Mathematics, Computations and Coordinate Geometry
Mathematics, Computations and Coordinate Geometry tests your ability to convert field observations into defensible positions and areas: bearings, azimuths, traverse closure, coordinate geometry, and curve computations.
Core rules
- Bearings are quadrant angles; convert to azimuths clockwise from north before computing components.
- Latitude = distance × cos(azimuth); departure = distance × sin(azimuth). Keep signs consistent.
- For a closed traverse, sum of latitudes and departures should be zero. Linear misclosure = √(Σlat² + Σdep²); precision is perimeter divided by misclosure.
- Coordinate area: 2A = Σ(XiYi+1 − Xi+1Yi). Take the absolute value and divide by 2. Convert square feet to acres by dividing by 43,560.
- Circular curve: T = R tan(Δ/2), L = πRΔ/180, C = 2R sin(Δ/2), M = R[1 − cos(Δ/2)].
Worked example
A closed square has coordinates A(1000,1000), B(1000,1100), C(1100,1100), D(1100,1000). Compute 2A: A-B = 100,000; B-C = −110,000; C-D = −110,000; D-A = 100,000; sum = −20,000. Area = |−20,000|/2 = 10,000 square feet = 10,000/43,560 = 0.2296 acre. Check the signs: a clockwise list gives a negative sum in this formula, but the absolute value gives the area.
Traps
Quadrant confusion (S 45° E is azimuth 135°, not 45°); mixing feet and meters; using the wrong sine/cosine for latitude and departure; forgetting to close a traverse before computing area; and rounding intermediate coordinates so a small misclosure is hidden. On curves, use the central angle Δ, not the bearing difference between tangents.