Question

Interior angles measured at the locations of a closed traverse ABCDA are given in table below.

Location	Interior angle
A	71° 1' 40"
B	104° 54' 23"
C	107° 54' 10"
D	76° 20' 42"

The total error in the measured angles (in degrees) is (rounded off to three decimal places).

Hint:

Theoretical angle sum for a closed traverse of \(n\) sides is \((n-2)\times180^\circ\); compare with the measured sum to get the error.

Answer 1

Correct Answer: 0.182

Instead of adding degrees, minutes and seconds in separate columns, let's convert every measured angle straight to seconds first, add them as plain numbers, and only convert back to degrees at the end. This avoids any carrying mistakes partway through.

Convert each angle to total seconds ($1^\circ = 3600''$, $1' = 60''$):

\[ A: 71^\circ1'40'' = 71(3600)+1(60)+40 = 255700'' \]\[ B: 104^\circ54'23'' = 104(3600)+54(60)+23 = 377663'' \]\[ C: 107^\circ54'10'' = 107(3600)+54(60)+10 = 388450'' \]\[ D: 76^\circ20'42'' = 76(3600)+20(60)+42 = 274842'' \]

Add these four values:

\[ 255700+377663+388450+274842 = 1296655'' \]

The theoretical sum for a 4-sided closed traverse is $(4-2)\times180^\circ = 360^\circ$, which in seconds is:

\[ 360^\circ \times 3600 = 1296000'' \]

The error, in seconds, is the difference between the measured total and the theoretical total:

\[ \text{Error} = 1296655'' - 1296000'' = 655'' \]

Convert this error back to degrees:

\[ \text{Error} = \frac{655}{3600} = 0.18194^\circ \]

Let's summarize:

• Working entirely in seconds avoids the degrees-minutes-seconds carrying arithmetic and gives the same result more directly.
• The measured angle sum exceeds the theoretical $360^\circ$ closed-traverse sum by 655 seconds.
• Converting 655 seconds back to degrees gives the total angular error.

So the total error in the measured angles, rounded to three decimal places, is $0.182^\circ$.