Five-Sided Boundary Traverse: A 72-Second Angular Misclosure Before a Closing
A crew is closing a five-sided boundary traverse around a 4.2-acre parcel in Warren County, New Jersey, using a 5-second total station. Every angle was turned in two direct and two reverse positions. The five interior angles sum to 540°01'12". The closed loop measures 2,118 feet. At station 3 the backsight distance is only 47 feet while the foresight to station 4 is 810 feet, because a woodline forced an offset setup. The party chief computes preliminary coordinates, finds a linear misclosure of 0.35 foot, and asks whether he should simply spread the 72 seconds equally among the five angles and send coordinates to the CAD technician that evening. He points out that the client needs a boundary exhibit before Friday's closing and that the field budget is nearly spent.
1. Is equal distribution of the 72-second angular misclosure defensible? Show the numbers you would rely on and state your decision.
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Compute what is expected. For a closed polygon with n = 5, the theoretical interior-angle sum is (n − 2) × 180° = 540°00'00". The observed sum is 540°01'12", so the misclosure is +72 seconds.
Compare to a plausible random-error budget. For a 5-second instrument, a common criterion is roughly the instrument accuracy times the square root of the number of angles: 5" × √5 ≈ 11". Even a generous doubling of that criterion gives about 22". A 72-second misclosure is three to six times the allowance, which is the signature of a blunder or a systematic error, not random noise.
Test the geometry. Station 3 has a 47-foot backsight. A centering or pointing error of only 0.005 ft at that distance produces arctan(0.005/47) ≈ 22 seconds of angular error. The same 0.005 ft at 810 ft produces about 1.3 seconds. The short backsight is therefore more than sufficient to explain the entire discrepancy, and station 3 is the obvious suspect.
Decision. Do not adjust. Spreading 14.4 seconds into each angle would force the figure to close, conceal the blunder, and rotate every subsequent leg. The coordinates would look internally consistent while individual boundary lines were wrong by tenths of a foot - unacceptable for a boundary exhibit that will support a deed or a Schedule B description. Re-observe station 3 with a longer backsight (occupy station 4 and turn the angle back, or set a temporary point), recompute the closure, and document the re-observation. The closing date is a scheduling problem, not a surveying one; tell the client the truth rather than deliver coordinates you cannot defend.
2. The re-observation produces an 8-second misclosure. Walk through the adjustment and quantify what the angular correction actually means on the ground. Does it account for the 0.35-foot linear misclosure?
Read the worked answer
Accept and distribute. Eight seconds is inside the roughly 11-second allowance, so it is consistent with random error. Distributing equally gives 8"/5 = 1.6 seconds per angle, applied with the opposite sign.
Convert to distance. The angular correction is 8 seconds, or 8/206,265 ≈ 3.9 × 10⁻⁵ radians. Applied over the longest leg (810 ft) that is 3.9 × 10⁻⁵ × 810 ≈ 0.031 ft, about 0.03 ft. Over the whole 2,118-ft loop the angular component is of the same order.
Conclusion on the 0.35 ft. The angle fix explains only a small fraction of the 0.35-ft misclosure (1 part in 6,050, or about 1:6,000). The remainder must come from distance errors or from the eccentric setup at station 3 - an offset occupation point requires the measured legs and the angle to be reduced to the true station, or the traverse will not close no matter how good the angles are. Other candidates: an incorrect prism constant, a temperature/pressure setting left at default, or a mistaped leg.
What to expect after correction. Once the eccentric setup is properly reduced and the distances verified, a 2,118-ft loop with a 5-second instrument should close on the order of 1:10,000 or better (about 0.2 ft or less), which is defensible for this class of boundary work. Recompute and re-close before releasing anything.