The easiest way to think about it is measure from the part of the base the moved the most. The rule we use is it takes 6.5cm for a stand to do a 180 turn about its center. Turning on one of its corner 180 is about 12.5cm.
Technically, distance from a center of a stand to its corner is 4,47cm, which, if we follow that logic makes it 7cm, rather than 6,5 for a stand to make a 180 turn.
As I perceive it (I'm not saying it's correct), a stand is not a fixed formation of troops. As the rules say:
The area occupied by a regiment of models is assumed to encompass hundreds of warriors including all of their supporting elements such as mules laden with spare ammunition, surgeons’ wagons, preachers, servants, messengers, scouts and all the paraphernalia of warfare.
Therefore, a stand turning on the spot might mean the units switching places rather than the actual formation making a circle around its center.
For it to make more sense, i'd like to talk about a unit moving and turning at the same time, rather than a unit turning on the spot. To me, the ideal example would be an infantry unit (4cm frontage) moving through a 2cm gap and finishing its move facing the same direction as it started the move.
Garp rule implies, that the unit has tu be turned so it can pass through the gap.
I see two possible ways to approach measuring this;
1. Move the unit and then measure a "zig-zag" path through the gap for wach of the units flanks, making sure it didn't exceed the units movement. In other words, measure the distance of each part of the stand (in reality just the parts that moved the furthest) to its original destination.
2. Move the unit towards the gap, turning it, move it through the gap and then move it to the final position, turning it again. In other words, measure each part of the movement separately and do the math.
I prefer approach #1 as it's more simple and to me, it corresponds better to the above mentioned quote from the rulebook.
What follows is just a bit of good old speculating.

It just came into my mind while I was thinking about the whole thing, so I'm putitng it down.
If we follow that approach #1, I think we should follow the same if the stands are just turning on the spot (meaning if they turn 180, the straight corner-to corner distance is 4,47cm).
It doesn't make any difference when an infrantry or cavalry is turning on the spot. But it does make a difference if, for example an artilery unit in irregular formation would want to turn 180. If a stand had to circle around its center, it would not be allowed to do that using the 5cm half-pace movement (because making a circle would mean going 7cm).
What's your opinion about it?
Dave, you said that you're using 6,5cm to represent a unit turning 180. How do you do it when it's turning during a move, not just on the spot?