The check you can run before you pack up
Every closed figure with n corners has interior angles summing to exactly (n minus 2) times 180 degrees. A triangle gives 180, a quadrilateral 360, a five-sided figure 540. That is true of any shape whatsoever — convex, re-entrant, long and thin, it makes no difference — and it does not involve a single distance. So the moment the last angle is booked you can add the column and see how far off you are, standing at the last station with the instrument still set up.
The default five angles in the box sum to a few tens of seconds away from 540 degrees. That leftover is the angular misclosure. It costs nothing to compute and it catches the two things most likely to have gone wrong: an angle booked with the wrong number of degrees, and an angle turned to the left when it should have gone right.
Interior, exterior, deflection
An angle to the right is swung clockwise from the backsight to the foresight, always, with no exceptions and no sign to remember. Whether those angles come out as the interior angles of the figure or the exterior ones depends entirely on which way round you walked it. Go counter-clockwise round a closed loop and your angles to the right fall inside and sum to (n minus 2) times 180. Go clockwise and the same angles fall outside and sum to (n plus 2) times 180.
Getting that setting wrong is not a subtle error. The two targets differ by 720 degrees, so the misclosure comes out as two full revolutions rather than a handful of seconds, and nobody mistakes it for a bad day of angle work.
Deflection angles are the third pattern and they suit a route rather than a loop. At each station you record how far the line turns away from straight ahead, right positive, left negative, and the values stay small. Round any closed figure the deflections sum to 360 whatever the number of stations, because you have turned through exactly one revolution by the time you are pointing the way you started.
Why the tolerance goes as the square root
Written specifications express angular closure as some number of seconds times the square root of the number of angles. That form is not arbitrary. If each angle carries an independent random error, the errors partly cancel as they accumulate, and the expected total grows as the square root of the count rather than in proportion to it. Twenty angles allow about 1.41 times the closure that ten angles do, not twice. A specification that scaled linearly would be far too generous on long traverses.
The value of the multiplier is a matter for whatever specification governs the work. It differs by the class of survey, by jurisdiction, by client and over time, and this page has none of its own — the field takes yours and prints the comparison beside the computed figure. Whether the traverse satisfies it is a professional judgement about the survey rather than a result of the arithmetic.
Balancing, and its limits
The standard treatment is to spread the misclosure evenly: divide it by the number of angles and take that much off each one. It is defensible when every angle was turned the same way with the same instrument and the same number of repetitions, which is the usual case. Where some stations were short sights and others long, or some angles were doubled and others turned once, an even spread charges the good angles for the bad ones.
The page prints the correction in seconds and also in multiples of the instrument least count, and that second figure is often the honest one. Thirty seconds spread over five angles is six seconds each. On an instrument reading to five seconds that is a little over one division and means something. On one reading to twenty seconds it is less than a third of a division, and the even spread is a bookkeeping gesture rather than a measurement.
What closing the angles does not prove
It proves the angles agree with each other. It says nothing about the distances, which is a separate closure entirely. It says nothing about where the figure sits on the ground, because a correctly shaped traverse can be sited anywhere. And it says nothing about orientation: start from a wrong azimuth and every angle can be perfect while the whole figure is rotated. Only a tie to known control tests those last two, which is the argument for running a link traverse between two control points rather than a loop whenever the control exists.
Questions people ask
What should the interior angles of a traverse add up to?
For a closed figure with n corners, exactly (n minus 2) times 180 degrees — 180 for a triangle, 360 for a quadrilateral, 540 for five sides, and so on. It holds for any shape at all and involves no distances, which is why the angular closure can be checked in the field before the instrument comes down.
What is an angle to the right?
The angle swung clockwise from the backsight to the foresight at an occupied station. It is always clockwise, so there is no sign to remember and no left-hand case. Whether those angles turn out to be the interior or the exterior angles of a closed figure depends on which direction you walked round it, not on how the angles were measured.
Why is angular tolerance written as seconds times the square root of n?
Because independent random errors accumulate as the square root of the count. Twenty angles allow about 1.41 times the closure ten angles do, not twice as much. A tolerance that scaled linearly would be far too loose on a long traverse. The multiplier itself comes from whatever specification governs the work — it varies by class of survey and by jurisdiction, and it is a value you supply rather than one this page holds.
How do I get the azimuth of the next course?
Add 180 to the azimuth of the course you came in on, which gives the azimuth of the backsight, then add the angle to the right, then reduce the result into 0 to 360. That one line covers every station in the traverse with no cases. For deflection angles it is simpler still: add the deflection, signed, to the current azimuth.
My angular misclosure is about 720 degrees. What happened?
You have almost certainly told the page the wrong angle type. Angles to the right walked clockwise round a loop are the exterior angles and sum to (n plus 2) times 180, which is 720 degrees more than the interior sum. Change the setting rather than the angles. That the error is enormous rather than small is a feature — it can never be mistaken for a bad day in the field.
Does balancing the angles make the traverse right?
No. It makes the angles consistent, which they already almost were. It does not find a blunder, it does not test the distances, and it does not test where the figure sits or which way it is oriented. A traverse with perfectly balanced angles started from a wrong azimuth is the right shape in the wrong direction, and only a tie to known control reveals that.