Which is the following relative error of closure ?

Question

Examoogle · Accepted Answer

The correct answer is Option B: $\frac{1}{\frac{P}{e}}$. The relative error of closure is defined as the ratio of the total error of closure ($e$) to the perimeter of the traverse ($P$). Standard surveying practice expresses relative error as a unit fraction in the form of $\frac{1}{\left(\frac{P}{e}ight)}$, where the denominator represents the perimeter per unit error.

Step-by-Step Solution & Concepts:
The relative error of closure is defined as the ratio of the total error of closure ($e$) to the perimeter of the traverse ($P$). Standard surveying practice expresses relative error as a unit fraction in the form of $\frac{1}{\left(\frac{P}{e}ight)}$, where the denominator represents the perimeter per unit error.

Working Principle: In traverse surveying, linear closing error $e = \sqrt{(\sum L)^2 + (\sum D)^2}$ arises from small operational measurement inaccuracies in latitude and departure. Dividing the total error $e$ by the total length of the traverse $P$ yields the relative error $\frac{e}{P}$, which is conventionally rewritten in the standard reciprocal form $\frac{1}{P/e}$ (e.g., $1/5000$) to easily indicate accuracy.

Key Formulas: $	ext{Relative Error of Closure} = \frac{e}{P} = \frac{1}{\frac{P}{e}}$ — Ratio expressing closing error relative to perimeter; $e = \sqrt{(\sum L)^2 + (\sum D)^2}$ — Total linear error of closure

Key Points: The error of closure ($e$) is the distance by which the closing line fails to meet the starting station.; Perimeter ($P$) is the sum of lengths of all sides of the traverse polygon.; Relative error is expressed as a unit fraction with numerator 1, such as $1/1000$ or $1/5000$.

Advantages: Allows direct comparison of surveying precision across traverses of different total lengths.; Provides a standardized metric to check compliance with specified surveying standards.

Disadvantages: Does not isolate individual source errors (angular vs. linear).; Requires complete distance measurement of all sides before calculation.

Applications: Checking precision in theodolite and total station traverse surveys.; Determining necessity for traverse adjustment (e.g., via Bowditch's or Transit rule).

Additional Info: Option A (P/e) represents the reciprocal ratio, not the relative error itself.. Option C (1 / (e/P)) simplifies mathematically back to e/P, but standard engineering convention mandates expressing it with 1 in the numerator and (P/e) in the denominator to state degree of accuracy directly as 1 in 'N'.

Answer: B is correct — Relative error of closure is conventionally expressed as a unit fraction $\frac{1}{\frac{P}{e}}$, representing one unit of closing error per $P/e$ units of perimeter length.

Exam Tip: Always remember that in traverse accuracy specifications, a higher denominator in $1/(P/e)$ indicates a more precise survey (e.g., $1/10000$ is more precise than $1/1000$).

Examoogle · Answer

Option A: $\frac{P}{e}$

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Option C: $\frac{1}{\frac{e}{P}}$

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Option D: All of the above

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