Length of soap film?

Question

Two coaxial rings of diameter c = 5cm are connected
by thin soap film. Diameter of narrow neck of the
film is a = 4cm.

What is distance b between the rings measured along
the film (as ant travels)?

Answer 1

Once it's known that the minimal surface is the revolution of the Cosh(x) function (this is a classic minimal surface problem), the answer to this is straightfoward:

L = ∫ √(1+(d(Cosh(x)/dx)²) dx from -ArcCosh(5/4) to ArcCosh(5/4)

and it works out to a neat 3/2

What's also neat is that the indefinite integral is Sinh(x), so that the length of the curve Cosh(x) is the same as the slope at x. I wonder if this is the only function with this property? Let me see...