Math Is Fun Forum

Hello,
I would like to know if anyone can help me derive a function for estimating the maximum angle between two planes.

Here's the situation:
I have a tribrach, which is a piece of survey equipment that is used to plumb a level instrument atop a tripod.  It is essentially constructed of two equilateral triangles, one above the other, and connected at their vertices via turnbuckles that can be turned to change the vetical distance between them.  Please see uploaded image for a top and side view.

I would like to somehow measure the separation of the two planes (the two plates of the tribrach) with calipers, and compute the maximum angle between them.

I know that if I take any three measurements of distance, 120 degrees apart (lets call them z1, z2, and z3), their average is the distance between the two plates at their center (call it L).

I can come up with three functions describing the three angles relative to each of the three measuring points:
Say r is the distance between the center of the tribrach and the side of the tribrach that was measured.  I can come up with:

rsinθ1 = abs[L-z1]
rsinθ2 = abs[L-z2]
rsinθ3 = abs[L-z3]

Is there a function that can use this information to solve for the maximum angle between these two planes?

Thank you in advance for any insights!