Fundamentals of Surveying Exam (FS) Mathematics, Statistics, and Geodesy Syllabus

Mathematics, Statistics, and Geodesy syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics, Statistics, and Geodesy in Fundamentals of Surveying Exam (FS), not a summary of it.

1. 1

   Surveying Mathematics and Trigonometry

   4 topics

   • Plane Trigonometry
     • Right-triangle relationships (sine, cosine, tangent)
     • Law of sines and law of cosines for oblique triangles
     • Angle and distance computations in traverses
   • Coordinate Geometry (COGO)
     • Inverse computation: bearing and distance from coordinates
     • Forward computation: coordinates from bearing and distance
     • Line-line, line-arc, and arc-arc intersections
   • Analytic and Plane Geometry
     • Slope, perpendicular and parallel lines
     • Areas of polygons, circles, and circular segments
     • Volumes of prisms, pyramids, and frustums
   • Algebra and Basic Calculus Applications
     • Solving simultaneous equations
     • Derivatives and integrals for area and slope problems
2. 2

   Statistics and Error Theory

   4 topics

   • Measures of Central Tendency and Dispersion
     • Mean, median, and mode of observations
     • Standard deviation and variance
     • Standard error of the mean
   • Probability and Distributions
     • Normal (Gaussian) distribution of random errors
     • Confidence intervals and probability levels (50%, 90%, 95%)
   • Error Classification
     • Systematic, random, and gross (blunder) errors
     • Accuracy versus precision
   • Error Propagation
     • Propagation in sums, differences, and products
     • Error of a series and error of a mean
3. 3

   Geodesy and Earth Models

   4 topics

   • Figure of the Earth
     • Geoid, ellipsoid, and reference surfaces
     • Geoid undulation and deflection of the vertical
   • Geodetic Datums
     • Horizontal datums (NAD 27, NAD 83)
     • Vertical datums (NGVD 29, NAVD 88)
     • Geometric reference frame modernization concepts
   • Geodetic Coordinate Systems
     • Latitude, longitude, and ellipsoid height
     • Earth-centered, Earth-fixed (ECEF) coordinates
   • Curvature, Refraction, and Convergence
     • Effects of Earth curvature and atmospheric refraction
     • Convergence of meridians and grid versus geodetic north

Mathematics, Statistics, and Geodesy flashcards for Fundamentals of Surveying Exam (FS)

18 of 54 cards from the Mathematics, Statistics, and Geodesy deck — real questions with worked answers.

1. In a right triangle, state the definitions of $\sin\theta$, $\cos\theta$, and $\tan\theta$ in terms of the opposite side, adjacent side, and hypotenuse.

   $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

2. State the Law of Sines for a triangle with sides $a, b, c$ opposite angles $A, B, C$.

   $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$ where $R$ is the circumradius.

3. State the Law of Cosines used to find side $c$ given sides $a$, $b$ and included angle $C$.

   $$c^{2} = a^{2} + b^{2} - 2ab\cos C$$

4. What is the fundamental Pythagorean trigonometric identity?

   $$\sin^{2}\theta + \cos^{2}\theta = 1$$

5. Give the two area formulas for a triangle: (a) using base and height, and (b) using two sides and the included angle.

   (a) $A = \dfrac{1}{2}bh$; (b) $A = \dfrac{1}{2}ab\sin C$.

6. In surveying azimuths (measured clockwise from north), how do you compute a departure and a latitude of a line of length $L$ with azimuth $\alpha$?

   Departure (East–West) $= L\sin\alpha$; Latitude (North–South) $= L\cos\alpha$.

7. Given two points $(N_1, E_1)$ and $(N_2, E_2)$, what is the formula for the horizontal distance between them?

   $$D = \sqrt{(N_2 - N_1)^{2} + (E_2 - E_1)^{2}}$$

8. In COGO, how do you compute the azimuth (clockwise from north) of a line from point 1 to point 2 using northing and easting differences?

   $$\alpha = \arctan\!\left(\frac{\Delta E}{\Delta N}\right) = \arctan\!\left(\frac{E_2 - E_1}{N_2 - N_1}\right)$$ then adjust to the correct quadrant (0°–360°) using the signs of $\Delta N$ and $\Delta E$.

9. Given a starting coordinate $(N_1, E_1)$, a distance $D$, and an azimuth $\alpha$, write the inverse (forward computation) equations for the new point $(N_2, E_2)$.

   $$N_2 = N_1 + D\cos\alpha, \qquad E_2 = E_1 + D\sin\alpha$$

10. State the coordinate (shoelace) formula for the area of a closed polygon with vertices $(E_i, N_i)$.

    $$A = \frac{1}{2}\left|\sum_{i=1}^{n}\left(E_i N_{i+1} - E_{i+1} N_i\right)\right|$$ with indices taken cyclically ($n+1 \equiv 1$).

11. In quadrant bearings, how is a bearing such as N 30° E converted to an azimuth measured clockwise from north?

    NE bearings: azimuth $=$ bearing angle (e.g., $30°$). SE: azimuth $= 180° - \text{angle}$. SW: azimuth $= 180° + \text{angle}$. NW: azimuth $= 360° - \text{angle}$.

12. What is the general equation of a straight line in analytic geometry, and what does each coefficient relate to?

    $y = mx + b$, where $m$ is the slope (rise over run) and $b$ is the $y$-intercept. General form: $Ax + By + C = 0$.

13. What is the slope condition for two lines to be (a) parallel and (b) perpendicular?

    (a) Parallel: $m_1 = m_2$. (b) Perpendicular: $m_1 m_2 = -1$, i.e., $m_2 = -\dfrac{1}{m_1}$.

14. Write the standard equation of a circle of radius $r$ centered at $(h, k)$.

    $$(x - h)^{2} + (y - k)^{2} = r^{2}$$

15. Give the formula for the perpendicular distance from a point $(x_0, y_0)$ to the line $Ax + By + C = 0$.

    $$d = \frac{|A x_0 + B y_0 + C|}{\sqrt{A^{2} + B^{2}}}$$

16. State the quadratic formula for the roots of $ax^{2} + bx + c = 0$.

    $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

17. What is the derivative interpretation used in surveying for finding the slope/grade of a curve $y = f(x)$ at a point?

    The grade (slope) equals the first derivative $\dfrac{dy}{dx} = f'(x)$ evaluated at that point; setting $f'(x) = 0$ locates high/low points (turning points).

18. How is the area under a curve $y = f(x)$ between $x = a$ and $x = b$ expressed in calculus, and how does this relate to end-area volume computations?

    $$A = \int_{a}^{b} f(x)\,dx$$ Integration sums infinitesimal areas; in earthwork, accumulating cross-sectional areas along a length yields volume (e.g., average-end-area method).

See more Mathematics, Statistics, and Geodesy flashcards →

Planning Mathematics, Statistics, and Geodesy for Fundamentals of Surveying Exam (FS)

Mathematics, Statistics, and Geodesy (Fundamentals of Surveying Exam (FS)) FAQ