3.3 Unadjusted Closure & Angular Closure

What Rule 7.5 requires

Rule 7.5(1): "Any time the survey utilizes a conventional traverse loop or traverse between known control monuments, the following parameters shall be satisfied."

Two words in the table do the work. Unadjusted means the ratio is computed from the raw field measurements before any balancing. Minimum means the survey must be at least this good; better is always acceptable.

Step 1: angular misclosure

For a closed polygon with N angles, the interior angles should sum to (N − 2) × 180°. The angular misclosure is the difference between the measured sum and that theoretical sum. The allowable misclosure is k√N, where k is 15″ (A), 25″ (B), or 30″ (C).

If the misclosure exceeds the limit, look for a blunder (a mis-booked angle, the wrong backsight, or a centering error) and re-observe. Do not distribute the error.

Step 2: latitudes, departures, and linear closure

After the angles check, balance them (commonly equally among the stations), compute azimuths, and resolve each course into components:

• Latitude = distance × cos(azimuth), the north-south component
• Departure = distance × sin(azimuth), the east-west component

In a perfect loop the latitudes and departures each sum to zero. Real sums leave misclosures ΣLat and ΣDep. Then:

• Linear error of closure: E = √(ΣLat² + ΣDep²)
• Precision ratio: 1 : (P ÷ E), where P is the traverse perimeter (sum of the course lengths)

Worked example: Class A warehouse site

A five-station loop (N = 5) is run for a commercial warehouse site the client says will be highly developed, so the surveyor classifies it as Class A.

1. Theoretical interior-angle sum: (5 − 2) × 180° = 540°00′00″. Measured sum: 540°00′22″. Misclosure is 22″.
2. Class A limit: 15″ × √5 = 15″ × 2.236 = 33.5″. Since 22″ ≤ 33.5″, the angles pass.
3. Perimeter P = 3,600.00 ft; ΣLat = +0.21 ft; ΣDep = −0.28 ft.
4. E = √(0.21² + 0.28²) = √(0.0441 + 0.0784) = √0.1225 = 0.35 ft.
5. Ratio = 3,600.00 ÷ 0.35 = 10,286, so the closure is 1:10,286, which meets the 1:10,000 minimum.

Failure variant. Suppose a centering error makes ΣLat = +0.32 ft and ΣDep = −0.38 ft on the same perimeter. Then E = √(0.1024 + 0.1444) = √0.2468 = 0.4968 ft, and the ratio is 3,600.00 ÷ 0.4968 = 7,246, or 1:7,246:

Because this is a Class A job, the crew must find and correct the error in the field. Relabeling the survey as Class C to make it pass would misrepresent the classification required by Rule 7.4.

Why "unadjusted" matters

Adjustment methods are for distributing small random errors after the survey passes:

• Compass (Bowditch) rule: each course's latitude correction is −ΣLat × (course length ÷ P), and likewise for departure. It assumes angles and distances are of comparable precision and is the most common method for boundary loops.
• Transit rule: corrections are proportional to each course's latitude or departure magnitude. It assumes angles are more precise than distances and is rarely preferred with modern EDMs.
• Least squares: weights observations by their precision and is the preferred method for networks with redundant ties and GNSS vectors.

Any of these will make the adjusted coordinates close perfectly on paper. That is why Rule 7.5 measures the unadjusted closure. A survey that closes at 1:8,200 unadjusted is not a Class A survey however it is balanced. Certifying it as Class A misstates the work, which Rule 6.3 (truthful documents) and Rule 7.6 (enforcement) prohibit.

When the traverse table does not apply

Rule 7.5(1) applies when a conventional traverse loop or traverse between known control monuments is used. When boundary points are located by radial methods such as GNSS or total-station radial ties, Rule 7.5(2) requires acceptable procedures and adequate quality control so that the allowable positional accuracy is not exceeded (Section 3.4). Surveyors still commonly verify radial work with redundant measurements and closed checks, because the distance and bearing tolerances in the Rule 7.5 table still apply.