as lost seconds on every question that contains it, which is most of them.
Vector reasoning appears in the language of components and directions more often than in explicit vector notation. Be able to resolve a quantity into components, add components, and compute a dot product. Calculus appears lightly: differentiation to find a maximum or minimum, such as the turning point on a vertical curve, and integration in the context of areas and volumes. This is not a calculus examination, and effort spent on analytical technique is effort taken from trigonometry, where the return is far higher.
Statistics of observations
Measurement statistics is a small, self-contained topic that is disproportionately worth preparing, because it is unfamiliar enough to be neglected and simple enough to learn quickly.
σ = √( Σ(xᵢ − x̄)² / (n − 1) )σ_total = √(σ₁² + σ₂² + … )- Distinguish systematic error, random error and blunder, and know that only random error is what these formulas describe.
- Understand weighting: an observation with smaller variance carries more weight, and weight is inversely proportional to variance.
- Understand the meaning of a confidence level, and why an accuracy statement without one is incomplete.
- Understand the concept of a least squares adjustment and what its residuals tell you, even if you are not asked to perform one.
- Know the difference between precision and accuracy well enough to answer a question designed to confuse the two.
How to test your own fluency
Set a timer for ten minutes and, from a blank page, convert five angles between formats, invert between three pairs of coordinates, and compute one area from coordinates. If you finish comfortably and correctly, your mathematics is not the constraint and your study time belongs elsewhere. If you do not, that is the most valuable thing you could have learned this week, and it should reshape your entire plan.