Compass Rule on a Closed Lot Traverse
A closed traverse around a proposed 0.69-acre building lot in Cumberland County was run in feet with a 1-second total station. Starting at A (1000.000, 1000.000), the observed latitudes and departures are: AB +200.000, +0.000; BC +0.000, +150.000; CD -199.700, +0.000; DA +0.000, -149.600. The total perimeter is 699.300 ft. No external control is available, so the surveyor treats the loop as a closed traverse and plans to apply the compass rule before computing coordinates and area by the shoelace method. The architect needs adjusted coordinates for A, B, C, and D plus the area to check a 20-ft setback from the northerly line. The surveyor also must decide whether the closure meets a 1:5,000 relative precision standard stated in the firm's contract. Field notes show the angles closed 12 seconds and distances were not corrected for temperature.
1. Compute the linear misclosure and relative precision, and state whether the traverse meets the contract standard.
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Sum the latitudes: +200.000 + 0.000 - 199.700 + 0.000 = +0.300 ft. Sum the departures: 0.000 + 150.000 + 0.000 - 149.600 = +0.400 ft.
Linear misclosure = √(0.300² + 0.400²) = 0.500 ft. Relative precision = perimeter / misclosure = 699.300 / 0.500 = 1,398.6, conventionally stated as 1:1,399. That fails the 1:5,000 contract standard. The angle closure of 12 seconds does not cure the distance misclosure; the surveyor should consider reobservation or a more rigorous adjustment before relying on the coordinates.
2. Apply the compass rule and provide adjusted coordinates for A, B, C, and D.
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Under the compass rule, each side's latitude and departure correction is proportional to its length: correction = -misclosure × (side length / perimeter). For latitude the correction total is -0.300 ft; for departure it is -0.400 ft.
Adjusted values, to 0.001 ft:
| Side | Length | Adjusted latitude | Adjusted departure |
|---|---|---|---|
| AB | 200.000 | +199.914 | -0.114 |
| BC | 150.000 | -0.064 | +149.914 |
| CD | 199.700 | -199.786 | -0.114 |
| DA | 149.600 | -0.064 | -149.686 |
Coordinates from A (1000.000, 1000.000): B = (1199.914, 999.886) C = (1199.850, 1149.800) D = (1000.064, 1149.686) A check returns to approximately (1000.000, 1000.000).
3. Compute the area and explain how the adjustment affects the architect's 20-ft setback check.
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Using the adjusted coordinates in the shoelace formula gives an area of approximately 29,938 sq ft, or 29,938 / 43,560 = 0.687 acre. The unadjusted figure would be 200 × 150 = 30,000 sq ft, so the adjustment changes the area by about 62 sq ft.
The northerly line is C-D. The 20-ft setback must be measured perpendicular to the adjusted C-D line, not scaled from the unadjusted sketch. Because the traverse fails the 1:5,000 standard, the architect should be told that the coordinates are adjusted and not field-verified to that standard. A reobservation or a connection to external control would be appropriate before relying on the setback for construction.