8.2 Error of Closure

8.2 Error of Closure

Quick Answer: Linear error of closure is E = √((Σlat)² + (Σdep)²). Precision ratio is 1 : P / E. For ABCD, Σlat = −0.19 ft, Σdep = +0.17 ft, P = 1,770.20 ft, so E = 0.255 ft and the ratio is 1:6,943. Interior angles of a closed polygon should sum to (n−2)×180°—360° for four sides; this section uses interiors, not (n+2)×180° exteriors. A level loop's vertical misclosure is ΣBS − ΣFS when you return to the same benchmark.

Linear (horizontal) closure

Section 8.1 left two residuals: the loop came up 0.19 ft short of north and 0.17 ft long to the east. Those are orthogonal components of one closing line from computed A′ back toward held A.

E = √((Σlat)² + (Σdep)²)
E = √((−0.19)² + (0.17)²) = √(0.0361 + 0.0289) = √0.0650 = 0.25495 ft, reported 0.255 ft.

The precision ratio (relative precision, relative accuracy—shops use all three names) is the dimensionless comparison of perimeter to that linear error:

1 : P / E = 1 : 1,770.20 / 0.25495 = 1:6,943

Write it as 1:6,943, not as 0.000144. Larger N in 1:N is tighter. 1:10,000 is tighter than 1:5,000.

The direction of misclosure is the azimuth of the vector (Σlat, Σdep). Here that vector is slightly south of east (ΔN negative, ΔE positive). You do not need that azimuth to apply Compass rule in Section 8.3, but you do need it if a CES item asks which way the traverse failed to close.

Practice comparisons—not a Board-required CES ratio

The 2022 CES test plan asks you to evaluate error of closure (horizontal and vertical). The Board candidate information bulletin does not publish a single required ratio such as “CES traverses must close 1:10,000.” Do not invent one. On the exam, either the stem gives the comparison, or you report E and 1:N and judge them against a stated office or contract limit.

Two practice examples (labeled as such) using this perimeter:

Same field loop, two different office bars. That is the point of the comparison, not a hidden BPELSG cut score. A control survey for a mapped corridor may be specified much tighter than a short on-site assumed loop locating a catch basin. Domain I.E (accuracy and precision) is where those project decisions start; Domain III.B is where you compute the number you will compare.

If a stem gives only Σlat and Σdep and asks for E, do not divide by n, do not add the residuals, and do not report P/Σlat (1,770 / 0.19 ≈ 1:9,300)—that ignores the departure residual.

Angular misclosure of a closed polygon

This chapter uses interior angles. For a simple closed polygon of n sides, the sum of interior angles should be:

(n − 2) × 180°

For n = 4, that is 360°. If you instead measured exterior angles (the turning angles as you walk around the outside), those should sum to (n + 2) × 180° = 1,080° for a quadrilateral. State which set you measured. Adding interiors and comparing them to 1,080° is a guaranteed bust.

ABCD was traversed clockwise, interior to the right. Field interiors (before balance):

Expected sum = 360°00'. Angular misclosure = +4' (+240").

A typical hand-method distribution (not a Board-mandated CES method) is equal split: 4' / 4 stations = 1' per angle, subtracted because the sum is large. Adjusted interiors:

Carry azimuths from held AB = 42°18', adding each clockwise interior:

• Az BC = 42°18' + 70°00' = 112°18'
• Az CD = 112°18' + 96°18' = 208°36'
• Az DA = 208°36' + 95°30' = 304°06'
• Az AB check = 304°06' + 98°12' = 402°18' − 360° = 42°18'

Those are the azimuths used in Section 8.1. Balance angles before you compute latitudes. If you skip the angular check, every cosine and sine inherits a systematic twist, and Compass rule in 8.3 will try to hide an angle bust inside distance corrections.

Practice angular comparisons—again, not a CES published cut

Common office comparisons (examples only):

• 30" √n: for n = 4, 30" × 2 = 60" = 1'
• 1' √n: for n = 4, 2'

Raw +4' exceeds both of those example bars. After the 1' per-angle balance, the azimuths are internally consistent. A CES stem may give you the measured sum and ask only for the misclosure versus (n−2)×180°, without asking you to invent a Board tolerance.

Vertical closure of a level circuit

Horizontal E does not speak to elevation. A level circuit that starts and ends on the same benchmark should return the same elevation. For a two-wire (or single-wire) differential loop:

Vertical misclosure = ΣBS − ΣFS
(equivalently, computed closing elevation minus published starting elevation)

Worked loop on the same assumed job, four setups, start and end on BM-A, published elevation 412.385 ft:

ΣBS − ΣFS = 18.462 − 18.479 = −0.017 ft. Computed close = 412.385 − 0.017 = 412.368 ft.

The circuit length is the same 1,770 ft of walking ≈ 0.335 mile. A common office comparison 0.05 √M (result in feet, M in miles) gives 0.05 × √0.335 = 0.029 ft. The 0.017 ft misclosure is inside that example. It is not a BPELSG-published CES vertical ratio. If the stem cites a different specification (a contract 0.01 √M, a 12 mm √K metric form, or a stated hundredth), use that number.

Check the notes the way Chapter 7 teaches: page arithmetic (ΣBS − ΣFS) must match the elevation difference you carried station to station. A 0.01 ft page bust is an arithmetic error, not a compensator error.

Order of work on a CES traverse item

1. Sum interiors (or exteriors—one set) and compare with (n−2)×180° or (n+2)×180°.
2. Distribute the angular residual if the stem asks you to adjust.
3. Compute azimuths, then latitudes and departures.
4. Form E and 1:P/E; compare only with a stated or labeled-example limit.
5. If elevations are in the stem, close the level circuit separately. Do not combine 0.255 ft horizontal with 0.017 ft vertical into one “resultant” unless the item specifically asks for a 3-D misclosure vector.

Exam traps

• Reporting E = Σlat + Σdep = −0.02 ft, or E = 0.19 + 0.17 = 0.36 ft, instead of the hypotenuse 0.255 ft.
• Comparing interiors with (n+2)×180°, or exteriors with (n−2)×180°.
• Inventing “BPELSG requires 1:10,000” when the bulletin does not say that.
• Using the practice-bank count or any unpublished item count as if it were a closure spec.
• Calling a 1:6,943 loop a fail because a different job’s 1:10,000 example exists in your memory.