Mathematics, Measurement, and Computations
What this domain tests
Computational fluency: unit conversions, traverse closures, curve geometry, areas and volumes, and propagating observational error into a defensible result.
Core constants and conversions
- 1 Gunter's chain = 66 ft = 100 links; 1 acre = 43,560 ft² = 10 square chains; 1 rod = 16.5 ft; a full section = 640 acres.
- 1 hectare = 10,000 m² = 2.4711 acres.
Traverse computation
Latitude = distance × cos(bearing or azimuth); departure = distance × sin(bearing or azimuth). Sum the latitudes and departures, compute linear misclosure, and report relative precision as misclosure ÷ perimeter (e.g., 0.15 ft in 2,500.00 ft = 1:16,700). The compass (Bowditch) rule distributes misclosure in proportion to course length; the transit rule distributes by latitude and departure magnitude.
Horizontal circular curves
- Tangent T = R tan(Δ/2)
- Arc length L = RΔ, with Δ in radians
- Long chord LC = 2R sin(Δ/2)
- External E = R[1/cos(Δ/2) − 1]; middle ordinate M = R[1 − cos(Δ/2)]
Worked example
R = 500.00 ft, Δ = 24°00′. With Δ/2 = 12°, T = 500 × tan 12° = 500 × 0.21256 = 106.28 ft. L = 500 × 24 × 0.0174533 = 209.44 ft. LC = 2 × 500 × sin 12° = 1,000 × 0.20791 = 207.91 ft. The 1.53 ft gap between L and LC is the arc-versus-chord difference and is the reason stations are set by arc definition.
Traps that cost marks
- Mixing the arc and chord definitions of degree of curve; the radius differs by about 0.005 ft per degree, but layout blunders are larger.
- Entering Δ in degrees into L = RΔ without converting to radians.
- Sign errors on NW and SE bearings (negative latitude, negative departure).
- Reporting misclosure precision without stating the perimeter used.